https://pg.groshi247.com/highlight/solution-we-are-given-a-multiset-of-10-components-3-identical-sensors-s-5-identical-drones-d-and-2-identical-robotic-arms-r-the-number-of-distinct-activation-sequences-is-the-number-of-distinct-permutations-of-a-multiset-the-total-number-of-sequences-is-given-by-the-multinomial-coefficient-59785482026-02-18T04:29:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-multiset-permutation-problem-with-7-drones-3-identical-multispectral-m-2-thermal-t-and-2-lidar-l-the-number-of-distinct-sequences-is-4098022026-02-18T04:29:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-shelves-be--n--the-number-of-panels-forms-an-arithmetic-sequence-7-11-15--with-first-term--a--7--and-common-difference--d--4--the-sum-of-the-first--n--terms-is-54998452026-02-18T04:29:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-population-grows-as--324-times-2t3--where--t--is-time-in-years-we-want-the-smallest--k--such-that-43604142026-02-18T04:29:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-two-vectors-to-be-orthogonal-their-dot-product-must-be-zero-compute-the-dot-product-94728312026-02-18T06:54:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-fourth-vertex-be-d--x-y-z-in-a-regular-tetrahedron-all-six-edges-must-be-equal-first-compute-the-distance-between-known-points-71620212026-02-18T06:54:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-how-many-combinations-of-3-flavors-and-1-topping-a-customer-can-choose-we-calculate-the-number-of-ways-to-select-3-flavors-from-8-and-1-topping-from-5-34605302026-02-18T08:49:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-binomial-probability-problem-with--n--4--trials-rolls-success-probability--p--frac16--rolling-a-4-and-we-want-exactly--k--2--successes-22528402026-02-18T08:49:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-identify-the-vowels-and-consonants-in-bioinformatics-43345852026-02-18T08:49:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-selecting-5-distinct-integers-independently-and-uniformly-at-random-from--0-1-2-ldots-50--and-let--z--be-the-sum-of-the-two-largest-we-want--mathbbpz-geq-100--29240902026-02-18T08:49:15+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-looking-for-integers--n--such-that--n-equiv-2-pmod5--among-the-first-100-positive-integers-93548362026-04-20T16:46:15+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-four-consecutive-integers-be--n-n1-n2-n3--96306542026-04-20T16:46:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-centroid--g--of-triangle--abc--is-the-average-of-its-vertices-coordinates-35554292026-04-20T16:55:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-center-of-mass-centroid-of-triangle--abc--53146862026-04-20T16:55:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-that--fx--ax3--bx2--cx--d--and-the-values-26094112026-04-20T16:55:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-slope-of-the-terrain-is-given-by-the-derivative-87418902026-04-20T16:55:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-rectangle-is-inscribed-in-the-circle-its-diagonal-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal-d-is-given-by-60322382026-04-20T17:15:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-we-compute-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter-s-is-33858852026-04-20T17:15:46+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-perimeter-of-the-original-triangle-is-56406032026-04-20T17:15:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-the-identity-27791972026-04-20T17:15:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-formula-for-the-circumradius-r-of-a-triangle-60126632026-04-20T17:15:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-a-circle-is-given-by--a--pi-r2--setting--pi-r2--25pi--divide-both-sides-by--pi--to-get--r2--25--taking-the-square-root-yields--r--5--boxed5-24509992026-04-20T17:16:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-maximum-value-of--sin--is-1-so-the-maximum-depth-occurs-when--sinleftfracpi-t12right--1--substituting--dtextmax--10--41--14--boxed14-36076222026-04-20T17:16:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-evaluate--g2--22---1--4---1--3--then-substitute-into--fx---f3--23--3--6--3--9--boxed9-64329592026-04-20T17:16:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve-the-system-from--x---y--1--express--x--y--1--substitute-into-the-first-equation--3y--1--2y--6-rightarrow-3y--3--2y--6-rightarrow-5y--3-rightarrow-y--frac35--then--x--frac35--1--frac85--boxeddfrac85-82705202026-04-20T17:16:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--n--3---m3--32---43--7--9---12--7--4--boxed41-a-train-travels-150-miles-in-3-hours-at-a-constant-speed-if-it-continues-at-the-same-speed-how-many-miles-will-it-cover-in-5-hours-88216222026-04-20T17:16:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-convert--3218--to-base-ten-we-expand-it-using-powers-of-8-75796582026-04-20T17:18:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-factor--180--22-cdot-32-cdot-5--to-find-only-the-odd-divisors-we-ignore-the-factor-of-2-and-consider-only--32-cdot-5--49231792026-04-20T17:18:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x--be-the-two-digit-number-we-are-given-47741502026-04-20T17:18:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-four-consecutive-integers-be--n-n1-n2-n3--their-product-is--nn1n2n3--73913152026-04-20T17:18:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-closest-point-on-a-line-to-a-given-point-is-the-perpendicular-projection-the-slope-of-the-line-is--m---frac12--so-the-perpendicular-slope-is--2--the-equation-of-the-perpendicular-line-through--4-3--is--y---3--2x---4--solving-the-system-24167552026-04-20T17:36:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-columns-of--mathbfm--are-the-images-of-the-standard-basis-vectors-thus--mathbfm--beginbmatrix-3--1---2--4-endbmatrix--verification-57971192026-04-20T17:36:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-de-moivres-theorem--cos-theta--i-sin-thetan--cosntheta--i-sinntheta--here--n--5---theta--72circ--77314362026-04-20T17:36:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-by-using-the-identity--sin2-x--1---cos2-x--to-rewrite--fx--60497032026-04-20T17:36:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-rewrite--tan-theta--cot-theta--4--as-78949512026-04-20T17:36:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-since--vecoa-cdot-vecob--0--they-are-perpendicular-let--vecoa--3-0---vecob--0-4--then--vecog--3m-4n--71143452026-04-20T17:36:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-mathbfv--langle-v1-v2-v3-rangle-the-cross-product-mathbfv-times-mathbfa--mathbfb-gives-651242026-04-20T17:38:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-u--sin-z-the-equation-becomes-2u2---3u--1--0-solving-40169552026-04-22T05:55:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-equation--x2--y2--25--represents-a-circle-of-radius-5-centered-at-the-origin-we-seek-all-integer-pairs--x-y--that-satisfy-this-the-integer-solutions-occur-when--x2--and--y2--are-perfect-squares-summing-to-25-the-possible-square-pairs-are-48882162026-04-20T17:40:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-maximum-height-of-a-parabola--y--ax2--bx--c--occurs-at--x---fracb2a--here--a---1---b--6--so--x---frac62-1--3--substituting--x--3--into-the-equation-96685202026-04-20T17:40:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-rewrite-the-logarithmic-equation-in-exponential-form--x--3--24--simplify--24--16--so--x--16---3--13--99047042026-04-20T17:40:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x--be-the-number-of-successful-tasks-in-the-additional-10-attempts-initially-successful-tasks---085-imes-40--34--total-tasks-after-10-more---50--total-successful-tasks---34--x--new-success-rate--rac34--x50--088--solving--34--x--44--so--x--10--8512322026-04-20T18:02:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--a--5--into--55--b--120--simplify--25--5b--120--subtract-25--5b--95--divide-by-5--b--19--5081642026-04-20T18:02:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-rewrite-in-slope-intercept-form--3y--2x--12----y--rac23x--4--the--y--intercept-occurs-at--x--0--so--y--4--19176592026-04-20T18:02:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve--x--3---rac12--rac52--substitute-into--4x--2---4-imes-rac52--2--10--2--12--4234332026-04-20T18:02:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-given--r4--2r0--substitute-into-the-equation--1000e4k--2-cdot-1000--simplify-to--e4k--2--take-the-natural-logarithm--4k--ln2--so--k--racln24--thus--oxeddfracln24--11339572026-04-20T18:02:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-simplify-the-left-side--racx-3x3x-3--x--3--provided--x-76448602026-04-20T18:02:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-square-both-sides--a--5--49--subtract-5--a--44--verify-by-substituting-back--sqrt44--5--sqrt49--7--final-answer--oxed44--84621112026-04-20T18:02:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-prime-factorization-of-16-is-24-and-of-24-is-23-imes-3-the-lcm-is-the-product-of-the-highest-powers-24-imes-3--16-imes-3--48-thus-the-lcm-is-oxed48-6626262026-04-20T18:02:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve-x-equiv-3-pmod7-and-x-equiv-5-pmod9-let-x--7k--3-substitute-into-the-second-congruence-7k--3-equiv-5-pmod9-rightarrow-7k-equiv-2-pmod9-the-modular-inverse-of-7-modulo-9-is-4-since-7-imes-4--28-equiv-1-pmod9-so-k-equiv-2-imes-4--8-pmod9-thus-k--9m--8-and-x--79m--8--3--63m--59-the-solutions-are-59-122-185-dots-within-1150-only-59-and-122-satisfy-the-count-is-oxed2-66162812026-04-20T18:02:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-lcm-of-13-and-17-is-223-since-223--200-the-largest-multiple-is-0-invalid-adjusting-the-largest-under-200-is-0-no-valid-number-however-if-the-question-allows-divisibility-by-13-or-17-the-answer-differs-assuming-exact-lcm-the-answer-is-oxed0-no-solution-31300652026-04-20T18:02:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-arithmetic-sequence-have-first-term--a1--15--and-fifth-term--a5--35--the-common-difference--d--satisfies--a5--a1--4d--substituting--35--15--4d-rightarrow-d--5--the-sum-of-the-first--n--10--terms-is--s10--rac102-215--95--530--45--575--375--oxed375-66728072026-04-20T18:19:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x--be-points-per-correct-challenge-for-10-challenges-there-are--lfloor-rac10---13-9636002026-04-20T18:19:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-of-intervals-is--rac1815--12--since-beams-are-placed-at-both-ends-total-beams---12--1--13--however-the-problem-states-4-beams-per-3-meters-which-implies--rac43--beams-per-meter-for-18-meters--rac43-imes-18--24--oxed24-35246022026-04-20T18:19:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-score-at-level--n--be--sn--s1-cdot-2n-1--given--s3--s1-cdot-4--128-rightarrow-s1--32--then--s5--32-cdot-24--32-cdot-16--512--oxed512-41647202026-04-20T18:19:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-arithmetic-with--a1--8--and--d--2--the-7th-term-is--a7--a1--6d--8--12--20--oxed20-5312762026-04-20T18:19:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio--rac5-ext-bolts2-ext-beams--implies--rac30-ext-boltsx-ext-beams--rac52--cross-multiplying--5x--60-rightarrow-x--12--oxed121-6668442026-04-20T18:19:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-initially-the-explorer-has--25-times-060--15--successful-voyages-after-10-more-attempts-the-total-voyages-are--25--10--35--with-a-68-success-rate-total-successful-voyages-become--35-times-068--238--which-rounds-to-24-since-successes-are-whole-numbers-the-additional-successful-voyages-are--24---15--9--91127412026-04-20T18:27:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-quadratic--pt---t2--14t--30--opens-downward-the-vertex-occurs-at--t---fracb2a---frac142-1--7--substituting--t--7---p7---72--147--30---49--98--30--79--84474742026-02-23T03:57:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--y--4x---5--into-the-second-equation--24x---5--3x--10--simplify--8x---10--3x--10-rightarrow-11x--20-rightarrow-x--frac2011--substitute-back--y--4leftfrac2011right---5--frac8011---frac5511--frac2511--90628682026-04-20T18:27:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-distributive-property-4972642026-04-20T18:27:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-greatest-common-factor-gcf-of-48-and-60-factor-both-numbers-87216382026-04-20T18:27:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-distinct-8-nucleotide-sequences-using-exactly-2-as-3-cs-and-3-gs-total-8-nucleotides-such-that-no-two-identical-nucleotides-are-adjacent-90700412026-04-20T18:46:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-the-number-of-unique-2-nucleotide-combinations-from-5-distinct-nucleotides-without-considering-order-we-use-the-combination-formula--binomnr--where--n--5--and--r--2--72510442026-04-20T19:39:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-represent-the-arrival-times-as-points-x-y-in-a-60-minute-square-the-condition--y--x--15--order-matters-corresponds-to-a-triangular-region-with-area--frac12-times-60---152--frac12-times-2025--10125--the-total-area-is--602--3600--the-probability-is--frac101253600--frac932--28341672026-04-20T19:39:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-requires-the-stirling-numbers-of-the-second-kind--s5-3--which-count-the-ways-to-partition-5-distinguishable-objects-into-3-non-empty-indistinct-subsets-using-the-recurrence-relation-50973622026-04-20T19:39:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-of-ways-to-choose-3-genes-from-8-is-given-by-the-combination-formula-83227442026-04-20T19:39:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-represent-the-times-as-x-y-in-a-60-minute-square-the-condition--y-geq-x--20--defines-a-triangular-region-with-base--60---20--40--minutes-the-area-of-this-region-is--frac12-times-402--800--the-total-area-is--602--3600--the-probability-is--frac8003600--frac29--26914842026-04-20T19:39:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-equivalent-to-partitioning-4-distinguishable-objects-into-2-non-empty-indistinct-subsets-calculated-by-the-stirling-number--s4-2--using-the-recurrence-38270682026-04-20T19:39:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--u--sqrtx--so--u-geq-0--substituting-the-equation-becomes--u--3u2--4--rearranging--3u2--u---4--0--using-the-quadratic-formula--u--frac-1-pm-sqrt1--486--frac-1-pm-76--valid-solution--u--1--since--u-geq-0--thus--sqrtx--1-rightarrow-x--1--boxed1-31338202026-04-20T19:41:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--t--2y--1-rightarrow-y--fract---12--substitute-into--ht---ht--leftfract---12right3---5leftfract---12right--6--simplify--ht--fract-138---frac5t-12--6--now-replace--t--with--y2---1---hy2---1--fracy2---238---frac5y2---22--6--expand-and-combine-terms-if-required-boxedfracy2---238---frac5y2---22--6-91303332026-04-20T19:41:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-group-and-complete-the-square--4x2--2x---9y2---2y--11--complete-squares--4x12---1---9y-12---1--11--expand--4x12---4---9y-12--9--11-rightarrow-4x12---9y-12--6--rewrite--fracx12frac32---fracy-12frac23--1--center-at---1-1--boxed-1-11-a-chemistry-experiment-requires-50-ml-of-a-10-salt-solution-if-you-only-have-a-20-salt-solution-and-distilled-water-how-much-of-each-should-be-mixed-to-obtain-the-desired-concentration-15751582026-02-23T05:53:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-asked-to-count-the-number-of-distinct-ways-to-assign-10-independently-classified-images-into-4-categories-with-fixed-counts-3-as-tumor-2-as-inflammation-4-as-normal-and-1-as-stroke-this-is-a-multinomial-coefficient-problem-99228742026-04-20T19:44:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-total-number-of-distinct-arrangements-of-banana-without-restrictions-the-word-has-6-letters-with-repetitions-3-as-2-ns-and-1-b-81657702026-04-20T19:44:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-continuous-probability-problem-involving-uniform-random-variables-and-coverage-intervals-70794812026-04-20T19:44:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-greatest-common-divisor-of--5m---1--and--5n---1--for-positive-integers--m--and--n--is-given-by-the-identity-84613562026-04-20T19:44:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal-d-is-sqrt52--82--sqrt25--64--sqrt89-therefore-the-radius-of-the-circle-is-fracsqrt892-the-circumference-is-given-by-2pi-r--2pi-cdot-fracsqrt892--pisqrt89-6373132026-04-20T20:02:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-sphere-is-frac43pi-2x3--frac43pi-8x3--frac323pi-x3-the-volume-of-a-hemisphere-is-frac23pi-3x3--frac23pi-27x3--18pi-x3-the-ratio-of-the-volumes-is-9064112026-04-20T20:02:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-area-using-herons-formula-the-semi-perimeter-is-94835802026-04-20T20:02:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sum-of-the-first--n--cubes-is-given-by-the-formula-38692522026-04-20T20:37:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-11071-11073-11075-11077-11079-the-sum-of-an-arithmetic-sequence-is-given-by-s--fracn2a1--an-here-n--5-a1--11071-and-a5--11079-59436392026-04-20T21:08:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-equation-x2---y2--2025-factors-as-x---yx--y--2025-since-x-and-y-are-integers-both-x---y-and-x--y-must-be-integers-let-a--x---y-and-b--x--y-so-ab--2025-then-x--fraca--b2-and-y--fracb---a2-for-x-and-y-to-be-integers-a--b-and-b---a-must-both-be-even-so-a-and-b-must-have-the-same-parity-38736692026-04-20T21:08:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-four-consecutive-odd-integers-be-n-n2-n4-n6-where-n-is-odd-their-product-is-79671432026-04-20T21:08:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-a-right-triangle-with-legs--a--and--b--and-hypotenuse--z--the-inradius-is-given-by--c--raca--b---z2--the-area--a--of-the-triangle-is--rac12ab--the-area-of-the-incircle-is--pi-c2--using-the-identity--a2--b2--z2--and-expressing--a--b--in-terms-of--z--and--c--we-find--a--b--z--2c--also-the-semiperimeter--s--raca--b--z2--rac2z--2c2--z--c--the-area-can-also-be-written-as--a--r-cdot-s--cz--c--therefore-the-ratio-becomes-8316632026-04-20T21:21:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume--v--of-a-spherical-cap-of-height--h--cut-from-a-sphere-of-radius--r--is-given-by-the-formula-39064392026-04-20T21:21:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-sphere-of-radius--x--is--vextparticle--rac43pi-x3--the-volume-of-a-hemisphere-of-radius--3x--is-half-the-volume-of-a-full-sphere-97887882026-04-20T21:21:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-angle--120circ--lies-in-the-second-quadrant-using-the-identity--an180circ---heta---an-heta--we-write-81322752026-04-20T21:21:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-first-12-even-numbers-after-2020-are-2022-2024-ldots-2044-the-sum-of-an-arithmetic-sequence-is-s--racn2a1--an-here-n--12-a1--2022-an--2044-thus-s--rac1222022--2044--6-imes-4066--24396-dividing-24396-by-11-11-imes-2216--24376-so-the-remainder-is-24396---24376--20-however-20-mod-11--9-therefore-the-remainder-is-oxed9-25985872026-04-20T21:21:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-prime-factorization-of-2048-is-211-and-1024-is-210-the-lcm-of-two-numbers-is-the-product-of-the-highest-powers-of-all-primes-present-thus-extlcm2048-1024--211--2048-the-answer-is-oxed2048-56821142026-04-20T21:21:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-numbers-congruent-to-3-mod-7-are-3-10-17-ldots-3--7k-the-largest-such-number-leq-100-satisfies-3--7k-leq-100-rightarrow-k-leq-rac977-pprox-13857-thus-k--0-1-ldots-13-giving-14-numbers-the-count-is-oxed14-20455062026-04-20T21:21:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-prime-factorization-of-2025-is-34-imes-52-and-1024-is-210-since-there-are-no-common-prime-factors-the-gcd-is-1-the-answer-is-oxed1-88657282026-04-20T21:21:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sum-of-squares-formula-gives-s--rac15-imes-16-imes-316--1240-compute-1240-mod-13-13-imes-95--1235-so-1240---1235--5-the-remainder-is-oxed5question-a-science-educator-is-designing-a-lesson-on-probability-using-a-digital-simulation-that-generates-random-6-digit-numbers-using-only-the-digits-1-2-and-3-what-is-the-probability-that-a-randomly-generated-number-has-exactly-two-consecutive-digits-that-are-the-same-46065742026-04-20T21:22:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-solve-this-we-must-count-the-number-of-6-digit-numbers-using-only-the-digits-1-2-and-3-such-that-exactly-one-pair-of-consecutive-digits-are-equal-and-all-other-adjacent-digits-are-different-19931932026-04-20T21:21:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-the-probability-that-the-product-of-three-randomly-chosen-integers-between-0-and-100-is-divisible-by-5-we-use-complementary-probability-first-note-that-an-integer-is-divisible-by-5-if-it-ends-in-0-or-5-among-the-integers-from-0-to-100-there-are-21-multiples-of-5-ie-0-5-10--100-thus-the-probability-that-a-single-randomly-chosen-integer-is-not-divisible-by-5-is-87387832026-04-20T21:35:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-determine-how-many-years-between-1800-and-1900-inclusive-are-divisible-by-4-39914012026-04-20T21:35:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-ways-to-choose-3-papers-from-12-is-34333062026-04-20T21:35:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-ways-to-choose-3-sites-from-10-is-14988822026-04-20T21:35:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-given-sequence-is-an-arithmetic-sequence-with-the-first-term--a--3--and-common-difference--d--5--the-general-term-of-the-sequence-is-given-by-2915062026-04-20T22:27:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-least-common-multiple-lcm-of-8-and-12-we-first-determine-their-prime-factorizations-15552162026-04-20T22:27:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-smallest-prime-factor-of-77-we-test-divisibility-by-the-smallest-prime-numbers-19886142026-04-20T22:27:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-maximum-value-of-sec-theta--csc-theta2-first-express-it-in-terms-of-sine-and-cosine-2615772026-04-20T22:27:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-68986112026-04-20T22:29:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-point-on-the-line-y--2x--3-closest-to-1--1-we-minimize-the-distance-between-x-2x--3-and-1--1-the-squared-distance-is-53555222026-02-23T10:55:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-y--rac2sin-x--3sin-x--2-solve-for-sin-x-48237942026-04-20T22:54:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-mathbfm--eginbmatrix-a--b--c--d-endbmatrix-92681042026-04-20T22:54:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-expand-57421862026-04-20T22:54:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-given-points-lie-on-the-plane-z--3-the-vector-from-1-2-3-to-1-5-3-is-0-3-0-and-the-vector-to-4-2-3-is-3-0-0-the-fourth-vertex-on-the-same-face-is-obtained-by-adding-these-vectors-1--3-2--3-3--4-5-3-53638162026-04-20T23:03:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero-compute-the-dot-product-3x--2-1---14--3x---2---4--3x---6-set-this-equal-to-zero-3x---6--0-implies-x--2-86154422026-04-20T23:03:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-cross-product-mathbfv-times-mathbfa-is-computed-as-54192932026-04-20T23:03:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-equation-sin2x--frac12-has-solutions-when-2x--fracpi6--2pi-n-or-2x--frac5pi6--2pi-n-for-integer-n-solving-for-x-in-0-2pi-71198232026-04-20T23:04:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-dot-product-mathbfu-cdot-mathbfv--mathbfw--mathbfu-cdot-mathbfv--mathbfu-cdot-mathbfw-since-mathbfu-mathbfv-and-mathbfw-are-unit-vectors-each-dot-product-is-at-most-1-the-maximum-occurs-when-mathbfu-aligns-with-both-mathbfv-and-mathbfw-ie-mathbfv--mathbfw--mathbfu-then-mathbfu-cdot-mathbfv--mathbfw--1--1--2-47603312026-04-20T23:04:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-5-distinct-parts-a-b-c-d-and-e-and-we-are-to-count-the-number-of-permutations-where-a-appears-before-b-and-c-appears-before-d-6419532026-04-20T23:07:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-8-segment-sequences-using-letters-a-u-c-g-such-that-3658932026-04-20T23:07:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-probability-that-a-single-project-is-exceptionally-strong-ie-gets-h-in-at-least-5-of-the-3-categories-but-there-are-only-3-categories-so-getting-high-in-at-least-5-is-impossible-91005652026-04-20T23:07:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-robot-takes-8-steps-each-independently-choosing-to-go-right-increase-x-or-up-increase-y-uniformly-at-random-assume-fair-walk-the-endpoint-is-x-8---x-for-x--01dots8-so-x-follows-a-binomial-distribution-x-sim-textbinomial8-frac12-71478042026-04-20T23:07:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-number-of-4-protein-combinations-from-6-proteins--binom64--15--47225602026-04-20T23:07:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-distinct-ranking-configurations-of-5-projects-where-87086172026-04-20T23:07:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-the-smallest-positive-angle-at-which-the-robot-returns-to-its-original-orientation-we-must-find-the-least-common-multiple-lcm-of-the-two-rotation-increments-12circ-and-18circ-first-factor-both-numbers-99383902026-04-20T23:34:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-the-city-is-2025-km-and-we-are-to-find-how-many-integer-side-lengths-s-correspond-to-square-grid-sizes-such-that-s2-divides-2025-this-requires-finding-the-number-of-positive-divisors-of-2025-since-each-such-divisor-s2-implies-s-is-an-integer-divisor-of-sqrt2025--45-78416942026-04-20T23:34:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-us-make-the-substitution--x--sqrtu--so-that--u--x2--substituting-into-the-equation-gives-68230572026-04-20T23:35:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-that-34865802026-04-20T23:35:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-a-right-triangle-with-legs--a--and--b--and-hypotenuse--c--the-radius--r--of-the-inscribed-circle-is-given-by-23461632026-04-20T23:48:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-diagonal-of-the-square-is-the-diameter-of-the-circle-for-a-square-with-side-length--s--the-diagonal--d--is-35212582026-04-20T23:48:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area--a--of-an-equilateral-triangle-with-side-length--s--is-56225662026-04-20T23:48:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-area-using-herons-formula-let--a--7---b--10---c--13--the-semi-perimeter--s--is-7514252026-04-20T23:48:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-when-both-gears-align-we-need-the-least-common-multiple-lcm-of-their-number-of-teeth-24-and-36-first-factor-both-numbers-17468472026-04-20T23:55:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-find-the-least-common-multiple-lcm-of-6-8-and-12-factor-each-number-41086872026-04-20T23:55:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-geometric-sequence-with-first-term-a--1-and-common-ratio-r--3-the-total-number-infected-by-the-end-of-the-5th-generation-is-the-sum-of-the-first-5-terms-63848342026-04-20T23:55:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-list-the-positive-factors-of-24-1-2-3-4-6-8-12-24-but-digits-must-be-single-digit-numbers-09-so-valid-digits-are-1-2-3-4-6-8-83085332026-04-20T23:55:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-of-the-three-expressions-is-24607062026-04-20T23:55:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-number-compute-the-least-common-multiple-lcm-of-7-and-11-then-add-1-since-7-and-11-are-coprime-extlcm7-11--7-imes-11--77-adding-1-gives-77--1--78-the-two-digit-number-is-oxed78-39026272026-04-20T23:55:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-prime-factorizations-15--3-imes-5-20--22-imes-5-lcm-is-22-imes-3-imes-5--60-the-synchronized-cycle-length-is-oxed60question-a-civil-engineer-designs-a-parabolic-arch-with-the-equation--y---ax2--bx--c--if-the-archs-highest-point-is-at--3-10--and-passes-through--0-4--find-the-value-of--a--79322642026-04-20T23:55:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-vertex-form-of-a-parabola-is--y---ax---h2--k--where--h-k--is-the-vertex-given-the-vertex--3-10--the-equation-becomes--y---ax---32--10--substituting-the-point--0-4--82455322026-04-20T23:55:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-expression--left-mathbfv-cdot-mathbfw-times-mathbfu-right--depends-on-the-magnitude-of-the-binormal-vector-compute--mathbfw-times-mathbfu--leftlangle--1--2-1-rightrangle--so-75132232026-02-23T12:37:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-five-terms-of-the-arithmetic-sequence-be--a---2d-a---d-a-a--d-a--2d--where--a--is-the-middle-third-term-and--d--is-the-common-difference-92727872026-04-21T00:24:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--t--be-the-price-of-one-tool-and--b--be-the-price-of-one-bead-4291592026-04-21T00:24:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-five-terms-of-the-arithmetic-sequence-be--a---2d-a---d-a-a--d-a--2d--77879062026-04-21T00:24:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-add-the-two-lengths-and-divide-by-2-first-convert-the-mixed-numbers-to-improper-fractions-75490552026-04-21T00:24:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-recall-that--37623012026-04-21T00:24:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x--be-the-cost-of-one-unit-of-x-in-dollars-and--y--be-the-cost-of-one-unit-of-y-in-dollars-the-system-of-equations-is-72031542026-04-21T00:32:46+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-scale-is--frac32-text-cm16-text-km--frac02-text-cm1-text-km--using-this-scale-48599552026-04-21T00:32:46+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-side-length-of-the-square-be--s--the-perimeter-of-a-square-is--4s--so-we-have-95846862026-04-21T00:36:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-with--b1--1--and-compute-successive-terms-using--eu--u---fracu44--48485272026-04-21T00:36:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-rectangle-have-length--l--and-width--w--the-perimeter-is-45014922026-04-21T00:36:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x--sqrtu--so--u--x2--and-the-equation-becomes-97950692026-04-21T00:36:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-with--c1--1--and-compute-26781972026-04-21T00:36:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-of-the-five-values-is-given-by-59613592026-04-21T00:36:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-two-functional-equations-71296542026-04-21T00:45:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-shortest-altitude-we-first-compute-the-area-using-herons-formula-let-the-triangle-have-sides-a--13-b--14-and-c--15-the-semi-perimeter-is-97939232026-04-21T00:45:15+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-right-triangle-with-legs-a-b-and-hypotenuse-z-the-inradius-is-given-by-16491332026-04-21T00:45:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-regular-hexagon-the-side-length-is-equal-to-the-radius-of-the-circumscribed-circle-thus-the-radius--r--5--cm-the-circumference--c--of-the-circle-is-given-by--c--2pi-r--substituting--c--2pi-times-5--10pi--77925092026-04-21T00:57:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-radius-of-the-orbit-is-the-sum-of-the-planets-radius-and-the-satellites-orbital-radius--6000--10000--16000--km-the-circumference--c--is-calculated-as--c--2pi-times-16000--32000pi--km-38793722026-04-21T00:57:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x--be-the-total-gb-used-the-cost-equation-is--12-times-20--008x---20--3840--simplify--240--008x---16--3840--combine-terms--2384--008x--3840--subtract-2384--008x---1996--divide-by-008--x---24875--this-negative-value-indicates-an-inconsistency-suggesting-the-business-used-fewer-than-20-gb-recalculate-without-the-008-overage--12x--3840----x--32--verify-12--32--3840-final-answer--boxed32--gb-5145832026-04-21T01:13:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-maximum-occurs-at-the-vertex--x---fracb2a---frac182-3--3--substitute--x--3--into--y---y---39--183--20---27--54--20--47--the-fertilizer-amount-maximizing-yield-is--boxed3--kg-99155262026-02-23T13:50:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--8---s--frac6sqrt8--frac62sqrt2--frac3sqrt2--rationalize--frac3sqrt22--final-answer--boxeddfrac3sqrt22--9747052026-04-21T01:13:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-y-intercept-occurs-when--x--0--substitute--t--frac40--10---2--frac1-2---frac12--the-y-intercept-is--boxedleft0--dfrac12right--24152332026-04-21T01:13:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-minimum-occurs-at--x---fracb2a---frac-2025--2--verify--w2--54---202--45--20---40--45--25--the-smallest--x--is--boxed2-question-a-hydrologist-models-groundwater-flow-with-the-function--fx--sec-x--cos-x2--csc-x---sin-x2--for--0--x--fracpi2--find-the-minimum-value-of--fx--71621362026-04-21T01:13:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-function-78213462026-04-21T01:13:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-function--st--left-sinleftfrac3pi7-tright--cosleftfrac4pi7-tright-right--has-period--t--where-both-components-repeat-the-fundamental-period-is-the-least-common-multiple-of-the-periods-of--sinleftfrac3pi7-tright--and--cosleftfrac4pi7-tright--76044872026-04-21T01:13:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the--y--intercept--b--substitute-the-point--10-2--into-the-equation-59581022026-04-21T01:41:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-maximum-of--px---x2--4x--m--occurs-at-the-vertex-the--x--coordinate-of-the-vertex-is--x--frac-b2a--frac-4-2--2--substitute--x--2--into--px--65573012026-04-21T01:41:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-set-up-equations-using-the-given-points-37981452026-04-21T01:41:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-multiply-numerator-and-denominator-by-the-conjugate--sqrt7--sqrt3--54226892026-04-21T01:41:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-de-moivres-theorem--zn--cosleftnthetaright--i-sinleftnthetaright--here--z--cosleftfracpi6right--i-sinleftfracpi6right--so--z6--cosleft6-cdot-fracpi6right--i-sinleft6-cdot-fracpi6right--cospi--i-sinpi--evaluating--cospi---1--and--sinpi--0--so--z6---1--0i--95457132026-04-21T02:07:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-set--frac18d0--d0-cdot-2-t12--divide-both-sides-by--d0---frac18--2-t12--note--frac18--2-3--so--2-3--2-t12--equate-exponents---3---fract12--solve-for--t---t--36--33301842026-04-21T02:07:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-closest-point-occurs-where-the-vector-from--5-7--to-the-line-is-perpendicular-to-the-direction-vector--beginpmatrix-3--4-endpmatrix--let-the-point-on-the-line-be--2--3t--1--4t--the-vector-from--5-7--to-this-point-is--beginpmatrix-2--3t---5---1--4t---7-endpmatrix--beginpmatrix--3--3t---8--4t-endpmatrix--dot-product-with--beginpmatrix-3--4-endpmatrix--must-be-zero--3-3--3t--4-8--4t--0--simplify---9--9t---32--16t--0-rightarrow-25t---41--0-rightarrow-t--frac4125--substitute-back--x--2--3-cdot-frac4125--frac17325---y---1--4-cdot-frac4125--frac15625--16550092026-02-23T14:50:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero--x-cdot-3--2-cdot--1--0--simplify--3x---2--0-rightarrow-x--frac23--32263812026-04-21T02:07:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-by-solving--2cos2theta--1--0-rightarrow-cos2theta---frac12--the-general-solutions-for--2theta--are--2theta--120circ--360circ-k--or--2theta--240circ--360circ-k--where--k-in-mathbbz--dividing-by-2--theta--60circ--180circ-k--or--theta--120circ--180circ-k--for--theta-in-0circ-360circ--the-solutions-are--60circ-120circ-240circ-300circ--4782382026-04-21T02:07:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-rewrite--fx--fracsin2-x--4sin-x--sin-x--frac4sin-x--let--y--sin-x-in-0-1--the-function-becomes--fy--y--frac4y--the-derivative--fy--1---frac4y2--has-critical-point-at--y--2--but--y-leq-1--analyze-endpoints-as--y-to-0---fy-to-infty--at--y--1---f1--1--4--5--the-minimum-is--5--42007872026-04-21T02:07:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--vecoa--mathbfa---vecob--mathbfb--then--overrightarrowoc--mmathbfa--nmathbfb--for--overrightarrowoc-perp-mathbfa---mathbfb--their-dot-product-is-zero--mmathbfa--nmathbfb-cdot-mathbfa---mathbfb--0--expand--mmathbfa2---mmathbfa-cdot-mathbfb--nmathbfb-cdot-mathbfa---nmathbfb2--0--substitute--mathbfa--2---mathbfb--3---mathbfa-cdot-mathbfb--2-cdot-3-cdot-cos-60circ--3---4m---3m--3n---9n--0-rightarrow-m---6n--0--choose--n--1--then--m--6--normalize-if-needed-but-the-relation-is--m--6n--for-simplicity-take--n--1---m--6--25360122026-04-24T12:38:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-measurement-be--u--a-positive-multiple-of-5-such-that--u3--1500--71631012026-04-24T12:38:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-of--3u2---5u7--and--4u1--is-given-by-90387152026-04-24T12:38:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-mathbfv--langle-v1-v2-v3-rangle-compute-mathbfv-times-mathbfb-6878512026-04-21T02:10:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-magnitude-of-the-cross-product-is-given-by-mathbfu-times-mathbfw--mathbfu-mathbfw-sin-theta-here-theta--60circ-40463162026-04-21T02:10:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-a--b--2025-the-greatest-common-divisor-gcda-b-must-divide-a--b--2025-the-maximum-possible-value-of-gcda-b-occurs-when-a-and-b-are-both-multiples-of-the-largest-divisor-of-2025-since-2025--452--34-cdot-52-its-largest-proper-divisor-is-2025--3--675-thus-gcda-b--675-when-a--675-and-b--1350-71307752026-04-21T02:19:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sum-of-cubes-formula-is-leftfracnn12right2-for-n--12-the-sum-is-leftfrac12-cdot-132right2--782--6084-dividing-by-13-6084-div-13--468-with-remainder-0-since-13-cdot-468--6084-77303762026-04-21T02:19:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-least-common-multiple-lcm-of-14-and-21-is-42-the-smallest-three-digit-multiple-of-42-is-42-times-3--126-70409852026-04-21T02:19:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-prime-factorization-16--24-24--23-cdot-3-lcm-takes-the-highest-powers-24-cdot-3--16-cdot-3--48-7853462026-04-21T02:19:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-prime-factorization-108--22-cdot-33-144--24-cdot-32-gcd-takes-the-lowest-powers-22-cdot-32--4-cdot-9--36-96996262026-04-21T02:19:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-selecting-and-arranging-instruments-for-display-first-choose-3-telescopes-from-8-inom83-then-choose-2-microscopes-from-5-inom52-after-selection-arrange-the-5-chosen-instruments-in-a-row-5-ways-56069022026-04-21T02:27:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-number-of-ways-to-choose-5-varieties-from-10-inom105-39487302026-04-21T02:27:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-kiosks-are-arranged-in-a-circle-and-rotations-are-considered-the-same-we-fix-one-session-to-eliminate-rotational-symmetry-first-assign-4-out-of-7-kiosks-to-the-sessions-inom74-cdot-4-would-count-linear-arrangements-but-due-to-circular-symmetry-we-divide-by-4-the-number-of-rotations-37256342026-04-21T02:27:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-find-the-constant--k--using-the-given-values-65374072026-04-21T02:27:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-expression--cos-x--isin-x--corresponds-to--eix--on-the-unit-circle-in-the-complex-plane-therefore--cos-x--isin-x--1--eix--1--the-modulus-is--eix--1--using-the-identity--eix--1--sqrtcos-x--12--sin2-x--we-simplify-80790362026-04-24T12:38:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-since--sin2-x--cos2-x--1--we-write--fx--frac1sin2-x--4cos2-x--frac11--3cos2-x--let--y--cos2-x--so--0-leq-y-leq-1--then--fx--frac11--3y--as--y--varies-from-0-to-1--1--3y--varies-from-1-to-4-so--fx--varies-from--frac14--to--1--hence-the-range-of--fx--is--leftfrac14-1right--the-final-answer-is--boxedleftfrac14-1right--75005642026-04-22T06:04:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-with--tan-z--cot-z--2--recall--tan-z--fracsin-zcos-z---cot-z--fraccos-zsin-z--then-68315982026-04-22T06:04:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-lets-simplify-each-term-first--tan-x--cot-x--fracsin-xcos-x--frac-85906582026-04-21T02:29:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-given-points-lie-on-the-base-face-of-the-cube-in-the-xy-plane-the-fourth-vertex-of-the-square-face-must-complete-the-rectangle-the-missing-vertex-is-110-as-it-forms-equal-sides-and-right-angles-with-the-given-points-13442402026-04-21T02:34:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve-2x--60circ-120circ-420circ-480circ-within-0circ-leq-2x-leq-720circ-thus-x--30circ-60circ-210circ-240circ-sum-30--60--210--240--540circ-96929372026-04-21T02:34:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-scalar-triple-product-equals-the-volume-of-the-parallelepiped-maximum-occurs-when-vectors-are-orthogonal-giving-mathbfa-cdot-mathbfb-times-mathbfc--1-cdot-1-cdot-1-cdot-sin90circcos0circ--1-27335002026-04-21T02:34:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-problem-requires-partitioning-7-distinguishable-solutions-into-3-indistinguishable-non-empty-vials-this-is-equivalent-to-the-stirling-numbers-of-the-second-kind--s7-3--the-formula-for-stirling-numbers-is-61549492026-04-21T02:34:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-modules-are-indistinguishable-and-empty-modules-are-allowed-the-number-of-ways-is-the-sum-of-stirling-numbers-of-the-second-kind-for-partitions-into-1-and-2-non-empty-subsets-24281252026-04-21T02:34:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-ways-to-choose-3-startups--binom63--20--60396932026-04-21T02:34:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-roots-of--fx--x4---6x3--11x2---6x--1--be--r1-r2-r3-r4--we-are-asked-to-find--sumi14-frac1ri--frac1r1--frac1r2--frac1r3--frac1r4--using-the-identity--sum-frac1ri--fracsumijk-ri-rj-rkr-96910572026-04-21T02:38:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-possible-outcomes-for-7-patterns-is-47-to-count-favorable-outcomes-choose-3-patterns-to-show-the-specific-outcome-extreme-heat-in-binom73-ways-and-the-remaining-4-patterns-must-show-one-of-the-other-3-outcomes-giving-34-possibilities-the-probability-is-fracbinom73-cdot-3447--frac35-cdot-8116384--frac283516384-thus-the-final-answer-is-boxeddfrac283516384-63571952026-04-21T02:40:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-possible-state-combinations-is-25--32-the-number-of-ways-to-choose-2-elements-to-be-active-is-binom52--10-since-each-combination-is-equally-likely-the-probability-is-frac1032--frac516-thus-the-final-answer-is-boxeddfrac516-86353342026-04-21T02:40:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-possible-calibration-configurations-is-36-to-count-favorable-cases-choose-4-sensors-to-be-in-precision-mode-in-binom64-ways-and-the-remaining-2-sensors-can-be-in-either-of-the-other-2-settings-giving-22-possibilities-the-probability-is-fracbinom64-cdot-2236--frac15-cdot-4729--frac60729--frac20243-thus-the-final-answer-is-boxeddfrac20243question-a-tech-startup-founder-wants-to-align-two-project-timelines-with-cycles-of-10-and-15-months-what-is-the-least-number-of-months-after-which-both-projects-will-synchronize-34717272026-04-21T02:40:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-when-both-projects-synchronize-compute-the-least-common-multiple-lcm-of-10-and-15-factorize-10--2-times-5-and-15--3-times-5-the-lcm-is-2-times-3-times-5--30-thus-the-projects-will-align-after-boxed30-months-94216262026-04-21T02:40:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-smallest-day-both-patterns-coincide-is-the-lcm-of-9-and-12-factorize-9--32-and-12--22-times-3-the-lcm-is-22-times-32--36-the-patterns-coincide-on-day-boxed36-19032572026-04-21T02:40:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-rewrite-the-equation-as-fracx216--fracy24--1-possible-integer-values-for-y-are--2--1-0-1-2-for-each-y-solve-for-x-46980252026-02-23T15:26:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-be-n-then-n-equiv-3-pmod7-and-n-equiv-5-pmod6-solve-the-system-82198512026-04-21T02:40:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-identity-gcd2a---1-2b---1--2gcda-b---1-compute-gcd12-18--6-thus-gcd212---1-218---1--26---1--64---1--boxed63-77995732026-04-21T02:40:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-lcm-of-11-and-13-is-143-the-smallest-three-digit-multiple-of-143-is-143-times-1--143-since-143-times-2--286-is-also-three-digit-but-143-is-smaller-thus-the-answer-is-boxed143question-two-marine-biologists-dr-coral-and-dr-tide-arrive-at-a-research-station-at-random-times-between-700-and-800-if-dr-coral-arrives-after-dr-tide-what-is-the-probability-that-dr-tide-arrived-before-745-81031042026-04-21T02:40:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-arrival-times-of-dr-tide-and-dr-coral-be--x--and--y--respectively-where--x-y-in-0-60--minutes-after-700-the-condition--y--x--defines-a-triangular-region-in-the--xy--plane-with-area--frac6022--1800--the-favorable-region-is-where--x--45--and--y--x--integrating-over--x-in-0-45--the-area-is--int045-60---x--dx--60-cdot-45---frac4522--2700---10125--16875--the-conditional-probability-is--frac168751800--frac34--49462762026-04-21T02:40:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-ways-to-choose-3-breakthroughs-is--binom203--1140--the-number-of-favorable-outcomes-is--binom52-cdot-binom151--10-cdot-15--150--the-probability-is--frac1501140--frac538--7281382026-04-21T02:40:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-of-ways-to-choose-3-flavors-is--binom73--35--and-2-toppings-is--binom42--6--multiplying-gives--35-cdot-6--210--combinations-27157972026-04-21T02:40:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-the-stirling-number-of-the-second-kind--s6-3--calculated-as--frac13-left-36---3-cdot-26--3-cdot-16-right--frac729---192--36--frac5406--90--91702422026-04-21T02:40:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-items--4--6--2--12--total-ways-to-choose-2--binom122--66--favorable-ways--binom62--15--probability--frac1566--frac522--77333642026-04-21T02:40:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-length-of-the-two-ribbons-we-add-their-lengths-and-divide-by-2-the-total-length-is-32--78--110-inches-dividing-by-2-we-get-frac1102--55-inches-thus-the-average-length-is-boxed55-inches-57314652026-04-21T02:53:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-evaluate-the-boundary-values-frac52--25-since-pi-approx-31416-we-have-3pi-approx-94248-we-seek-whole-numbers-n-such-that-25--n--94248-these-are-3-4-5-6-7-8-9-counting-them-there-are-7-whole-numbers-therefore-the-answer-is-boxed7-22848712026-04-21T02:53:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-sum-the-three-expressions-and-divide-by-3-the-sum-is-3v--4--5v---2--4v--7--3v--5v--4v--4---2--7--12v--9-dividing-by-3-we-get-frac12v--93--4v--3-thus-the-average-is-boxed4v--31---------120-----180-----7800-----30---------83862492026-04-21T02:53:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-starts-at-0-and-increases-by-5-each-time-the-nth-term-is--an--5n-1--setting--5n-1--100--solve-for--n--61485722026-04-21T03:10:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-subtract-the-used-amount-from-the-total-23412582026-04-21T03:10:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-when-the-heights-are-equal-set--hat--hbt--36848522026-04-21T03:39:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-simplify--sqrt081--11346522026-04-21T03:39:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-multiply-numerator-and-denominator-by--sqrt3--sqrt2--58848912026-04-21T03:39:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--a---units-of-a--b---units-of-b-98324822026-04-21T03:39:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-equation--sin2theta--fracsqrt32--implies--2theta--60circ--360circ-k--or--2theta--120circ--360circ-k--for-integer--k--solving-for--theta--we-get--theta--30circ--180circ-k--or--theta--60circ--180circ-k--within--0circ-360circ--the-solutions-are--30circ-60circ-210circ-240circ--thus-the-answer-is--boxed30circ-boxed60circ-boxed210circ-boxed240circ--25375002026-04-21T03:40:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--mathbfm--beginpmatrix-a--b--c--d-endpmatrix--multiplying--mathbfm--with--beginpmatrix-1--2--3--4-endpmatrix--gives-45532002026-04-21T03:40:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero-compute-58116952026-04-21T03:40:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--theta--be-the-angle-between--mathbfa--and--mathbfb--so--costheta--frac12-rightarrow-theta--60circ--let--phi--be-the-angle-between--mathbfb--and--mathbfc--so--cosphi--fracsqrt32-rightarrow-phi--30circ--to-maximize--mathbfa-cdot-mathbfc--cosalpha--where--alpha--is-the-angle-between--mathbfa--and--mathbfc--arrange--mathbfa-mathbfb-mathbfc--in-a-plane-the-maximum-occurs-when--mathbfa--and--mathbfc--are-aligned-but-constrained-by-their-angles-relative-to--mathbfb--the-minimum-angle-between--mathbfa--and--mathbfc--is--60circ---30circ--30circ--so--cos30circ--fracsqrt32--however-if-they-are-aligned--alpha--0circ--but-this-requires--theta--phi--0circ--which-contradicts-the-given-dot-products-instead-use-the-cosine-law-for-angles--cosalpha-leq-cos60circ---30circ--cos30circ--fracsqrt32--thus-the-maximum-is--boxedfracsqrt32--58542762026-04-22T06:06:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--mathbfv--beginpmatrix-a--b--c-endpmatrix--the-cross-product-is-96706702026-04-22T06:06:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-cone-is-given-by--vtextcone--frac13pi-r2-h--substituting--h--2r--we-get-26600172026-04-21T03:52:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-cylinder-is--vtextcyl--pi-r2-h--for-the-cylinder--r--3r--and--h--4r--so-53120422026-04-21T03:53:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-hemisphere-is--vtexthemisphere--frac23-pi-r3--77575082026-04-21T03:53:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-the-triangle-is-28598212026-04-21T03:53:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-right-triangle-the-circumradius--r--is-half-the-hypotenuse-63726782026-04-21T03:53:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-power-output-we-sum-the-three-expressions-and-divide-by-3-92191892026-04-21T03:53:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-total-volume-we-add-the-three-fractions-first-find-a-common-denominator-the-least-common-denominator-of-4-8-and-16-is-16-55601372026-04-21T03:53:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-efficiency-we-sum-the-three-expressions-and-divide-by-3-35280152026-04-21T03:53:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve--t2---5t--6--0--factor--t---2t---3--0--the-solutions-are--t--2--and--t--3--thus-the-times-are--oxed2--and--oxed3--44370852026-04-21T04:20:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-vector-mathbfv--eginpmatrix-v1--v2--v3-endpmatrix-such-that-mathbfv-imes-mathbfc--mathbfd-compute-the-cross-product-mathbfv-imes-mathbfc-95818692026-04-21T04:23:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-k-such-that-mathbfp--kmathbfq-is-perpendicular-to-mathbfr-we-use-the-dot-product-condition-63428732026-04-21T04:29:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-mathbfw--langle-w1-w2-w3-45116822026-04-21T04:30:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-line-through-a-and-b-is-parametrized-as-84971732026-04-24T12:38:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-given-equation-is--80292762026-04-21T04:30:22+00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:44:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given--mt--t---fract33--and--b1--1--compute-successive-terms-22986272026-04-21T06:50:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--u--sqrtx--so--x--u2--and--px--u2-cdot-u---5u2--6u--u3---5u2--6u--17898402026-04-21T06:50:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-given--st--t---fract55---c1--frac12--compute-recursively-21575352026-04-21T06:50:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--hx--ax2--bx--c--from--h1--a--b--c--5--and--h-1--a---b--c--3--adding-gives--2a--2c--8--so--a--c--4--the-sum-of-roots-is---racba--4--so--b---4a--substituting--b---4a--into--a--b--c--5---a---4a--c--5-----3a--c--5--since--a--c--4--subtracting-gives---4a--1--so--a---rac14--then--c--4---a--4--rac14--rac174--and--b---4a--1--thus--hx---rac14x2--x--rac174--multiplying-by-4-to-eliminate-fractions--hx---x2--4x--17--verifying--h1---1--4--17--20--wait-inconsistency-rechecking--a---14---c--174---b--1--then--h1---14--1--174---1--4--174--204--5--correct--h-1---14--1--174----1--4--17-4--124--3--correct-sum-of-roots---ba---1---14--4--correct-final-answer--oxed-x2--4x--dfrac174--or--oxed-dfrac14x2--x--dfrac174--71369872026-02-23T20:10:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-set--x--y--1---k2--k1--k1---2k1-cdot-1--1--1---21--0--verify-consistency-try--x--2-y--1---k3--k2--k1---2k2--0--1---0--1--try--x--y--2---k4--k2--k2---2k4----k4--2k4--0--0----3k4--0----k4--0--assume--kx--0--for-all--x--but--k1--1-93594352026-04-21T06:53:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-note--x2---2x--1--x---12--use-polynomial-division-or-remainder-theorem-for-repeated-roots-let--fx--x5---3x3--2x---1--the-remainder--rx--has-degree--2-so--rx--ax--b--since--x---12--divides--fx---rx--we-have--f1--r1--and--f1--r1--compute--f1--1---3--2---1---1---fx--5x4---9x2--2--so--f1--5---9--2---2---rx--ax--b--so--r1--a--b---1---rx--a--so--a---2--then---2--b---1----b--1--thus-remainder-is---2x--1--final-answer--oxed-2x--1-question-a-plant-biologist-is-studying-a-genetic-trait-that-appears-in-every-12th-plant-in-a-rows-of-crops-planted-in-a-120-plant-grid-if-the-trait-is-expressed-only-when-the-plants-position-number-is-relatively-prime-to-12-how-many-plants-in-the-first-120-positions-exhibit-the-trait-98013832026-02-23T20:10:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-how-many-integers-from-1-to-120-are-relatively-prime-to-12-this-is-given-by-eulers-totient-function-applied-to-the-set-of-positions-but-since-the-condition-is-relative-primality-with-12-we-compute--phi12--the-number-of-integers-from-1-to-12-that-are-coprime-to-12-and-then-multiply-by-the-number-of-full-cycles-in-120-67234882026-04-21T06:53:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-two-gears-will-realign-at-their-starting-position-after-a-time-equal-to-the-least-common-multiple-lcm-of-their-rotation-periods-we-compute--extlcm18-30--73557522026-04-21T06:54:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-triangle-with-sides-9-12-and-15-is-a-right-triangle-since-92--122--81--144--225--152-the-area-is-frac12-times-9-times-12--54-the-longest-altitude-corresponds-to-the-shortest-side-9-units-using-the-formula-for-altitude-h--frac2-times-textareatextbase-we-calculate-h--frac2-times-549--12-boxed12-74111622026-04-21T06:57:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-capsule-consists-of-a-cylinder-of-height-10---2-times-2--6-units-subtracting-the-hemispheres-and-two-hemispheres-equivalent-to-one-sphere-volume-of-the-cylinder-pi-r2-h--pi-times-22-times-6--24pi-volume-of-the-sphere-frac43pi-r3--frac43pi-times-8--frac323pi-total-volume-24pi--frac323pi--frac1043pi-boxeddfrac1043pi-43751132026-04-21T06:57:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-herons-formula-the-semi-perimeter-s--frac7--8--92--12-the-area-is-sqrt1212-712-812-9--sqrt12-times-5-times-4-times-3--sqrt720--12sqrt5-the-inradius-r--fractextareas--frac12sqrt512--sqrt5-boxedsqrt5-59779692026-04-21T06:57:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-herons-formula-s--frac10--13--142--185-area--sqrt185185-10185-13185-14--sqrt185-times-85-times-55-times-45-simplify-185-times-45--8325-85-times-55--4675-so-area--sqrt8325-times-4675-approx-sqrt38909375-approx-6238-the-shortest-altitude-corresponds-to-the-longest-side-14-units-h--frac2-times-623814-approx-891-exact-calculation-yields-h--frac2-times-sqrt185-times-85-times-55-times-4514-simplify-the-expression-under-the-square-root-185-times-45--8325-85-times-55--4675-product--38909375-exact-area-frac14-sqrt185--10--13-185--10--13185---10--13185--10---13--frac14-sqrt415-times-45-times-215-times-55-this-is-complex-but-using-exact-values-the-altitude-simplifies-to-frac8414--6-however-precise-calculation-shows-the-exact-area-is-84-so-h--frac2-times-8414--12-wait-conflicting-results-correct-approach-for-sides-10-13-14-semi-perimeter-s--185-area--s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aily0.8https://pg.groshi247.com/highlight/solution-first-evaluate--g2--24541252026-04-21T07:41:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-find-the-slope--m--of-the-line-38345522026-04-21T07:41:15+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-equation--x2---y2--2025--factors-as--x---yx--y--2025--since--x--and--y--are-integers-both--x---y--and--x--y--must-be-integers-let--a--x---y--and--b--x--y--so--ab--2025--then--x--raca--b2--and--y--racb---a2--for--x--and--y--to-be-integers--a--b--and--b---a--must-both-be-even-meaning--a--and--b--must-have-the-same-parity-since--2025--34-cdot-52--it-has--4121--15--positive-divisors-each-pair--a-b--such-that--ab--2025--gives-a-solution-but-only-those-with--a--and--b--of-the-same-parity-are-valid-since-2025-is-odd-all-its-divisors-are-odd-so--a--and--b--are-both-odd-ensuring--x--and--y--are-integers-each-positive-divisor-pair--a-b--with--a-leq-b--gives-a-unique-solution-and-since-2025-is-a-perfect-square-there-is-one-square-root-pair-there-are-15-positive-divisors-so-15-such-factorizations-but-only-those-with--a-leq-b--are-distinct-under-sign-and-order-considering-both-positive-and-negative-factor-pairs-each-valid--ab--with--a-16601922026-02-23T20:58:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-events-be-35637002026-04-21T07:46:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-when-the-gears-align-again-we-compute-the-least-common-multiple-lcm-of-their-rotation-periods-since-they-rotate-at-48-and-72-rpm-rotations-per-minute-the-time-until-alignment-is-the-time-it-takes-for-each-to-complete-a-whole-number-of-rotations-such-that-both-return-to-start-simultaneously-this-is-equivalent-to-the-lcm-of-the-number-of-rotations-per-minute-in-terms-of-cycle-time-first-find-the-lcm-of-the-rotation-counts-over-time-or-convert-to-cycle-periods-the-time-for-one-rotation-is--frac148--minutes-and--frac172--minutes-so-we-find--mathrmlcmleftfrac148-frac172right--frac1mathrmgcd48-72--compute--mathrmgcd48-72--39140912026-04-21T07:50:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-the-entire-spherical-tumor-modeled-as-a-circle-in-the-cross-section-with-radius-6-cm-is-68693772026-04-21T07:51:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter-is-61826822026-04-21T07:51:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-the-hemisphere-is-94672002026-04-21T07:51:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-area-of-the-region-enclosed-by-the-circle-x2--y2--16-and-the-line-y--x--2-we-first-determine-the-points-of-intersection-substitute-y--x--2-into-the-circle-equation-79165612026-04-21T08:20:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--fx--ax3--bx2--cx--d--using-the-given-values-42267322026-04-21T08:36:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-minimize-the-expression-79864502026-04-21T08:48:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-recurrence-40708022026-04-21T08:48:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-the--y--intercept--b--represents-the-initial-value-when--x--0--here-the-folding-rate-starts-at-23-units-so--b--23--62471582026-04-21T08:52:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--s--be-the-sensor-price-and--m--the-meter-price-the-system-is-20774372026-04-21T08:52:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-set---3x--18--4x---2--solve--18--2--7x-rightarrow-x--rac207--substitute-into--y---3-cdot-rac207--18---rac607--rac1267--rac667--intersection-at--left-rac207-rac667-81815352026-04-21T08:52:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--v--sqrtu--so--u--v2--substituting-into-the-equation-gives--v3---4v2--3v--0--factoring-yields--vv2---4v--3--0--which-simplifies-to--vv-1v-3--0--thus--v--0-1-3--corresponding-to--u--0-1-9--the-sum-of-the-roots-is--0--1--9--10--oxed10-69466682026-04-21T08:52:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--t--x2---3--so--x2--t--3--substitute-into-the-equation--ft--2t--32---12t--3--13--expanding--2t2--6t--9---12t---36--13--2t2--12t--18---12t---36--13--2t2---5--thus--fx2--1--2x2--12---5--2x4--4x2--2---5--2x4--4x2---3--oxed2x4--4x2---3-4109352026-04-21T08:52:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-sides-be--a--and--b--by-the-pythagorean-theorem--a2--b2--252--625--the-area--a--ab--to-maximize--a--use-the-identity--a--b2--a2--2ab--b2--but-it-is-more-direct-to-note-that-for-fixed--a2--b2---ab--is-maximized-when--a--b--thus--2a2--625-rightarrow-a--rac25sqrt2--the-maximum-area-is--a--leftrac25sqrt2-51109962026-04-21T08:52:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-maximum-possible-value-of-mathbfu-cdot-mathbfw-we-start-by-using-the-cauchy-schwarz-inequality-mathbfu-cdot-mathbfw-leq-mathbfu-mathbfw--8-this-gives-the-bound---8-leq-mathbfu-cdot-mathbfw-leq-8-however-we-must-incorporate-the-given-dot-products-mathbfu-cdot-mathbfv--1-and-mathbfv-cdot-mathbfw--6-6518612026-02-23T22:17:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--pt--at2--bt--c--using-the-given-data-9339812026-04-22T06:15:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--et--at3--bt2--ct--d--use-the-given-values-68695952026-04-22T06:15:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-system-36108712026-04-21T09:29:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-sum-44251102026-04-21T09:46:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-a--b--2024-and-suppose-d--gcdab-since-d-mid-a-and-d-mid-b-then-d-mid-a--b--2024-so-d-is-a-divisor-of-2024-73693632026-04-21T09:46:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-formula-for-the-sum-of-cubes-4372402026-04-21T09:46:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-property-35395692026-04-21T09:46:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-travel-time-we-compute-the-arithmetic-mean-of-the-two-times-125-minutes-and-8-minutes-the-formula-for-the-average-is-65297042026-04-21T10:20:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-convert-2-hours-into-seconds-44107752026-04-21T10:20:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-spacecraft-travels-from-the-sun-to-venus-67-million-miles-then-from-venus-to-earth-the-remaining-distance-since-earth-is-93-million-miles-from-the-sun-and-venus-is-67-million-miles-away-the-distance-from-venus-to-earth-is-59285082026-04-21T10:20:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-time-until-alignment-is-the-least-common-multiple-lcm-of-88-and-4333-first-factor-both-numbers-10248232026-04-21T10:20:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-of-12-15-and--x--is-given-by--frac12--15--x3--14--multiply-both-sides-by-3--27--x--42--subtract-27--x--15--boxed15-51476312026-04-21T10:20:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-set-up-the-proportion--frac74--frac21x--cross-multiply--7x--84--divide--x--12--boxed12-96719432026-04-21T10:20:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-integers-between-447-and-707-are-5-6-7-total-3-boxed3-63389692026-04-21T10:20:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-total-days-be--x--ratio--frac38--frac15x--cross-multiply--3x--120--divide--x--40--boxed40-91406792026-04-21T10:20:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-total-articles-be--x---06x--45--divide--x--frac4506--75--boxed75question-if-fraca--ba---b--fraca---ba--b--3-find-the-value-of-fraca2--b2a2---b2-24606622026-04-21T10:20:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-selecting-3-out-of-5-distinct-data-types-and-arranging-them-in-order-since-order-matters-and-no-repetition-is-allowed-this-is-a-permutation-problem-the-number-of-permutations-of-5-distinct-items-taken-3-at-a-time-is-given-by-76881152026-04-21T11:30:58+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-counting-the-number-of-6-day-sequences-using-4-experiment-types-p-c-b-a-where-each-type-appears-at-least-once-and-sequences-are-ordered-this-is-a-classic-inclusion-exclusion-problem-counting-the-number-of-surjective-functions-from-a-6-day-sequence-to-4-experiment-types-60077172026-04-24T12:40:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-code-has-5-positions------the-middle-3rd-character-is-fixed-as-b-we-must-choose-4-distinct-letters-from-a-b-c-d-e-with-b-fixed-in-position-3-so-the-remaining-4-positions-use-the-other-4-letters-but-b-is-already-placed-so-we-use-a-c-d-e-for-the-other-4-positions-78068312026-04-21T11:31:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-9-tokens-in-total-where-there-are-4-identical-gold-3-identical-silver-and-2-identical-bronze-tokens-the-number-of-distinct-permutations-is-given-by-the-multinomial-coefficient-38983632026-04-21T11:31:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-7-badges-where-3-are-identical-robotics-r-2-are-identical-coding-c-and-2-are-identical-data-science-d-the-number-of-distinct-sequences-is-the-multinomial-coefficient-43928412026-04-21T11:31:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-given-equation-in-spherical-coordinates-is--64217002026-04-21T11:44:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-vector-equation-mathbfv-imes-mathbfa--mathbfb-has-a-solution-only-if-mathbfa-cdot-mathbfb--0-since-the-cross-product-is-perpendicular-to-mathbfa-56553722026-04-21T11:44:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-identity-sin2z--2sin-z-cos-z-so-59298712026-04-21T11:44:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-set--3t--9t-1--since--9--32--rewrite-the-equation-as--3t--32t-1--simplify-the-righ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aily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-sphere-with-radius--2r--is-34752572026-04-21T12:12:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--s--raca--ba---b--raca---ba--b--combine-the-fractions-31361102026-04-21T12:17:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--px--ax2--bx--c--using-the-given-values-2377752026-04-21T12:17:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-recurrence--an1--an---racan36--resembles-the-taylor-series-for--rctanu--where--racddu-rctanu--rac11--u2--however-the-recurrence-is-not-exact-assume-the-limit--l--exists-then--l--l---racl36-rightarrow-racl36--0-rightarrow-l--0--to-confirm-convergence-note--a1--pi2-pprox-157--1--and--an1--an1---racan26--since--a1--sqrt6---an--is-decreasing-and-bounded-below-by-0-by-monotone-convergence--an-o-0--29221022026-04-21T12:17:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-radius-of-the-circular-lake-be--r--km-the-chord-length-is-10-km-so-half-the-chord-length-is--5--km-the-perpendicular-distance-from-the-center-to-the-chord-is-3-km-by-the-pythagorean-theorem-applied-to-the-right-triangle-formed-by-the-radius-half-the-chord-and-the-perpendicular-from-the-center-to-the-chord-we-have-49039932026-04-21T12:38:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-central-angle-corresponding-to-the-arc-is--120circ--or--rac2pi3--radians-the-chord-length--c--subtended-by-a-central-angle--heta--in-a-circle-of-radius--r--is-given-by-81686402026-04-21T12:38:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-an-equilateral-triangle-with-side--s--the-circumradius--r--is-given-by-35760702026-04-21T12:38:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--r--be-the-radius-of-the-forest-the-chord-length-is-14-km-so-half-is-7-km-the-perpendicular-distance-from-the-center-to-the-chord-is-5-km-using-the-pythagorean-theorem-64208072026-04-22T06:19:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-chord-length--c--1000--km-radius--r--500sqrt2--59204022026-04-22T06:19:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-surface-area-of-a-regular-hexagonal-prism-consists-of-the-area-of-the-two-hexagonal-bases-and-the-six-triangular-lateral-faces-each-face-is-equilateral-with-side-length--s--4--cm-6279322026-04-21T12:52:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter--s--frac10--13--152--19--km-the-area--a--sqrt1919-1019-1319-15--sqrt19-cdot-9-cdot-6-cdot-4--sqrt4104--6407--km-approximate-the-shortest-altitude-corresponds-to-the-longest-side-15-km-using--a--frac12-cdot-textbase-cdot-textheight--the-altitude--h--frac2a15--frac2-cdot-640715-approx-854--km-for-exact-value-simplify--sqrt4104--sqrt4-cdot-1026--2sqrt1026--but-numerical-approximation-suffices-here-thus-the-shortest-altitude-is--boxed854--km-94889722026-04-21T13:00:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-hemisphere-is--frac23pi-r3--frac23pi-2x3--frac163pi-x3--the-cylinders-volume-is--pi-r2-h--pi-x2-cdot-4x--4pi-x3--the-ratio-is--fracfrac163pi-x34pi-x3--frac163-div-4--frac43--thus-the-ratio-is--boxeddfrac43--56478052026-04-21T13:00:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-diagonal-of-the-rectangle-is-the-circles-diameter-using-the-pythagorean-theorem--textdiagonal--sqrt32--42--5--cm-the-circumference-is--pi-cdot-textdiameter--5pi--cm-thus-the-circumference-is--boxed5pi--cm-28216232026-04-21T13:00:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-diameter-of-the-semicircular-drum-is-12-meters-so-the-radius-is-half-the-diameter-thus--r--frac122--6--meters-the-radius-is--boxed6--metersquestion-find--x--such-that-the-vectors--beginpmatrix-x--3-endpmatrix--and--beginpmatrix-2---x-endpmatrix--are-orthogonal-5073982026-02-24T02:42:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-two-vectors-are-orthogonal-if-their-dot-product-equals-zero-compute-the-dot-product--x-cdot-2--3-cdot--x--2x---3x---x--set-this-equal-to-zero---x--0--solving-gives--x--0--boxed0-80936802026-02-24T02:42:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--y--frac2sin-x--1sin-x---3--solve-for--sin-x---ysin-x---3--2sin-x--1-rightarrow-ysin-x---3y--2sin-x--1-rightarrow-sin-xy---2--3y--1--thus--sin-x--frac3y--1y---2--since--sin-x-leq-1--solve--left-frac3y--1y---2-right-leq-1--this-leads-to---1-leq-frac3y--1y---2-leq-1--sol-35286462026-02-24T02:42:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circumscribed-circle-using-the-pythagorean-theorem-the-diagonal--d--sqrt102--242--sqrt100--576--sqrt676--26--km-the-circumference--c--pi-times-d--26pi--km-thus-the-circumference-is--boxed26pi--87663392026-04-21T13:04:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-the-incircle-is--pi-c2--for-a-right-triangle-the-semiperimeter--s--fraca--b--z2--and-the-inradius--c--fraca--b---z2--the-area-of-the-triangle-is--r-times-s--c-times-z--c--therefore-the-ratio-is--fracpi-c2cz--c--fracpi-cz--c--the-answer-is--boxeddfracpi-cz--c--11438442026-04-21T13:04:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-a-right-triangle-with-legs-7-and-24-and-hypotenuse-25-the-hypotenuse-is-the-diameter-of-the-circumscribed-circle-the-radius--r--frac252--125--units-thus-the-radius-is--boxed125--54447582026-04-21T13:04:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-length-be--3x--and-width--2x--the-perimeter--23x--2x--10x--14322132026-04-21T13:04:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-assigning-5-distinct-tracking-devices-to-8-bird-species-such-that-each-species-gets-at-least-one-device-this-is-equivalent-to-counting-the-number-of-onto-functions-from-a-set-of-8-species-to-a-set-of-5-devices-with-the-condition-that-no-device-is-left-unused-49880412026-04-22T06:19:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-sequences-of-length-5-one-per-layer-where-each-element-is-an-epoch-from-1-to-4-and-no-two-consecutive-layers-have-the-same-epoch-82131902026-04-21T13:06:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-total-number-of-6-digit-numbers-with-digits-only-3-or-7-each-digit-has-2-choices-so--26--64--but-6-digit-numbers-cannot-start-with-0-but-since-digits-are-only-3-or-7-all-64-are-valid-86809872026-04-21T13:06:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-multiply--frac34--by--frac52--57111202026-04-21T13:22:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-multiply-the-original-height-by--frac32--31087672026-04-21T13:22:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-subtract--frac23--from--frac73--95368592026-04-21T13:22:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-equation-simplifies-to--a---3--5--so-18927042026-04-21T13:22:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-smallest-4-digit-number-divisible-by-11-is--1001--since--1001-div-11--91--and-the-largest-is--9999--since--9999-div-11--909--the-count-is--909---91--1--819--19929712026-04-21T13:22:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-each-of-the-6-rods-there-are-3-choices-of-connectors-the-total-configurations-are-36--729-therefore-the-answer-is-boxed729-57425412026-04-21T13:22:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-ways-to-choose-3-items-is-binom163--560-the-favorable-cases-are-binom72-times-binom91--21-times-9--189-the-probability-is-frac189560--frac2780-thus-the-answer-is-boxeddfrac2780-99755992026-04-21T13:22:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-ways-to-select-4-items-binom254--12650-the-unfavorable-case-no-turtles-is-binom154--1365-the-probability-of-at-least-one-turtle-is-1---frac136512650--1---frac2732530--frac22572530-the-answer-is-boxeddfrac22572530-94417132026-04-21T13:22:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-items-are-5--7--12-with-5-red-and-7-blue-the-number-of-distinct-sequences-is-frac1257--792-therefore-the-answer-is-boxed792question-a-robotic-engineer-is-programming-a-machine-to-assemble-components-using-exactly-3-different-colored-17709182026-04-21T13:22:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-distributing-6-identical-panels-into-4-distinguishable-rooftops-with-each-rooftop-receiving-at-least-one-panel-this-is-a-classic-stars-and-bars-problem-with-the-constraint-of-positive-integers-53040732026-04-21T13:51:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-9-organisms-where-there-are-repeated-types-4-thermophiles-t-3-psychrophiles-p-and-2-barophiles-b-the-number-of-distinct-permutations-of-a-multiset-is-given-by-95930272026-04-21T13:51:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-identify-the-prime-numbers-on-a-six-sided-die-2-3-5--3-primes-non-primes-1-4-6--3-non-primes-70878712026-04-21T13:51:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve--2sin2z--1--0-rightarrow-sin2z---rac12--the-general-solutions-for--2z--are--2z--210circ--360circ-k--and--2z--330circ--360circ-k--where--k-in-mathbbz--dividing-by-2--z--105circ-165circ-285circ-345circ--within--0circ-360circ--the-sum-is--105--165--285--345--900circ--oxed900-11302112026-04-21T14:03:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-transformation-swaps-components-and-negates-one-the-standard-matrix-is--eginpmatrix-0---1--1--0-endpmatrix--oxedeginpmatrix-0---1--1--0-endpmatrix-16992732026-02-24T03:51:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-closest-point-minimizes-distance-parametrize-the-line-as--t-2t--1--the-squared-distance-to--3-4--is--t---32--2t--1---42--t---32--2t---32--differentiate--2t---3--8t---12--0-rightarrow-10t---18--0-rightarrow-t--18--substituting-the-point-is--18-46--oxed18-46-5547272026-04-21T14:03:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-simplify--fn--by-factoring-the-numerator--n3---8--n---2n2--2n--4--thus--fn--racn---2n2--2n--4n---2--n2--2n--4--for--n-15336932026-04-21T14:03:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--y--rac3t4---t2--for--t--2--denominator--4---t2--0--so--y--0--rewrite--y--rac3t-t2---4---rac3tt---2t--2--let--t--2--epsilon---epsilon--0--but-instead-analyze--y--as--t-o-2---y-o--infty--as--t-o-infty---y-o-0---the-minimum-value-of-26164402026-02-24T03:51:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-1-5-9-13-17-21-25-29-33-37-the-sum-of-an-arithmetic-sequence-is-given-by--s--fracn22a--n-1d--here--n--10---a--1---d--4--38680642026-04-21T14:04:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-looking-for-integers--n--such-that--n-equiv-3-pmod7--among-the-first-200-positive-integers-these-numbers-form-an-arithmetic-sequence--3-10-17-ldots--with-first-term-3-and-common-difference-7-72671822026-04-21T14:04:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-problem-requires-counting-the-number-of-ways-to-partition-4-distinct-items-into-2-non-empty-identical-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind--s4-2--the-formula-for--sn-k--is--sn-k--sn-1-k-1--k-cdot-sn-1-k--using-known-values--s4-2--7--thus-the-number-of-ways-is--boxed7--37033822026-04-21T14:04:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-ways-to-choose-3-guidelines-from-5-is--binom53--10--the-number-of-favorable-outcomes-2-welfare-1-non-welfare-is--binom32-cdot-binom21--3-cdot-2--6--the-probability-is--frac610--frac35--thus-the-probability-is--boxeddfrac35--4710302026-04-21T14:04:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-circumference-of-the-circle-in-which-a-rectangle-is-inscribed-we-first-recognize-that-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal-d-of-a-5-cm-by-8-cm-rectangle-is-93530202026-04-21T14:04:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-standard-trigonometric-values-62591132026-04-21T14:04:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-atextcircle-of-the-inscribed-circle-is-29411252026-04-21T14:04:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-ramanujans-approximation-for-the-circumference-of-an-ellipse-56217652026-04-21T14:04:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-radius-r-of-the-inscribed-circle-of-a-triangle-with-sides-a--13-b--14-and-c--15-we-use-the-formula-90364782026-04-21T14:04:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-us-define--fu--such-that--fx2--2--x4--4x2--4--observe-that-the-right-hand-side-can-be-rewritten-as-85420612026-04-21T14:08:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given--p--q--5--and--pq--6--we-use-the-identity-45992852026-04-21T14:08:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-observe-that-the-right-hand-side-is--x3--12--2--however-we-can-directly-write-45351112026-04-21T14:08:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-us-compute--x2--x--13-mod-x2---x--1--60339442026-04-21T14:09:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-identify-the-prime-numbers-between-1-and-6-2-3-5-so-there-are-3-primes-and-3-non-primes-1-4-6-on-each-die-88621552026-04-21T14:12:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-partitioning-5-distinguishable-objects-proposals-into-3-indistinguishable-non-empty-subsets-panels-this-is-given-by-the-stirling-numbers-of-the-second-kind-denoted-s53-65381652026-04-21T14:12:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-distributing-4-distinguishable-satellites-into-2-indistinguishable-orbital-slots-where-each-slot-can-hold-any-number-including-zero-and-the-order-within-a-slot-does-not-matter-since-slots-are-indistinct-and-not-ordered-15037462026-04-21T14:13:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-partitioning-6-distinguishable-objects-proposals-into-3-non-empty-indistinct-subsets-teams-this-is-given-by-the-stirling-number-of-the-second-kind-s63-51216532026-04-21T14:13:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-rewrite-the-equation-in-standard-form-64985802026-04-21T14:20:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area--a--of-a-regular-hexagon-with-side-length--s--is--a--frac3sqrt32s2--given--54sqrt3--frac3sqrt32s2--solving-for--s2--yields--s2--36--so--s--6--the-new-side-length-is--6---2--4--the-new-area-is--frac3sqrt3242--24sqrt3--the-decrease-in-area-is--54sqrt3---24sqrt3--30sqrt3--boxed30sqrt3-30178802026-04-21T14:21:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-a-circular-sector-is--a--fractheta360-pi-r2--original-area--frac90360-pi-122--36pi--new-radius--12--4--16--meters-new-area--frac90360-pi-162--64pi--the-increase-is--64pi---36pi--28pi--boxed28pi-2658652026-04-21T14:20:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-original-area--frac12-times-15-times-20--150--km-new-shorter-leg--15--5--20--km-new-area--frac12-times-20-times-20--200--km-the-increase-is--200---150--50--km-boxed50-55923352026-04-21T14:21:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-original-area--8-times-15--120--m-new-dimensions--8---3--5--meters-and--15---3--12--meters-new-area--5-times-12--60--m-the-decrease-is--120---60--60--m-boxed60question-a-geologist-is-studying-the-periodic-patterns-of-seismic-waves-and-notes-that-a-specific-wave-pattern-repeats-every-24-seconds-if-the-current-time-is-2025-06-28-182116-what-is-the-smallest-positive-time-in-seconds-after-182116-when-the-wave-pattern-will-repeat-such-that-this-time-is-congruent-to-6-mod-11-88672212026-04-21T14:21:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--t--be-the-time-in-seconds-after-182116-when-the-pattern-repeats-and-satisfies--t-equiv-6-pmod11--since-the-wave-pattern-repeats-every-24-seconds-we-are-looking-for-the-smallest--t--such-that-30742452026-04-21T14:21:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--px--ax3--bx2--cx--d--we-are-given-88561292026-04-21T14:46:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-functional-equation-25499282026-04-21T14:46:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-fourth-vertex-of-a-regular-tetrahedron-with-the-given-three-vertices-a--1-2-3-b--4-5-6-and-c--7-8-9-we-first-compute-the-pairwise-distances-between-the-given-points-58370682026-04-21T14:53:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-mathbfv--beginpmatrix-a--b-endpmatrix-be-any-vector-orthogonal-to-beginpmatrix-3---4-endpmatrix-ie-35172222026-04-21T14:53:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-with--b1--2--compute--b2--q2--22---rac244--4---rac164--4---4--0--then-compute--b3--q0--02---rac044--0---0--0--thus--b3--oxed0--70903092026-04-21T14:55:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve--3k---rac32k2---1--let--x--3k--note--x--0--the-inequality-becomes--x---racx22---1--or---racx22--x--1--0--multiply-by--2-reversing-inequality--x2---2x---2--0--solve--x2---2x---2--0--roots-are--x--1-pm-sqrt3--since--x--0--critical-point-is--x--1--sqrt3-pprox-2732--the-quadratic-is-positive-when--x--1--sqrt3--since--x--3k--find-smallest--k--such-that--3k--1--sqrt3--test--k--1---31--3--2732--check--r1--3---rac92---15---1--thus--k--oxed1--84679942026-02-24T04:43:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-ways-to-choose-3-distinct-letters-is-inom263-the-favorable-cases-involve-selecting-3-letters-that-correspond-to-exactly-one-placeholder-symbol-assuming-each-symbol-is-defined-by-a-unique-combination-of-3-letters-with-no-overlap-in-symbol-definitions-there-are-10-favorable-combinations-thus-the-probability-is-rac10inom263--rac102600--rac1260-oxeddfrac1260-41721272026-04-21T14:55:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-treat-the-two-letters-as-a-single-entity-this-gives-4-arrangements-for-the-entities-the-letter-pair-and-3-manuscripts-the-letters-within-the-pair-can-be-ordered-in-2-ways-total-favorable-arrangements-4-imes-2-total-possible-arrangements-5-the-probability-is-rac4-imes-25--rac25-oxeddfrac25-46799262026-04-21T14:55:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-there-are-11-integers-010-of-which-6-are-even-0-2-4-6-8-10-the-number-of-ways-to-choose-4-distinct-even-integers-is-inom64-the-total-number-of-ways-to-choose-4-distinct-integers-is-inom114-the-probability-is-racinom64inom114--rac15330--rac122-oxeddfrac122-34342182026-04-21T14:55:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-letters-26-vowels-5-consonants-21-total-3-letter-arrangements-with-distinct-letters-26-imes-25-imes-24-unfavorable-cases-no-vowels-21-imes-20-imes-19-favorable-probability-1---rac21-imes-20-imes-1926-imes-25-imes-24--1---rac798015600--1---rac133260--rac127260-oxeddfrac127260-13268252026-04-21T14:55:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-of-ways-to-choose-3-18th-century-manuscripts-is-inom53-total-ways-to-choose-3-manuscripts-from-15-is-inom153-the-probability-is-racinom53inom153--rac10455--rac291-oxeddfrac291question-if-the-combined-population-of-two-marine-mammal-species-in-a-region-is-10-and-the-sum-of-their-squared-populations-is-50-find-the-sum-of-their-cubes-59707782026-04-21T14:55:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-populations-be--a--and--b--given--a--b--10--and--a2--b2--50--first-compute--ab--using--a--b2--a2--2ab--b2--substituting--100--50--2ab--so--ab--25--the-sum-of-cubes-is--a3--b3--a--b3---3aba--b--1000---3-cdot-25-cdot-10--1000---750--oxed250--44476012026-04-21T14:55:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-given--x--y--45--multiplying-by-5-yields--5x--5y--5x--y--5-cdot-45--oxed225--16544992026-04-21T14:55:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-multiply-numerator-and-denominator-by--sqrt7--sqrt3--2076522026-04-21T14:55:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-recognize-it-as-a-difference-of-cubes--2a3---3b3--8a3---27b3--alternatively-expand-directly-68426252026-04-21T14:55:58+00:00daily0.8https://pg.groshi247.com/highlight/solution-set--3x--2---2x--12--solve--5x--10--so--x--2--substitute-into--y--32--2--8--the-intersection-is--oxed2-8-question-let--gx--be-a-polynomial-such-that--g2x--3--4x2--12x--5--find--gx2---1--85457552026-02-24T04:43:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--u--2x--3--then--x--racu---32--substitute-into--gu--4leftracu---32-15069212026-04-21T14:55:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-group-terms--9x2---2x---16y2--4y--144--complete-the-square--9x---12---1---16y--22---4--144--expand--9x---12---9---16y--22--64--144--simplify--9x---12---16y--22--89--the-center-is-at--1--2--final-answer-oxed-82117742026-02-24T04:43:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-must-be-divisible-by-the-least-common-multiple-lcm-of-7-11-and-13-since-these-are-all-primes-lcm--7--11--13--1001-the-smallest-four-digit-number-is-the-first-multiple-of-1001--1000-which-is-1001-imes-1--1001-thus-the-answer-is-oxed1001-32726312026-04-21T14:57:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-greatest-common-divisor-of-a-and-b-must-divide-their-sum-a--b--100-the-largest-divisor-of-100-is-50-to-achieve-gcda-b--50-set-a--50-and-b--50-thus-the-maximum-value-is-oxed50-29682932026-04-21T14:57:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-amount-of-saline-in-the-first-solution-be--020-times-3--06--liters-the-second-solution-contributes--050x--liters-of-saline-the-total-volume-is--3--x--liters-and-the-total-saline-is--0353--x--setting-up-the-equation-70992412026-04-21T15:06:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-model-states-that-each-hour-the-count-is-multiplied-by-3-after-1-hour--31--after-2-hours--32--so-after-4-hours--34--but-the-problem-states-the-population-is--36--times-initial-which-contradicts-unless-growth-rate-differs-however-assuming-the-model-is-correct-after-4-hours-30055532026-04-21T15:06:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--t--6--into-the-function-33120782026-04-21T15:06:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-local-extrema-of-the-function--pt--t3---6t2--9t--10--we-first-compute-its-derivative-29170752026-04-21T15:06:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-slant-asymptote-of--it--fract2--4t--2--we-perform-polynomial-long-division-of--t2--4--by--t--2--45222632026-04-21T15:07:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-maximum-of--ft--frac100tt2--4--we-compute-its-derivative-using-the-quotient-rule-65090562026-04-21T15:07:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-analyze-the-cubic-polynomial--rx--x3---3x--2--to-find-the-number-of-real-roots-we-first-attempt-rational-root-theorem-possible-rational-roots-are--pm1-pm2--testing-80114332026-04-21T15:07:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given--iu--u---fracu39--and--ixn--frac1n--so-42617682026-04-21T15:07:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-standard-form-of-a-parabola-with-vertical-axis-is--y--frac14px2--the-focus-is-at--0-p--so--p--5--the-width-of-20-units-implies-the-parabola-passes-through--10-5--substituting--5--frac120100-rightarrow-5--5--which-holds-thus-the-equation-is--y--frac120x2--boxedy--dfrac120x2-27462412026-04-21T15:12:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-partial-fractions--frac1kk2--frac12left-frac1k---frac1k2-right--the-sum-becomes--frac12-left-sumk150-frac1k---sumk150-frac1k2-right--frac12-left-sumk150-frac1k---sumk352-frac1k-right--telescoping-gives--frac12-left-frac11--frac12---frac151---frac152-right--frac12-left-frac32---frac1032652-right--frac12-cdot-frac39752652--frac13251768--boxeddfrac13251768-7042862026-04-21T15:12:15+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-identity--p2--q2--p--q2---2pq--substituting--10---24i--3--4i2---2pq--compute--3--4i2--9--24i---16---7--24i--then--10---24i---7--24i---2pq--rearranging--17---48i---2pq-rightarrow-pq--frac-17--48i2--boxed-dfrac172--24i-18686962026-02-24T04:59:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-probability-follows-a-binomial-distribution-with-parameters-n--4-trials-and-p--frac13-success-probability-the-formula-for-exactly-k-successes-is-binomnk-pk-1-pn-k-for-k--2-62961922026-04-21T15:15:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-equivalent-to-finding-the-number-of-ways-to-partition-6-distinct-elements-into-2-non-empty-subsets-which-is-the-stirling-number-of-the-second-kind-s6-2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tion-the-problem-involves-arranging-7-earthquakes-where-3-are-identical-magnitude-75-and-4-are-identical-magnitude-68-the-number-of-distinct-sequences-is-the-multinomial-coefficient-44236292026-04-21T16:43:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-distinct-rankings-is-calculated-by-treating-the-73540652026-04-21T16:43:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-when-the-concentration-is-increasing-we-analyze-the-derivative-of--ct--using-the-quotient-rule-77995122026-04-21T16:46:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-vertical-asymptotes-and-holes-we-examine-the-denominator-of--dt--ract2--1t---2--71241522026-04-21T16:46:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-evaluate--f6--12346872026-04-21T16:47:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-four-terms-be--a---frac3d2-a---fracd2-a--fracd2-a--frac3d2--this-ensures-the-terms-are-in-arithmetic-progression-with-common-difference--d--the-first-term-is--a---frac3d2--and-the-last-term-is--a--frac3d2--their-sum-of-squares-is-38935302026-04-21T16:47:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-gcf-of-42-and-63-factor-both-numbers-42--2-times-3-times-7-and-63--32-times-7-the-common-prime-factors-are-3-and-7-multiply-them-3-times-7--21-boxed21-47773292026-04-21T16:55:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-equation-simplifies-to-b---3--5-so-b---3--5-or-b---3---5-solving-gives-b--8-or-b---2-sum-8---2--6-boxed6-67043902026-02-24T06:46:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-increases-by-3-each-time-the-nth-term-is-2--n-1-times-3--3n---1-set-3n---1--32-3n--33-so-n--11-boxed11-question-1-24369132026-04-21T16:55:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-angle-between-the-diagonal-of-a-square-and-its-side-is-45circ-but-here-the-angle-is-150circ-which-lies-in-the-second-quadrant-using-the-reference-angle-30circ-cos-150circ---cos-30circ---fracsqrt32-boxed-dfracsqrt32-91497242026-02-24T07:02:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-triangle-with-sides-9-12-15-is-right-angled-since-92--122--152-the-area-is-frac12-cdot-9-cdot-12--54-the-altitudes-correspond-to-frac2-cdot-textareatextside-so-61870472026-04-21T17:07:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-volume-of-hemisphere-frac23pi-3x3--frac23pi-cdot-27x3--18pi-x3-61182202026-04-21T17:07:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-vertex-form-of-a-parabola--y--ax2--bx--c--has-its-vertex-at--x---fracb2a--here--a---2---b--8--so--x---frac82-2--2--substituting--x--2--into-the-equation--y---222--82---5---8--16---5--3--thus-the-vertex-is-at--2-3--boxed2-3-8948952026-02-24T07:03:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-multiply-both-sides-by-2--mv2--frac32kt--divide-by--m---v2--frac3kt2m--take-the-square-root--v--sqrtfrac3kt2m--boxedv--sqrtdfrac3kt2m-20874922026-04-21T17:07:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-maximum-height-occurs-at-the-vertex-for--y---x2--6x---8---a---1---b--6--so--x---frac62-1--3--substitute--x--3---y---32--63---8---9--18---8--1--the-maximum-height-is--1--unit-boxed1-56578222026-02-24T07:03:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--u--2x--1--so--x--fracu---12--substitute-into--fu---fu--4leftfracu---12right2--12leftfracu---12right--9--u---12--6u---1--9--u2---2u--1--6u---6--9--u2--4u--4--thus--fx2---3--x2---32--4x2---3--4--x4---6x2--9--4x2---12--4--x4---2x2--1--boxedx4---2x2--1-28653802026-04-21T17:07:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-subtract-1--frac2x---5x--3---1-geq-0-rightarrow-frac2x---5---x--3x--3-geq-0-rightarrow-fracx---8x--3-geq-0--critical-points-at--x--8--and--x---3--test-intervals-88404492026-02-24T07:03:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-factors-of-45-are-1-3-5-9-15-45-there-are-6-factors-the-probability-is-frac645--frac215-boxeddfrac215-53863652026-04-21T17:12:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-prime-factors-of-12-22-times-3-of-18-2-times-32-lcm-is-22-times-32--36-boxed36-8572472026-04-21T17:12:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-sum-is-5-times-22--110-sum-of-known-values-20--21--23--24--88-the-fifth-value-is-110---88--22-boxed22question-find-the-matrix-mathbfm-such-that-mathbfm-beginpmatrix-1--0-endpmatrix--beginpmatrix-2---1-endpmatrix-and-mathbfm-beginpmatrix-0--1-endpmatrix--beginpmatrix-3--4-endpmatrix-31409172026-02-24T07:08:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-matrix-mathbfm-is-constructed-by-placing-the-images-of-the-standard-basis-vectors-as-its-columns-thus-mathbfm--beginpmatrix-2--3---1--4-endpmatrix-verifying-mathbfm-beginpmatrix-1--0-endpmatrix--beginpmatrix-2---1-endpmatrix-and-mathbfm-beginpmatrix-0--1-endpmatrix--beginpmatrix-3--4-endpmatrix-confirming-correctness-boxedbeginpmatrix-2--3---1--4-endpmatrix-22877842026-02-24T07:08:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-closest-point-is-the-projection-of-4-3-onto-the-line-the-formula-for-the-projection-of-a-point-x0-y0-onto-ax--by--c--0-is-used-rewriting-the-line-as-frac12x--y---5--0-we-compute-the-projection-alternatively-parametrize-the-line-and-minimize-distance-let-x--t-then-y---frac12t--5-the-squared-distance-to-4-3-is-t---42--left-frac12t--5---3right2--t---42--left-frac12t--2right2-expanding-t2---8t--16--frac14t2---2t--4--frac54t2---10t--20-taking-derivative-and-setting-to-zero-frac52t---10--0-rightarrow-t--4-substituting-back-y---frac124--5--3-thus-the-closest-point-is-4-3-which-lies-on-the-line-boxed4-3-40491332026-04-21T17:12:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-angle-theta-between-mathbfa-and-mathbfb-is-given-by-costheta--fracmathbfa-cdot-mathbfbmathbfa-mathbfb-compute-the-dot-product-21---34--2---12---10-compute-magnitudes-mathbfa--sqrt22---32--sqrt13-mathbfb--sqrt12--42--sqrt17-thus-costheta--frac-10sqrt13sqrt17-rationalizing-theta--arccosleft-frac10sqrt221right-boxedarccosleft-dfrac10sqrt221right-86180702026-04-21T17:12:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-scalar-triple-product-mathbfu-cdot-mathbfv-times-mathbfw-represents-the-volume-of-the-parallelepiped-formed-by-the-vectors-its-maximum-absolute-value-occurs-when-the-vectors-are-mutually-orthogonal-since-they-are-unit-vectors-the-maximum-value-is-1-cdot-1-cdot-1--1-for-example-if-mathbfu--mathbfv--mathbfw-are-orthonormal-the-19694332026-04-21T17:12:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-first-100-even-numbers-form-an-arithmetic-sequence-2--4--6--dots--200-the-sum-is-s--frac1002-times-2--200--50-times-202--10100-to-find-10100-mod-7-divide-10100-by-7-7-times-1442--10094-so-the-remainder-is-10100---10094--6-boxed6-14589492026-04-24T12:42:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-x-be-the-integer-then-x-equiv--1-pmod5-and-x-equiv--1-pmod7-this-implies-x--1-is-divisible-by-both-5-and-7-so-x--1--textlcm5-7--35-thus-x--34-boxed34question-a-retired-engineer-designing-a-robotic-arm-observes-that-the-sum-of-two-gear-rotation-speeds--a--and--b--is-10-and-the-sum-of-their-squares-is-58-what-is--a3--b3--20759912026-02-24T07:15:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given--a--b--10--and--a2--b2--58--we-want-to-find--a3--b3--34011192026-04-24T12:42:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-initially-the-scientist-conducted-40-experiments-with-a-75-success-rate-44014872026-04-24T12:42:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given--x--frac1x--4--square-both-sides-50694562026-04-22T06:24:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x--sqrt7--2sqrt10--sqrt7---2sqrt10--we-compute--x2--49802822026-04-21T17:16:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-analyze-the-expression-by-considering-the-critical-points-where-expressions-inside-absolute-values-change-sign--x--frac52--and--x---3--we-consider-three-intervals-51632062026-04-21T17:16:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-the-sum-to-product-identity-73214232026-04-21T17:17:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-maximum-value-of--ft--t2---fract44--take-the-derivative--ft--2t---t3--set--ft--0---2t---t3--0-rightarrow-t2---t2--0--critical-points-at--t--0--and--t--pmsqrt2--evaluate--fsqrt2--sqrt22---fracsqrt244--2---frac44--1--the-maximum-value-is--boxed1--61205592026-04-21T17:24:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--a--b--10--and--a2--b2--58--compute--a--b2--100--a2--2ab--b2--58--2ab--solve-for--ab---2ab--42-rightarrow-ab--21--the-area-is--frac12ab--boxedfrac212--19071352026-04-21T17:24:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-simplify--pn--fracnnn2--2--frac1n2--2--as--n-to-infty---n2--2-to-infty--so--limn-to-infty-pn--boxed0--33655842026-04-21T17:24:58+00:00daily0.8https://pg.groshi247.com/highlight/solution-take-derivative--ax---frac4x2---2x--set--ax--0----frac4x2---2x--0-rightarrow--4---2x3--0-rightarrow-x3---2--no-positive-roots-analyze-behavior-as--x-to-0---a-to-infty--as--x-to-infty---a-to--infty--thus-no-maximum-exists-the-function-increases-without-bound-near--x--0--however-if-constrained-recheck-calculations-correct-approach-49258192026-02-24T07:32:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-count-of-integers--n--such-that--1-leq-n-leq-500--and--n-equiv-2-pmod7--we-note-the-sequence-starts-at-2-and-increases-by-7-each-time--2-9-16-dots--the-general-term-is--7k--2--solving--7k--2-leq-500--gives--k-leq-rac4987-pprox-7114--so--k--0-1-dots-71--this-yields--72--integers-oxed72-27515512026-04-21T17:44:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-divisors-of-120-are--1-2-3-4-5-6-8-10-12-15-20-24-30-40-60-120--those-divisible-by-5-are--5-10-15-20-30-40-60-120--summing-them--5--10--15--20--30--40--60--120--290--oxed290question-let--x--and--y--be-positive-real-numbers-representing-the-bandwidths-in-gbps-of-two-parallel-data-streams-find-the-minimum-value-of-67569442026-04-21T17:44:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--a--racx2--y2xy--and--b--racxyx2--y2--then-the-expression-becomes-83354592026-04-21T17:44:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--a--racp--qp---q--and--b--racp---qp--q--note-that--b--rac1a--and-since--p--q--0---a--1--then-38683932026-04-21T17:44:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-of-hours-in-a-week-is-7-times-24--168-the-prime-factorization-of-168-is-23-times-3-times-7-the-odd-divisors-are-the-divisors-of-3-times-7--21-which-are-1-3-7-21-their-sum-is-1--3--7--21--32-19994402026-04-21T18:23:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-word-atmospheric-has-4-vowels-a-o-a-i-and-mitigation-has-3-vowels-i-i-i-the-gcd-of-4-and-3-is-1-since-they-share-no-common-divisors-other-than-1-60836352026-04-21T18:23:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-word-climate-has-7-letters-and-modeling-has-8-letters-the-lcm-of-7-and-8-is-56-the-smallest-three-digit-multiple-of-56-is-112-since-56-times-2--112-17463632026-04-21T18:23:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-time-between-two-consecutive-orbits-is--frac5pi2--hours-since--pi-approx-31416--we-compute-6343822026-04-21T18:39:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-of-birds-follows-a-geometric-sequence--3-6-12-24-dots--with-first-term--a--3--and-common-ratio--r--2--51041552026-04-21T18:39:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given--rt--sqrtt---82--77986362026-04-21T18:39:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-original-total-cost-for-two-sensors-is-88-so-each-sensor-costs-15600782026-04-21T18:39:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-tasked-with-finding-the-number-of-integer-solutions-x-y-to-the-equation--x2---y2--2025--this-can-be-factored-as-80434852026-04-21T18:40:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-factor--1440--79064782026-04-21T18:40:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-an-arithmetic-series-of-odd-numbers-from-1001-to-1099-inclusive-98299092026-04-21T18:40:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-multinomial-probability-problem-let--x--xs-xp-xf--be-the-counts-of-each-outcome-in-12-independent-trials-with-probabilities--ps--04---pp--03--and--pf--03--the-probability-mass-function-of-a-multinomial-distribution-gives-8094382026-04-21T18:44:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-ways-to-choose-4-instruments-from-10-is-72316272026-04-21T18:44:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-multinomial-probability-let--x--xa-xb-xc--be-the-counts-of-each-letter-in-a-random-8-character-word-with-each-character-independently-chosen-from-a-b-c-with-equal-probability--rac13--51857112026-04-21T18:44:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-multinomial-distribution-with-10-independent-trials-and-3-categories-lets-define-the-counts-based-on-the-actual-route-types-51009932026-04-21T18:44:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-since--fx--is-a-polynomial-and-four-values-are-given-and-the-behavior-resembles-a-quadratic-pattern--3-8-15-24--we-first-examine-the-second-differences-48472162026-04-21T19:05:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-functional-equation-is--fxy--fx-y--2fx--2fy--this-is-a-known-variant-of-the-quadratic-functional-equation-24999172026-04-21T19:06:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--y--x2---1--then--x2--y--1--substitute-into--gy--2x4---5x2--1--47134842026-04-21T19:26:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--y--x2--2--then--x2--y---2--express--gy--in-terms-of--y--7126362026-04-21T19:26:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-point-on-the-line--y--frac34x--2--closest-to--0-0--we-minimize-the-square-of-the-distance-function-42431512026-04-21T19:26:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-expand--sin-x--cos-x2--sin2-x--2sin-x-cos-x--cos2-x--1--2sin-x-cos-x--50001492026-04-21T19:27:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-since--mathbfa-mathbfb-mathbfc--are-unit-vectors-we-use-the-identity-20028202026-04-21T19:27:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-three-given-points-be--a--111--77308922026-04-21T19:27:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-that--fx--is-a-cubic-polynomial-such-that-31107702026-04-21T19:34:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-that--dt--is-a-cubic-polynomial-satisfying-15420772026-04-21T19:34:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-arithmetic-mean-we-first-identify-the-positive-integers-less-than-30-that-are-divisible-by-both-3-and-5-these-numbers-are-multiples-of-the-least-common-multiple-of-3-and-5-which-is-15-the-multiples-of-15-less-than-30-are-19886392026-04-21T19:37:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-we-find-all-positive-integers-less-than-or-equal-to-50-that-are-factors-of-60-the-positive-divisors-of-60-are-63806222026-04-21T19:37:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-length-is-calculated-by-adding-the-two-lengths-and-dividing-by-2-13249882026-04-21T19:37:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-smallest-prime-factor-of-1211-we-test-divisibility-by-small-primes-first-1211-is-odd-so-not-divisible-by-2-the-sum-of-the-digits-is--1--2--1--1--5--not-divisible-by-3-it-doesnt-end-in-0-or-5-so-not-divisible-by-5-next-test-divisibility-by-7-6358732026-04-21T19:37:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-a-vector-orthogonal-to-both-compute-their-cross-product-2233922026-04-21T20:19:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-binomial-probability-problem-with--n--7---p--04--and-we-want--px-geq-5--px5--px6--px7--24934832026-04-21T20:19:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-binomial-probability-problem-with--n--6---p--03--and-we-want--px--3--px0--px1--px2--84500882026-04-21T20:19:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-circular-arrangements-of--n--distinguishable-people-the-number-of-distinct-arrangements-is--n-1--since-rotations-are-equivalent-51308852026-04-21T20:19:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-hypergeometric-probability-problem-24510062026-04-21T20:19:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-looking-for-the-smallest-two-digit-positive-integer--l--such-that-69234742026-04-21T20:31:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-arithmetic-mean-sum-the-values-and-divide-by-the-number-of-days-70019202026-04-21T20:36:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-square-root-of-a-square-simplifies-to-the-absolute-value-61947132026-04-21T20:36:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-prime-factorize-both-numbers-84219482026-04-21T20:36:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-simplify-the-equation-using-absolute-value-32188902026-04-21T20:36:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-solve-for--hx2---2--start-with-the-given-function-64599022026-04-21T20:50:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-length-and-width-of-the-rectangle-be--l--and--w--respectively-the-perimeter-is-given-by-69364162026-04-21T20:50:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-d--gcda-b-then-we-can-write-a--d-cdot-m-and-b--d-cdot-n-where-m-and-n-are-coprime-positive-integers-the-total-road-length-is-a--b--dm--n--1000-so-d-must-divide-1000-to-maximize-d-we-minimize-m--n-subject-to-m-and-n-being-coprime-positive-integers-the-smallest-possible-value-of-m--n-is-2-which-occurs-when-m--n--1-and-they-are-coprime-this-gives-d--frac10002--500-since-m--1-and-n--1-are-coprime-this-is-valid-therefore-the-largest-possible-value-of-gcda-b-is-boxed500-90380292026-04-21T20:59:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-find-the-least-common-multiple-lcm-of-18-and-30-first-factor-both-numbers-11847742026-04-21T20:59:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-d--gcda-b-so-a--d-cdot-m-b--d-cdot-n-with-m-and-n-coprime-positive-integers-then-20690682026-04-21T20:59:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-d--gcda-b-so-a--d-cdot-m-b--d-cdot-n-with-m-and-n-coprime-then-76378532026-04-21T20:59:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-sides-be-a-1-a-a1-this-is-a-triangle-only-if-triangle-inequalities-hold-35795992026-04-21T20:59:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute--g3--4i--3--4i--since--gz--returns--z--itself-then--fg3--4i--f3--4i--3--4i2--9--24i--16i2--9--24i---16---7--24i--the-result-is-boxed-7--24i-9917542026-04-21T21:12:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--alpha--sqrt1--i---beta--sqrt1---i--the-conjugate-pairs--alpha--and---alpha---beta--and---beta--must-both-be-roots-for-real-coefficients-but-since-the-polynomial-is-monic-of-degree-2-and-has-only-these-two-specified-roots-we-must-consider-symmetry-instead-compute-the-sum-and-product-note--1--i--1---i--2--and--1--i1---i--1--1--2--let--z2----alpha--beta-z--alphabeta--but-observing-that--alphabeta--sqrt1i1-i--sqrt2--also--alpha2--beta2--2--and--alpha2beta2--2--let--s--alpha--beta--then--s2--alpha2--beta2--2alphabeta--2--2sqrt2--but-to-find-a-real-polynomial-consider-that--alpha--sqrt1i--and--sqrt1i--sqrtsqrt2-eipi8--214-cos-fracpi8--isin-fracpi8--however-instead-of-direct-polar-form-consider-squaring-the-sum-alternatively-note-that--alpha--and--beta--are-conjugate-like-in-structure-but-realize--sqrt1i--and--sqrt1-i--are-not-conjugates-but-if-we-form-a-polynomial-with-both-and-require-real-coefficients-then-the-minimal-monic-polynomial-must-have-roots--sqrt1i--sqrt1i-sqrt1-i--sqrt1-i--unless-paired-but-the-problem-says-roots-at-these-two-so-assume--alpha--sqrt1i---beta--sqrt1-i--and-for-real-coefficients-must-include---alpha--beta--but-that-gives-four-roots-therefore-likely-the-polynomial-has-roots--sqrt1i--and--sqrt1-i--and-since-coefficients-are-real-it-must-be-invariant-under-conjugation-but--overlinesqrt1i--sqrt1---i--beta--so-if--alpha--sqrt1i--then--overlinealpha--beta--thus-the-roots-are--alpha--and--overlinealpha--so-the-monic-quadratic-is--z---alphaz---overlinealpha--z2---2operatornamerealpha-z--alpha2--now--alpha2--1i--so--alpha2--alpha2--1i--sqrt2--also--2operatornamerealpha--alpha--overlinealpha--but--alpha--overlinealpha2--alpha2--2alpha2--overlinealpha2--wait-better--operatornamerealpha--frac-alpha--overlinealpha-2--from--alpha2--1i--take-real-part--operatornamerealpha2--operatornamere1i--1--alpha2-cos2theta---operatornameimalpha2--sin2theta--1--so--cos2theta--1sqrt2---sin2theta--1sqrt2--so--2theta--pi4---theta--pi8--then--operatornamerealpha--alpha-costheta--sqrt2-cospi8--but--cospi8--sqrt2--sqrt2--2--so--operatornamerealpha--sqrt2-cdot-frac-sqrt2--sqrt2-2--frac-sqrt2-sqrt2--sqrt2-2--this-is-messy-instead-use-identity--alpha2--1i--so--alpha4--1i2--2i--but-for-the-polynomial--z---alphaz---beta--z2---alpha--betaz--alphabeta--note--alphabeta--sqrt1i1-i--sqrt2--now--alpha--beta2--alpha2--beta2--2alphabeta--1i--1-i--2sqrt2--2--2sqrt2--so--alpha--beta--sqrt2--2sqrt2--but-this-is-not-helpful-note--alpha--and--beta--satisfy-a-polynomial-whose-coefficients-are-symmetric-but-recall-the-minimal-monic-polynomial-with-real-coefficients-having--sqrt1i--as-a-root-must-also-have---sqrt1i--unless-we-accept-complex-coefficients-but-we-want-real-so-likely-the-intended-polynomial-is-formed-by-squaring-suppose--z--sqrt1i--then--z2---1--i--so--z2---12---1--so--z4---2z2--1---1-rightarrow-z4---2z2--2--0--but-this-has-roots--pmsqrt1i-pmsqrt1-i--check-if--z2--1i---z4---2z2--2--1i2---21i--2--12i-1--2--2i--2--0--2i---2i--0--0-wait--1i2--1--2i--1--2i--then--2i---21i--2--2i--2--2i--2--0--yes-so--z4---2z2--2--0--has-roots--pmsqrt1i-pmsqrt1-i--but-the-problem-wants-a-quadratic-however-if-we-take--z--sqrt1i--and---sqrt1-i--no-but-notice-the-root--sqrt1i--and-its-negative-is-also-a-root-if-polynomial-is-even-but--f-z--fz--only-if-symmetric-but--fz--z2---1---i--has--sqrt1i--but-not-symmetric-the-minimal-real-coefficient-polynomial-with--sqrt1i--as-root-is-degree-4-but-the-problem-likely-intends-the-monic-quadratic-formed-by--sqrt1i--and-its-conjugate--sqrt1-i--even-though-it-doesnt-have-real-coefficients-unless-paired-but--sqrt1-i--is-not---overlinesqrt1i--let--alpha--sqrt1i---overlinealpha--sqrt1-i--since--overlinesqrt1i--sqrt1-overlinei--sqrt1-i--yes-complex-conjugation-commutes-with-square-root-only-if-domain-is-fixed-but--overlinesqrtz--sqrtoverlinez--for--overlinez--in-domain-of-definition-assuming--sqrt1i--is-taken-with-positive-real-part-then--overlinesqrt1i--sqrt1-i--so-the-conjugate-is--sqrt1-i--overlinealpha--so-for-a-polynomial-with-real-coefficients-if--alpha--is-a-root-so-is--overlinealpha--so-the-roots-are--sqrt1i--and--sqrt1-i--overlinesqrt1i--therefore-the-monic-quadratic-is--z---sqrt1iz---overlinesqrt1i--z2---2operatornameresqrt1i-z--sqrt1i2--now--sqrt1i2--alpha2--1i--sqrt2--no--alpha2--alpha2--1i--sqrt2--no--alpha2---alpha2-1--no--alpha2--alpha2--and--alpha2--1i--so--alpha2--1i--sqrt12--12--sqrt2--yes-and--operatornamerealpha--frac-alpha--overlinealpha-2--from--alpha2--1i--take-modulus--alpha4--1i2--2--so--alpha22--2--thus--alpha4--2--so--alpha2--sqrt2--since-magnitude-positive-so--operatornamerealpha--frac-alpha--overlinealpha-2--but--alpha--overlinealpha2--alpha2--2alpha2--overlinealpha2--no--overlinealpha2--overlinealpha2--overline1i--1-i--so--alpha--overlinealpha2--alpha2--2alphaoverlinealpha--overlinealpha2--1i--1-i--2alpha2--2--2sqrt2--therefore--alpha--overlinealpha--sqrt2--2sqrt2--so-the-quadratic-is--z2---sqrt2--2sqrt2--z--sqrt2--but-this-is-not-nice-wait--theres-a-better-way-note-that--sqrt1i--fracsqrt221i12--but-perhaps-the-intended-answer-is-to-use-the-identity-the-polynomial-whose-roots-are--sqrt1pm-i--is--z4---2z2--2--0--but-we-want-quadratic-but-the-only-monic-quadratic-with-real-coefficients-having--sqrt1i--as-a-root-must-also-have---sqrt1i---overlinesqrt1i----overlinesqrt1i--and-if-its-degree-4-but-the-problem-asks-for-quadratic-unless--sqrt1i--is-such-that-its-minimal-polynomial-is-quadratic-but-its-not-as--mathbbqsqrt1imathbbq--4--but-perhaps-in-the-context-they-want--z---sqrt1iz---sqrt1-i--but-again-not-real-after-reconsideration-the-intended-solution-likely-assumes-that-the-conjugate-is-included-and-the-polynomial-is--z2---2cospi8sqrt2-z--sqrt2--but-thats-not-nice-alternatively-recognize-that--1i--sqrt2-eipi4--so--sqrt1i-51427962026-02-24T13:08:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-set--f2--g2--compute--f2--4---6--m---2--m--and--g2--4---6--5m---2--5m--equate---2--m---2--5m--simplify--0--4m--so--m--boxed0--28997302026-02-24T13:44:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-identity--a3--b3--a--b3---3aba--b--first-find--ab---a--b2--a2--2ab--b2-rightarrow-49--35--2ab-rightarrow-ab--7--then--a3--b3--343---3-cdot-7-cdot-7--343---147--boxed196--36767682026-04-21T21:35:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve-the-system-from-the-second-equation--x--y--2--substitute-into-the-first--2y--2--y--10-rightarrow-2y--4--y--10-rightarrow-3y--6-rightarrow-y--2--then--x--2--2--4--the-intersection-is--boxed4-2--22733942026-04-21T21:35:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-square-both-sides--x2--2--frac1x2--25-rightarrow-x2--frac1x2--23--multiply-by-2--2x2--frac2x2--46--final-answer--boxed46-question-in-a-sustainable-desert-agriculture-project-the-combined-water-efficiency-of-two-systems-is-modeled-by--a--b--10--and--a2--b2--58--determine-the-total-water-efficiency-cubed--a3--b3--66212832026-04-21T21:35:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-given--fracab--3--let--a--3b--substituting-into--a--b--16---3b--b--16--so--b--4--and--a--12--now--a2---b2--factors-as--a---ba--b--12---416--8-times-16--128--therefore-the-value-is--boxed128--15016892026-04-21T21:35:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-asked-to-count-how-many-of-the-first-100-positive-integers-satisfy-the-congruence-35058522026-04-21T22:08:15+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-find-the-least-common-multiple-of-5-and-7-92186642026-04-21T22:08:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x--be-the-two-digit-number-87203292026-04-21T22:08:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-100th-signal-occurs-at-minute-85129022026-04-21T22:08:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-divisibility-rule-for-9-the-remainder-when-a-number-is-divided-by-9-is-the-same-as-the-remainder-when-the-sum-of-its-digits-is-divided-by-9-89901472026-04-21T22:08:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-euclidean-algorithm-375702026-04-21T22:08:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-mathbfm--beginpmatrix-a--b--c--d-endpmatrix-multiply-mathbfm-with-beginpmatrix-1--2--3--4-endpmatrix-to-get-11622742026-04-21T22:42:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-pythagorean-theorem-since-overrightarrowoa-and-overrightarrowob-are-perpendicular-compute-overrightarrowoc2--2overrightarrowoa2--overrightarrowob2--425--144--100--144--244-thus-overrightarrowoc--sqrt244--2sqrt61-final-answer-boxed2sqrt61-94132792026-04-21T22:43:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-set-up-the-system-51229482026-04-21T22:43:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-us-define-a-function--fu--such-that-3791932026-04-21T22:54:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-that--gx--is-a-cubic-polynomial-99244622026-04-21T23:22:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-functional-equation-is-32442972026-04-21T23:22:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-since--px--is-a-cubic-polynomial-when-dividing-by--x4---1--the-remainder--rx--must-be-a-polynomial-of-degree-less-than-4-however-because--x4---1--has-roots--pm-1-pm-i--and--px--is-real-the-remainder-can-be-expressed-as-any-cubic-or-lower-but-here-we-are-to-find-the-actual-remainder-which-is-unique-and-of-degree---4--but-since--px--is-already-degree-3-and-division-by-degree-4-polynomial-yields-a-unique-remainder-of-degree---4--and-since--px--agrees-with-any-degree-3-polynomial-at-these-4-points-we-conclude-51926962026-04-21T23:22:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-apply-the-cauchy-schwarz-inequality-in-the-engel-form-also-known-as-titus-lemma-97382012026-04-21T23:22:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-the-value-of--k--substitute--x--3--into-the-function-and-set--f3--0--54476302026-04-22T00:16:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-population-stabilizes-at--t---6--this-means--p-6--0--substituting-into-the-equation-94885322026-04-22T00:16:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-rationalize-the-denominator-of--frac3x--4sqrtx---2--multiply-numerator-and-denominator-by-the-conjugate--sqrtx--2--41779402026-04-22T00:16:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-expression--x2--5x--6--is-a-quadratic-trinomial-we-look-for-two-numbers-that-multiply-to-6-and-add-to-5-these-numbers-are-2-and-3-44023842026-04-22T00:16:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-a-point--3-11--lies-on-the-line--y--2x--b--substitute--x--3--and--y--11--18680132026-04-22T00:16:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-vertex-of-a-parabola--hx--x2---4x--c--occurs-at--x--frac--421--2--which-matches-the-given-condition-now-substitute--x--2--and-set--h2--3--78849802026-04-22T00:16:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--fx--ax3--bx2--cx--d--we-use-the-given-values-to-form-a-system-of-equations-23573962026-04-22T00:16:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-that--a2--b2--1--and--ab--frac12--we-are-to-find--a4--b4--41999262026-04-22T00:41:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-minimize-83173202026-04-22T00:41:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--sqrtu--t--so--u--t2--substituting-into-the-equation-26982162026-04-22T00:41:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-spherical-coordinates-phi-is-the-polar-angle-measured-from-the-positive-z-axis-when-phi-is-constant-the-set-of-points-forms-a-cone-with-its-apex-at-the-origin-opening-symmetrically-around-the-z-axis-for-phi--fracpi4-the-cone-makes-a-45circ-angle-with-the-z-axis-thus-the-shape-is-a-cone-25502402026-04-22T00:42:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-theta-be-the-angle-between-any-pair-of-vectors-since-mathbfu-cdot-mathbfv--costheta--frac12-theta--60circ-the-magnitude-squared-is-mathbfu--mathbfv--mathbfw2--3--2mathbfu-cdot-mathbfv--mathbfu-cdot-mathbfw--mathbfv-cdot-mathbfw--3--23-cdot-frac12--3--3--6-thus-the-maximum-value-is-sqrt6-96007152026-04-22T00:42:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-orthogonality-their-dot-product-must-be-zero-1-cdot-x--x-cdot-3--2-cdot--1--0-simplifying-x--3x---2--0-rightarrow-4x--2-rightarrow-x--frac12-69483352026-04-22T00:42:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-angle-aob--90circ-overrightarrowoa-cdot-overrightarrowob--0-compute-overrightarrowoc2--4overrightarrowoa2--overrightarrowob2---4overrightarrowoa-cdot-overrightarrowob--44--9---0--25-thus-overrightarrowoc--5-75547192026-04-22T00:42:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-t--sin-x--csc-x-note-tan-x--cot-x--fracsin-xcos-x--fraccos-xsin-x--frac1sin-x-cos-x-let-y--sin-x-cos-x--frac12sin-2x-so-tan-x--cot-x--frac2sin-2x-the-expression-becomes-leftfrac2sin-2xright2--sin-x--csc-x2-let-z--sin-x--csc-x-geq-2-by-am-gm-substitute-u--sin-2x-then-minimize-frac4u2--leftfrac1sin-x--sin-xright2-after-calculus-or-substitution-the-minimum-occurs-at-x--fracpi4-yielding-1--12--sqrt2--frac1sqrt22--4--leftfrac3sqrt2right2--4--frac92--frac172-however-correct-simplification-shows-the-minimum-is-9-when-x--fracpi4-48823882026-04-22T00:42:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-integer-be--n--we-have--n-equiv-0-pmod16--and--n-equiv-1-pmod5--let--n--16k--substituting-into-the-second-congruence-32955452026-04-22T00:42:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-divisors-of--12--are--1-2-3-4-6-12--compute-the-sum-of-their-squares-71415142026-04-22T00:42:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-product-of--k--consecutive-integers-is-divisible-by--k--for-five-consecutive-integers-this-is--5--120--for-example--1-cdot-2-cdot-3-cdot-4-cdot-5--120--and--2-cdot-3-cdot-4-cdot-5-cdot-6--3420012026-04-22T00:42:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-surface-area-of-a-sphere-is--4pi-r2--for-the-pollen-grain-this-is--4pi-r2--the-surface-area-of-a-hemisphere-includes-the-curved-surface--2pi-3r2--and-the-flat-circular-base--pi-3r2--so-total-surface-area-is--2pi-9r2--pi-9r2--18pi-r2--9pi-r2--27pi-r2--the-ratio-is-2568402026-04-22T00:45:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circumscribed-circle-using-the-pythagorean-theorem-19975002026-04-22T00:45:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-cone-is--frac13pi-r2-h--and-the-volume-of-a-sphere-is--frac43pi-r3--setting-them-equal-78006522026-04-22T00:45:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-circle-has-radius--5--micrometers-so-its-area-is-62853612026-04-22T00:45:46+00:00daily0.8https://pg.groshi247.com/highlight/solution-volume-of-the-sphere-59542802026-04-22T00:45:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio-of-inscribed-to-plain-tablets-is-53-so-the-total-number-of-parts-is-5--3--8-since-there-are-48-tablets-each-part-represents--frac488--6--tablets-thus-the-number-of-inscribed-tablets-is-5-times-6--30-we-are-told-that-inscribed-tablets-must-be-displayed-in-groups-of-7-so-we-seek-the-greatest-multiple-of-7-that-is-less-than-or-equal-to-30-the-multiples-of-7-below-30-are-7-14-21-28-the-greatest-is-28-therefore-the-largest-number-of-inscribed-tablets-that-can-be-grouped-in-sevens-is-28-60984312026-04-22T00:45:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio-of-emissions-is-479-let-the-common-multiplier-be-x-then-the-second-borough-emitted-7x--147-metric-tons-solving-for-x-11122122026-04-22T00:45:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-area-of-the-hall-is-12-times-12--144-square-meters-each-rectangle-has-area-at-least-18-so-the-maximum-number-of-rectangles-is--leftlfloor-frac14418-rightrfloor--8--but-we-want-the-minimum-number-so-we-seek-the-largest-rectangle-area-that-divides-144-and-is--geq-18--and-check-if-tiling-with-such-rectangles-covers-the-square-11294292026-04-22T00:45:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-total-be-1-add-the-fractions-of-coniferous-and-deciduous-23593422026-04-22T00:46:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-of-five-values-is--frac58--so-the-total-rainfall-is-93943612026-04-22T00:46:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-sum--sumk16-k2--12--22--32--42--52--62--1--4--9--16--25--36--91--next-compute-the-product--prodj13-2j--1--2cdot1--12cdot2--12cdot3--1--3-cdot-5-cdot-7--105--now-multiply-the-results--91-times-105--9105--therefore-the-value-is--boxed9105--79185532026-04-22T00:46:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-slope--m--of-the-line-is--frac19---79---3--frac126--2--using-point-slope-form-with-point--3-7---y---7--2x---3--simplify--y--2x---6--7--2x--1--the--y--intercept-occurs-when--x--0--so--y--20--1--1--thus-the--y--intercept-is--boxed1--18024792026-04-22T00:46:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--u--x2--so-the-expression-becomes--u2---13u--36--factor--u---9u---4--substitute-back--x2---9x2---4--each-is-a-difference-of-squares--x---3x--3x---2-3193372026-04-22T00:46:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-dot-product-of-two-unit-vectors-is-mathbfu-cdot-mathbfv--costheta-given-costheta--fracsqrt32-the-angle-theta-satisfies-theta--arccosleftfracsqrt32right-this-corresponds-to-theta--30circ-or-fracpi6-radians-however-since-cosine-is-positive-in-both-the-first-and-fourth-quadrants-but-angles-between-vectors-are-typically-taken-in-0-pi-the-solution-is-boxeddfracpi6-25854012026-04-22T00:59:15+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-cosine-of-the-angle-between-vectors-mathbfa-and-mathbfb-is-given-by-costheta--fracmathbfa-cdot-mathbfbmathbfa-mathbfb-compute-the-dot-product-mathbfa-cdot-mathbfb--10--01--11--1-the-magnitudes-are-mathbfa--sqrt12--02--12--sqrt2-and-mathbfb--sqrt02--12--12--sqrt2-thus-costheta--frac1sqrt2-cdot-sqrt2--frac12-the-final-answer-is-boxeddfrac12-16025822026-04-22T00:59:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-magnitude-of-the-cross-product-is-mathbfp-times-mathbfq--mathbfp-mathbfq-sintheta-since-mathbfp-and-mathbfq-are-unit-vectors-mathbfp-times-mathbfq--sintheta--frac12-thus-theta--arcsinleftfrac12right-which-gives-theta--fracpi6-or-frac5pi6-however-the-angle-between-vectors-is-taken-as-the-acute-angle-so-theta--boxeddfracpi6-56629212026-04-22T00:59:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-vectors-to-be-orthogonal-their-dot-product-must-be-zero-mathbfm-cdot-mathbfn--x1--2-1--x---2--0-solving-gives-x--2-the-final-answer-is-boxed2-38451172026-02-24T18:42:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-w--z2-so-w2--w--1--0-solving-gives-w--frac-1-pm-sqrt-32--epm-2pi-i3-thus-z2--epm-2pi-i3-so-z--epm-pi-i3--pi-i-k2-for-k--0-1-the-roots-have-angles-pi3-5pi3-7pi3-11pi3-but-77292342026-04-22T00:59:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-hemisphere-is--frac23pi-3x3--frac23pi-27x3--18pi-x3--the-cylinders-volume-is--pi-x2-cdot-3x--3pi-x3--the-ratio-is--frac18pi-x33pi-x3--boxed6--95249362026-04-22T01:02:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-original-radius--r--sqrtfrac100pipi--10--textkm--new-radius--r--8--textkm--original-area--100pi--new-area--pi-82--64pi--the-decrease-is--100pi---64pi--boxed36pi--textkm2--86646042026-04-22T01:02:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-a-right-triangle-the-inradius--r--fraca--b---c2--where--c--is-the-hypotenuse-let--a--and--b--be-the-legs-then--r--fraca--b---252--6-rightarrow-a--b--37--the-area-of-the-triangle-is--fracab2--and-the-inradius-formula--r--fraca--b---c2--gives-consistency-using--a2--b2--625--and--a--b--37--solve-for--ab---a--b2--1369--a2--b2--2ab--625--2ab-rightarrow-ab--372--area---186--textm2--circle-area--pi-r2--36pi--ratio--frac36pi186--boxedfrac6pi31--38242462026-04-22T01:02:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-slope-forms-a-right-triangle-with-opposite-side-1-and-adjacent-side--sqrt3--thus--tan-theta--frac1sqrt3--boxedfracsqrt33-question-what-is-the-remainder-when-the-sum--13--23--33--dots--103--is-divided-by-9-88030142026-04-22T01:02:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-formula-for-the-sum-of-the-first--n--cubes-is--left-fracnn12-right2--for--n--10--we-compute-25552432026-04-22T01:02:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-convert--3217--to-base-ten-expand-using-powers-of-7-14899922026-04-22T01:02:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-gcd-take-the-minimum-exponent-for-each-common-prime-factor-63433112026-04-22T01:02:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-lcm-take-the-maximum-exponent-for-each-prime-factor-95831712026-04-22T01:02:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sum-of-the-divisors-of--23-cdot-32--is-given-by-7881872026-04-22T01:02:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-note-that--7-equiv-1-mod-6--so--7k-equiv-1k--1-mod-6--for-any-positive-integer--k--86288062026-04-22T01:02:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-prime-factorization-is-already-given--24-cdot-33-cdot-52--the-largest-prime-factor-is-clearly--boxed5--71325772026-04-22T01:02:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-least-common-multiple-of-12-and-18-factor-both-80142262026-04-22T01:02:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-greatest-common-factor-gcf-of-84-and-126-factor-both-numbers-73140182026-04-22T01:06:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-arithmetic-mean-is-the-sum-divided-by-3-77097102026-04-22T01:06:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-microscopes-be-x-the-ratio-54--15x-implies-74834182026-04-22T01:06:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-sum-the-expressions-76357482026-04-22T01:06:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-arithmetic-with-first-term-a1--2-and-common-difference-d--3-the-n-th-term-is-3332762026-04-22T01:06:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-original-side-length-of-the-square-is--frac404--10--textcm--the-original-area-is--102--100--textcm2--after-increasing-each-side-by--2--textcm--the-new-side-length-is--12--textcm--and-the-new-area-is--122--144--textcm2--the-increase-in-area-is--144---100--44--textcm2--70563302026-04-22T01:06:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-cosine-of--360circ--corresponds-to-the-x-coordinate-of-the-point-on-the-unit-circle-at-a-full-rotation-from-the-positive-x-axis-this-point-is--1-0--so--cos-360circ--1--24471172026-04-22T01:06:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-the-cylinder-is--vtextcyl--pi-r2-6r--6pi-r3--the-volume-of-the-cone-is--vtextcone--frac13-pi-2r2-3r--frac13-pi-4r23r--4pi-r3--the-ratio-is-84490342026-04-22T01:06:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-herons-formula-the-semi-perimeter-is-53215952026-04-22T01:06:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-surface-area-of-a-cone-is-the-sum-of-the-base-area-and-the-lateral-surface-area-the-base-area-is-80344512026-04-22T01:06:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-tangent-of--45circ--is-defined-as--fracsin-45circcos-45circ--fracfracsqrt22fracsqrt22--1--21275452026-04-22T01:06:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-diameter-of-the-inscribed-circle-is-equal-to-the-side-length-of-the-square-which-is--8--textcm--therefore-the-radius-is--frac82--4--textcm--the-area-of-the-circle-is-81831632026-04-22T01:06:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-side-length-of-the-hexagon-is--frac366--6--text-5268292026-04-22T01:06:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-sphere-is-given-by--v--frac43-pi-r3--differentiating-with-respect-to-time--t--we-get-33207292026-04-22T01:12:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-triangle-has-side-lengths--a--13---b--14--and--c--15--first-compute-the-semi-perimeter--s--85382782026-04-22T01:12:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-original-side-length-of-the-equilateral-triangle-is--frac363--12--cm-the-area-of-an-equilateral-triangle-is-given-by-28007872026-04-22T01:12:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-hypotenuse-be--z--25--m-and-the-inradius-be--c--5--m-for-a-right-triangle-the-area--a--is-given-by-58665572026-04-22T01:12:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--3--into--fx--57557332026-04-22T01:32:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--5--into--hx--3463872026-04-22T01:32:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--y--x2---3x--then-the-product-becomes--y--2my--5m--y2--7my--10m2--substitute-back-32639812026-04-22T01:32:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--m---3--54772962026-04-22T01:32:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x--be-the-liters-of-15-fertilizer-and--y--be-the-liters-of-40-fertilizer-the-system-is-52892722026-04-22T01:32:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-width-be--5k--and-height--3k--by-the-pythagorean-theorem-68096022026-04-22T01:32:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-probability-of-neither-test-being-positive-is--left1---rac23-71204492026-04-22T01:32:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-with-a1--1-compute-a2--ma1--1---rac122--rac12-next-compute-a3--ma2--rac12---racrac1222--rac12---rac18--rac38-thus-the-value-of-a3-is-oxeddfrac38-42760762026-04-22T01:32:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-cauchy-schwarz-inequality-22--32--42x2--y2--z2-geq-2x--3y--4z2-this-gives-29x2--y2--z2-geq-144-so-x2--y2--z2-geq-rac14429-equality-holds-when-racx2--racy3--racz4--k-leading-to-x--2k-y--3k-z--4k-substituting-into-2x--3y--4z--12-gives-4k--9k--16k--29k--12-so-k--rac1229-thus-the-minimum-value-is-oxeddfrac14429-14005342026-04-22T01:32:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-a-recursive-sequence-defined-by--bn1--gbn--where--gt--t---ract24--and--b1--1--we-are-to-find--limn-o-infty-bn--assuming-the-limit-exists-80262482026-04-22T01:39:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-a-cubic-polynomial--hx--x3--px--q--and-values-77708682026-04-22T01:39:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-equation-80916052026-04-22T01:39:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--a--2x---b--y--so-the-equation-becomes-58476152026-04-22T01:39:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-angles-are-30circ-60circ-and-90circ-using-the-law-of-sines-frac5sin-30circ--fraccsin-90circ-since-sin-30circ--05-and-sin-90circ--1-c--frac505--10-km-44567162026-04-22T01:44:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-circumference-c--2pi-r-so-r--fracc2pi--frac12pi2pi--6-nm-91766482026-04-22T01:44:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-longest-chord-in-a-circle-is-the-diameter-however-the-question-specifies-the-longest-chord-that-is-not-a-diameter-which-would-approach-the-diameter-for-a-circle-the-maximum-chord-length-is-the-diameter-so-the-answer-is-14-cm-51263162026-04-22T01:44:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-hypotenuse-is-sqrt92--122--15-cm-sin-theta--fractextoppositetexthypotenuse--frac915--frac35-58806972026-04-22T01:44:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-the-largest-number-of-identical-grants-compute-gcd2025-1515-43582652026-04-22T01:44:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-smallest-number-of-distinct-sequences-is-the-least-common-multiple-lcm-of-13-and-17-78851572026-04-22T01:44:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-alignment-occurs-at-the-least-common-multiple-of-2025-and-1515-14737482026-04-22T01:44:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-x2-equiv-25-pmod100-98714922026-04-22T01:44:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-problem-requires-calculating-the-number-of-distinct-permutations-of-10-items-where-there-are-5-identical-bees-3-identical-wasps-and-2-identical-beetles-the-formula-for-multinomial-coefficients-applies-90680902026-04-22T01:44:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-multinomial-permutation-problem-the-total-sequences-are-given-by-15565312026-04-22T01:44:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-total-number-of-ways-to-choose-3-species-binom203--1140-the-number-of-favorable-outcomes-is-binom102-cdot-binom61--45-cdot-6--270-the-probability-is-43321992026-04-22T01:44:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-treat-each-discipline-as-a-block-there-are-3-ways-to-arrange-the-blocks-around-the-table-within-the-biology-block-3-physics-2-and-chemistry-2-for-circular-permutations-fix-one-block-to-eliminate-rotational-symmetry-34264722026-04-22T01:44:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-ways-to-choose-4-species-binom134--715-favorable-outcomes-binom82-cdot-binom52--28-cdot-10--280-the-probability-is-26409972026-04-22T01:44:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-treat-the-two-consecutive-sessions-as-a-single-entity-there-are-4-ways-to-arrange-the-4-entities-the-pair-and-the-other-3-sessions-the-pair-can-be-ordered-in-2-ways-thus-total-arrangements-are-4-times-2--24-times-2--48-boxed48-34630052026-04-22T01:54:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-combinations-without-restrictions-dbinom94--126-subtract-combinations-where-both-problematic-startups-are-included-dbinom72--21-choosing-2-more-from-the-remaining-7-valid-combinations-126---21--105-boxed105-17314372026-04-22T01:54:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-inclusion-exclusion-total-assignments-36--729-subtract-assignments-missing-at-least-one-category-3-times-26--192-add-back-assignments-missing-two-categories-3-times-16--3-valid-assignments-729---192--3--540-boxed540question-an-electrical-engineer-tests-a-new-circuit-with-5-identical-resistors-each-with-a-10-probability-of-failing-independently-what-is-the-probability-that-exactly-2-resistors-fail-12308302026-04-22T01:54:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-probability-is-calculated-using-the-binomial-distribution-formula-93630912026-04-22T01:54:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-complementary-probability-of-no-scrolls-containing-the-symbol-is-0958-thus-13840292026-04-22T01:54:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-the-binomial-formula-15676022026-04-22T01:54:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-first-5-terms-are--a1--5---a2--8---a3--11---a4--14--and--a5--17--their-sum-is--5--8--11--14--17--55--dividing-55-by-7-gives-a-quotient-of-7-and-a-remainder-of--55---7-times-7--55---49--6--67810372026-04-22T01:54:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x-equiv-1-pmod9--and--x-equiv-1-pmod11--this-implies--x-equiv-1-pmod99--since-9-and-11-are-coprime-the-two-digit-numbers-satisfying-this-are--99--1--100--not-two-digit-and--1--however-the-next-smaller-solution-is--1--which-is-not-two-digit-thus-no-two-digit-number-satisfies-both-conditions-55818242026-04-22T01:54:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-first-4-primes-after-20-are-23-29-31-and-37-their-sum-is--23--29--31--37--120--dividing-120-by-5-gives-a-remainder-of--120---5-times-24--0--51409702026-04-22T01:54:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-area-of-a-triangle-with-known-side-lengths--a--7---b--10--and--c--13--we-use-herons-formula-4237652026-04-22T01:54:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given--a--b--c--1--with--a-b-c--0--so--1---a--b--c---1---b--a--c--and--1---c--a--b--the-expression-becomes-58064962026-04-22T02:36:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--fx--ax2--bx--c--use-the-given-conditions-43672472026-04-22T02:37:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--u--sqrtv--so--v--u2--and--u-geq-0--substitute-54252822026-04-22T02:37:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-slope-of-the-line-from-2-3-to-5-11-is-frac11---35---2--frac83-since-tan-theta-equals-the-slope-tan-theta--frac83-boxeddfrac83-99625962026-04-22T02:37:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve-2z--60circ-120circ-420circ-480circ-within-0circ-720circ-yielding--91745012026-04-22T02:37:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-smallest-positive-integer-divisible-by-both-7-and-5-we-compute-the-least-common-multiple-lcm-of-7-and-5-since-7-and-5-are-coprime-their-lcm-is-simply-their-product-23476072026-04-22T02:43:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-that-the-total-number-of-regions-is-12-and-the-number-of-clusters-must-be-an-integer-between-2-and-5-inclusive-we-want-to-maximize-the-number-of-regions-per-cluster-which-means-minimizing-the-number-of-clusters-while-satisfying-the-constraints-35242942026-04-22T02:43:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-180-dots-and-must-arrange-them-into-rows-with-the-same-number-of-dots-where-the-number-of-rows-is-a-prime-number-greater-than-3-to-maximize-the-number-of-dots-per-row-we-must-minimize-the-number-of-rows-among-valid-prime-numbers-2306222026-04-22T02:43:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-divide-both-48-and-72-regions-into-equal-sized-groups-with-no-regions-left-out-we-must-find-the-greatest-common-divisor-gcd-of-48-and-72-66791912026-04-22T02:43:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-smallest-number-of-samples-per-column-such-that-the-number-of-columns-is-a-multiple-of-both-6-and-10-we-compute-the-least-common-multiple-lcm-of-6-and-10-77597152026-04-22T02:43:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sine-of--theta--equals--fracsqrt32--at--60circ--and--120circ--in-the-interval--0circ-360circ--these-angles-correspond-to-the-reference-angle--60circ--in-the-first-and-second-quadrants-thus-the-solutions-are--boxed60circ--and--boxed120circ--18198912026-04-22T03:02:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-vectors-to-be-orthogonal-their-dot-product-must-be-zero--2x---13--0--solving--2x---3--0--gives--x--frac32--the-ordered-pair-is--boxedleft-frac32-3-right--24954982026-04-22T03:02:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-cosine-function-oscillates-between---1--and--1--so--5cosleftfracpi6xright--ranges-from---5--to--5--adding-10-shifts-the-range-to--5-15--the-range-is--boxed5-15--54786772026-02-24T20:42:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--theta--arccosleftfrac1sqrt2right--so--cos-theta--frac1sqrt2--then--theta--45circ--and--tan-theta--1--the-answer-is--boxed1--43668122026-04-22T03:02:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-expression-under-the-square-root-must-satisfy--4---x2-geq-0--so--x-in--2-2--the-maximum-value-of--sqrt4---x2--is--2--at--x--0--and-the-minimum-is--0--at--x--pm-2--the-range-is--boxed0-2-think-30643762026-02-24T20:42:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-solve-for-mathbfv-we-use-the-cross-product-equation-mathbfv-times-mathbfw--mathbfp-let-mathbfv--beginpmatrix-v1--v2--v3-endpmatrix-the-cross-product-mathbfv-times-mathbfw-is-calculated-as-beginpmatrix-v2-cdot-3---v3-cdot--1--v3-cdot-2---v1-cdot-3--v1-cdot--1---v2-cdot-2-endpmatrix--beginpmatrix-3v2--v3--2v3---3v1---v1---2v2-endpmatrix-setting-this-equal-to-mathbfp--beginpmatrix-5--0---2-endpmatrix-we-get-the-system-73767532026-02-24T20:42:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-mathbfv--beginpmatrix-v1--v2--v3-endpmatrix-the-cross-product-mathbfv-times-mathbfw-is-beginpmatrix-3v2--v3--2v3---3v1---v1---2v2-endpmatrix--beginpmatrix-3--0---2-endpmatrix-this-gives-the-system-14980942026-04-22T03:02:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-least-common-multiple-lcm-of-12-and-18-factorize-both-12--22-cdot-3-and-18--2-cdot-32-the-lcm-takes-the-highest-powers-of-all-primes-22-cdot-32--4-cdot-9--36-thus-the-lcm-is-boxed36-54025492026-04-22T03:08:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-minimum-depth-where-both-measurements-coincide-is-the-lcm-of-5-and-7-since-5-and-7-are-prime-lcm5-7--5-cdot-7--35-the-answer-is-boxed35-5458642026-04-22T03:08:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-arithmetic-mean-is-calculated-by-summing-the-values-and-dividing-by-the-count-frac10--14--163--frac403-approx-13overline3-the-exact-average-is-boxeddfrac403-55344272026-04-22T03:08:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-sum-the-expressions-4y---3--2y--5--y--7--7y--9-divide-by-3-and-set-equal-to-10-frac7y--93--10-multiply-both-sides-by-3-7y--9--30-subtract-9-7y--21-divide-by-7-y--boxed3-10281252026-04-22T03:08:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-cosine-of-45circ-is-given-by-cos-45circ--fracsqrt22-for-a-square-with-side-length-s--7-units-the-length-of-the-diagonal-d-can-be-calculated-using-the-pythagorean-theorem-d--ssqrt2-substituting-the-value-of-s-we-get-d--7sqrt2-therefore-the-length-of-the-diagonal-is-boxed7sqrt2-32091382026-04-22T03:10:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-a-of-a-regular-hexagon-with-side-length-s-is-given-by-the-formula-a--frac3sqrt32s2-setting-this-equal-to-54sqrt3-we-have-90487962026-04-22T03:10:46+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-a-of-the-triangle-is-given-by-83064652026-04-22T03:10:46+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-v-of-a-cone-is-given-by-64093512026-04-22T03:10:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-an-equilateral-triangle-with-side-length-s-the-radius-r-of-the-circumscribed-circle-is-given-by-8856532026-04-22T03:10:46+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--sqrtx--y--so--x--y2--substituting-into-the-equation-gives-79252032026-04-22T03:10:46+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-analyze-the-inequality--x--y--x---y-leq-8--this-expression-is-symmetric-and-represents-a-diamond-rhombus-centered-at-the-origin-54371912026-04-22T03:10:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--fx--ax3--bx2--cx--d--use-the-given-values-to-form-a-system-of-equations-55293412026-04-22T03:10:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--sqrtx--y--so--x--y2--substituting-gives-84187022026-04-22T03:11:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-one-marker-is-mandatory-choose-1-additional-marker-from-the-remaining-3-dbinom31-then-select-2-sequences-from-6-dbinom62-multiply-these-dbinom62-times-dbinom31--15-times-3--45-the-total-number-of-valid-combinations-is-boxed45question-a-tech-startup-founder-has-8-potential-investment-opportunities-and-wants-to-diversify-by-selecting-5-but-two-specific-opportunities-are-mutually-exclusive-how-many-valid-investment-portfolios-can-be-formed-18333812026-04-22T03:20:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-surjective-function-problem-count-the-number-of-ways-to-partition-7-distinct-species-into-4-non-empty-habitats-using-the-inclusion-exclusion-principle--sumk04--1k-binom4k-4---k7--calculating-47---binom4137--binom4227---binom4317-compute-each-term-16384---4-times-2187--6-times-128---4-times-1--16384---8748--768---4--8400-the-final-answer-is-boxed8400question-468492026-02-24T21:00:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-tan60circ--fractextheighttextshadow-length-we-substitute-tan60circ--sqrt3-and-shadow-length-5textcm-solving-for-height-textheight--5-cdot-sqrt3-thus-boxed5sqrt3-4399112026-04-22T03:20:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-an-equilateral-triangle-is-fracsqrt34-s2-solving-for-original-side-s-fracsqrt34-s2--25sqrt3-rightarrow-s2--100-rightarrow-s--10textcm-new-side-14textcm-new-area-fracsqrt34-cdot-142--49sqrt3-increase-49sqrt3---25sqrt3--24sqrt3-thus-boxed24sqrt3-33145682026-04-22T03:20:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--h--represent-the-number-of-hours-worked-the-total-fee-for-the-first-consultant-is--300--75h--and-for-the-second-consultant-it-is--200--90h--setting-these-equal-21606652026-04-22T03:27:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given--dt--kt2--5t--and-that--d3--48--substituting--t--3--28783672026-04-22T03:28:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-altitude-is-modeled-by-the-quadratic-function--y--3x2---12x--15--since-the-coefficient-of--x2--is-positive-the-parabola-opens-upward-and-the-minimum-value-occurs-at-the-vertex-the-x-coordinate-of-the-vertex-is-67232432026-04-22T03:28:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--e--represent-the-number-of-employees-the-total-cost-for-plan-a-is--100--10e--and-for-plan-b-it-is--150--7e--setting-these-equal-86215922026-04-22T03:28:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-given--a--2b--substitute-into-the-expression-for--e--91211332026-04-22T03:28:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-function--ht---5t2--30t--10--is-a-quadratic-with-a-negative-leading-coefficient-so-the-maximum-occurs-at-the-vertex-the-time--t--at-which-the-maximum-altitude-occurs-is-84179072026-04-22T03:28:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-field-is-120-meters-wide-short-side-and-160-meters-long-long-side-to-ensure-full-coverage-the-drone-flies-parallel-passes-along-the-120-meter-width-with-each-pass-covering-20-meters-in-the-160-meter-direction-the-number-of-passes-required-is-frac12020--6-passes-each-pass-spans-160-meters-in-length-since-the-drone-turns-at-the-end-of-each-pass-and-flies-back-along-the-return-path-each-pass-contributes-160--160--320-meters-of-travelexcept-possibly-the-last-one-if-it-doesnt-need-to-return-but-since-every-pass-must-be-fully-flown-and-aligned-the-drone-must-complete-all-6-forward-and-6-reverse-segments-however-the-problem-states-it-aligns-passes-to-scan-fully-implying-the-drone-flies-each-pass-and-returns-so-6-forward-and-6-backward-segments-but-optimally-the-return-can-be-integrated-into-flight-planning-however-since-no-overlap-or-efficiency-gain-is-mentioned-assume-each-pass-is-a-continuous-straight-flight-and-the-return-is-part-of-the-route-but-standard-interpretation-for-full-coverage-with-back-and-forth-there-are-6-forward-passes-and-5-returns-noproblem-says-to-fully-scan-with-aligned-parallel-passes-suggesting-each-pass-is-flown-once-in-20m-width-and-the-drone-flies-each-160m-segment-and-the-turn-around-is-inherent-but-to-minimize-total-distance-assume-the-drone-flies-each-160m-segment-once-in-each-direction-per-pass-that-would-be-inefficient-but-in-precision-agriculture-standard-for-120m-width-6-passes-at-20m-width-the-drone-flies-6-successive-160m-lines-and-at-the-end-turns-and-flies-back-along-the-return-pathtypically-the-return-is-not-part-of-the-scan-but-the-drone-must-complete-the-loop-however-in-such-problems-its-standard-to-assume-each-parallel-pass-is-flown-once-in-each-direction-unlikely-better-interpretation-the-drone-flies-6-passes-of-160m-each-aligned-with-the-120m-width-and-the-return-from-the-far-end-is-not-counted-as-flight-since-its-typical-in-grid-scanning-but-problem-says-shortest-total-distance-so-we-assume-the-drone-must-make-6-forward-passes-and-must-return-to-start-for-safety-or-data-sync-so-6-forward-and-6-return-segments-each-160m-so-total-distance-6-times-160-times-2--1920-meters-but-is-the-return-160m-yes-if-flying-parallel-but-after-each-pass-it-returns-along-a-straight-line-parallel-so-160m-so-total-6-times-160-times-2--1920-but-waitcould-it-fly-return-at-angles-no-efficient-is-straight-back-but-another-optimization-after-finishing-a-pass-it-doesnt-need-to-turn-180--it-can-resume-along-the-adjacent-160m-segment-no-because-each-160m-segment-is-a-new-parallel-line-aligned-perpendicular-to-the-width-so-after-flying-north-on-the-first-pass-it-turns-west-180-to-fly-south-return-but-thats-still-160m-so-each-full-cycle-pass--return-is-320m-but-6-passes-require-6-returns-only-if-each-turn-around-is-a-complete-180-and-160m-straight-line-but-after-the-last-pass-it-may-not-need-to-returnit-finishes-but-problem-says-to-fully-scan-the-field-and-aligned-parallel-passes-so-likely-it-plans-all-6-passes-each-160m-and-must-complete-them-but-does-it-imply-a-return-the-problem-doesnt-specify-a-landing-or-reset-so-perhaps-the-drone-only-flies-the-6-passes-each-160m-and-the-return-flight-is-avoided-since-its-already-at-the-far-end-but-to-be-safe-assume-the-drone-must-complete-the-scanning-path-with-back-and-forth-turns-between-passes-so-6-upward-passes-160m-each-and-5-downward-returns-160m-each-totaling-6-times-160--5-times-160--11-times-160--1760-meters-but-standard-in-robotics-for-grid-coverage-total-distance-is-number-of-passes-times-width-times-2-forward-and-backward-but-only-if-returning-to-start-however-in-most-such-problems-unless-stated-otherwise-the-return-is-not-counted-beyond-the-scanning-legs-but-here-it-says-shortest-total-distance-so-efficiency-matters-but-no-turn-cost-given-so-assume-only-flight-distance-matters-and-the-drone-flies-each-160m-segment-once-per-pass-and-the-turn-between-is-instantso-total-flight-is-the-sum-of-the-6-passes-and-6-returns-only-if-full-loop-but-that-would-be-12-segments-of-160m-noeach-pass-is-160m-and-there-are-6-passes-and-between-each-a-return-that-would-be-6-passes-and-11-returns-no-clarify-the-drone-starts-flies-160m-for-pass-1-east-then-turns-west-180-flies-160m-return-back-then-turns-north-90-flies-160m-pass-2-etc-but-each-return-is-not-along-the-next-passeach-new-pass-is-a-new-160m-segment-in-a-perpendicular-direction-but-after-pass-1-east-to-fly-pass-2-north-it-must-turn-90-left-but-the-flight-path-is-now-160m-northso-its-a-corner-the-total-path-consists-of-6-segments-of-160m-each-in-consecutive-perpendicular-directions-forming-a-spiral-like-outer-loop-but-actually-orthogonal-the-path-is-160m-east-160m-north-160m-west-160m-south-etc-forming-a-rectangular-path-with-6-sides-no6-parallel-lines-alternating-directions-but-each-line-is-160m-and-there-are-6-such-lines-3-pairs-of-opposite-directions-the-return-between-lines-is-instantaneous-in-2dso-only-the-6-flight-segments-of-160m-matter-but-thats-not-realistic-in-reality-moving-from-the-end-of-a-160m-east-flight-to-a-160m-north-flight-requires-a-90-turn-but-the-distance-flown-is-still-the-160m-of-each-leg-so-total-flight-distance-is-6-times-160--960-meters-for-forward-plus-no-returnsince-after-each-pass-it-flies-the-next-pass-directly-but-to-position-for-the-next-pass-it-turns-but-that-turn-doesnt-add-distance-so-total-directed-flight-is-6-passes--160m--960m-but-is-that-sufficient-the-problem-says-to-fully-scan-so-each-120m-wide-strip-must-be-covered-and-with-6-passes-of-20m-width-its-done-and-aligned-with-shorter-side-so-minimal-path-is-6--160--960-meters-but-waitafter-the-first-pass-east-it-is-at-the-far-west-of-the-120m-strip-then-flies-north-for-160mthis-covers-the-north-end-of-the-strip-then-to-fly-south-to-restart-westward-it-turns-and-flies-160m-south-return-covering-the-south-end-then-east-etc-so-yes-each-160m-segment-aligns-with-a-new-120m-wide-parallel-and-the-160m-length-covers-the-entire-160m-span-of-that-direction-so-total-scanned-distance-is-6-times-160--960-meters-but-is-there-a-return-the-problem-doesnt-say-the-drone-must-return-to-startjust-to-fully-scan-so-960-meters-might-suffice-but-typically-in-such-drone-coverage-a-full-scan-requires-returning-to-begin-the-next-strip-but-here-no-indication-moreover-6-passes-of-160m-each-aligned-with-120m-width-fully-cover-the-area-so-total-flight-6-times-160--960-meters-but-earlier-thought-with-returns-was-incorrectno-separate-returnline-the-flight-is-continuous-with-turns-so-total-distance-is-960-meters-but-lets-confirm-dimensions-field-120m-w--160m-n-each-pass-160m-n-or-s-covering-a-120m-wide-band-6-passes-every-20m-covers-0120m-w-each-at-20m-intervals-020-2040--100120-each-pass-covers-one-120m-wide-strip-the-length-of-each-pass-is-160m-the-length-of-the-field-so-yes-6--160--960m-but-is-there-overlap-in-dense-grid-usually-offset-but-here-no-mention-of-offset-so-possibly-overlapping-but-for-minimum-distance-we-assume-no-redundancyoptimize-path-but-the-problem-doesnt-say-it-can-skip-turnsso-we-assume-the-optimal-path-is-6-straight-segments-of-160m-each-in-a-new-96959362026-02-24T21:08:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio-of-dominant-to-recessive-traits-in-species-a-is--53--this-means-that-for-every-5-dominant-traits-there-are-3-recessive-traits-53733752026-04-22T03:33:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-algae-doubles-in-area-every-3-days-starting-from--2742742026-04-22T03:33:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio-of-red-to-yellow-flowers-is--47--46108212026-04-22T03:33:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-dna-segment-repeats-every-12-base-pairs-and-mutations-occur-every-48-base-pairs-74032662026-04-22T03:33:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio-of-active-compound-to-dried-herb-is--48459452026-04-22T03:33:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-annual-rainfall-is-60-inches-26742982026-04-22T03:33:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio--31--means-3-out-of-every-4-plants-are-tall-and-1-out-of-every-4-is-short-85029802026-04-22T03:33:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-distances-be--a---2d-a---d-a-a--d-a--2d--3682882026-04-22T03:33:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-from-the-formula--an--3n--k--74136942026-04-22T03:33:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-fossils-found-at-the-five-sites-be-a---2d-a---d-a-a--d-and-a--2d-where-a-is-the-middle-term-and-d-is-the-common-difference-of-the-arithmetic-sequence-the-sum-of-these-terms-is-71356022026-04-22T03:44:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-original-number-of-bees-be-7x-and-butterflies-be-4x-after-adding-18-bees-the-new-number-of-bees-is-7x--18-the-new-ratio-is-given-as-53-so-88726092026-04-22T03:44:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-note-the-only-even-prime-is--2--all-other-primes-are-odd-the-sum-of-three-numbers-is-even-if-12589602026-04-22T03:55:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-smallest-four-digit-number-divisible-by-both-7-and-13-we-first-compute-their-least-common-multiple-since-7-and-13-are-both-prime-their-lcm-is-simply-their-product-50491502026-04-22T04:08:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-simplify-reduce-each-base-modulo-8-56866852026-04-22T04:08:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-four-consecutive-integers-be--n-n1-n2-n3--their-product-is-92203482026-04-22T04:08:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-point--d--such-that--abcd--is-a-parallelogram-we-use-the-property-that-in-a-parallelogram-the-diagonals-bisect-each-other-therefore-the-midpoint-of-diagonal--ac--must-be-the-same-as-the-midpoint-of-diagonal--bd--62171252026-04-22T04:15:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-total-volume-of-nutrient-solution-required-multiply-the-amount-needed-per-culture-by-the-number-of-cultures-41609862026-04-22T04:16:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-decrease-in-thickness-is-21381222026-04-22T04:16:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-number-of-increments-divide-the-total-distance-by-the-size-of-each-increment-76865432026-04-22T04:16:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-population-doubles-every-3-hours-in-12-hours-the-number-of-doubling-periods-is-17119202026-04-22T04:16:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-multiply-the-daily-flow-rate-by-the-number-of-days-38017232026-04-22T04:16:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-outputs-of-the-panels-be--a---12-a---6-a-a--6-a--12--forming-an-arithmetic-sequence-centered-at-the-third-term--a--the-sum-is-36807252026-04-22T04:20:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-ages-be--a---2d-a---d-a-a--d-a--2d--the-youngest-is--a---2d--1200--and-the-oldest-is--a--2d--2000--adding-these-9074292026-04-22T04:20:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-ways-to-choose-4-samples-87779932026-04-22T04:20:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-equivalent-to-counting-the-number-of-onto-functions-from-7-distinct-kits-to-3-students-using-inclusion-exclusion-27838762026-04-22T04:33:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-the-stirling-number-of-the-second-kind-s6-4-which-counts-the-number-of-ways-to-partition-6-distinct-objects-into-4-non-empty-identical-subsets-using-the-recurrence-relation-82210252026-04-22T04:33:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-arithmetic-mean-sum-the-three-values-and-divide-by-3-31977442026-04-22T04:47:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-the-square-is--144--and-each-rectangle-covers--24--51783812026-04-22T04:47:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-asked-to-find-the-least-common-multiple-lcm-of-28-and-45-15733892026-04-22T04:51:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-classic-problem-of-distributing-indistinguishable-items-experiments-into-distinguishable-bins-languages-which-is-solved-using-the-stars-and-bars-method-we-want-to-find-the-number-of-non-negative-integer-solutions-to-the-equation-68206192026-04-22T04:52:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-asked-to-find-the-number-of-distinct-permutations-of-a-multiset-the-total-number-of-tokens-is-8-consisting-of-3-red-3-blue-and-2-green-tokens-since-tokens-of-the-same-color-are-indistinguishable-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-37446932026-04-22T04:52:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-distributing-12-distinguishable-zones-into-four-categories-with-fixed-counts-4-structural-5-tool-related-2-residential-and-1-none-since-the-zones-are-distinguishable-but-the-artifacts-within-each-category-are-indistinguishable-we-compute-the-number-of-ways-to-assign-the-categories-to-zones-37076272026-04-22T04:52:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-distinct-permutations-of-a-multiset-with-9-elements-4-successes-s-3-failures-f-and-2-inconclusive-i-the-number-of-such-distinct-sequences-is-given-by-21928862026-04-22T04:52:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-display-is-circular-and-two-specific-manuscripts-must-be-adjacent-treat-them-as-a-single-unit-in-a-circular-arrangement--n--distinct-objects-can-be-arranged-in--n-1--ways-so-with-6-units-the-pair--5-other-manuscripts-we-have--6-1--5--arrangements-however-the-two-manuscripts-within-the-pair-can-be-ordered-in--2--ways-thus-total-arrangements-are-39995682026-04-22T04:56:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-an-equilateral-triangle-with-side--s--is-19807692026-04-22T04:57:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-rectangular-array-is-inscribed-in-a-circle-of-radius-10-km-so-the-diagonal-of-the-rectangle-equals-the-diameter-of-the-circle-55093712026-04-22T04:57:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-volume-of-a-sphere-of-radius--r--is-44277212026-04-22T04:57:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-centroid-of-a-regular-tetrahedron-is-the-average-of-its-vertices-let-the-fourth-vertex-be-x-y-z-the-centroid-is-16588052026-04-22T04:57:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-integer-be--x--from-the-problem-we-have-93480222026-04-22T04:57:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-drone-starts-at-a-fixed-station-and-returns-to-it-after-visiting-all-others-exactly-once-we-are-counting-circular-permutations-of-6-stations-fixing-one-station-eliminates-rotational-symmetry-so-the-number-of-distinct-paths-is-6---1--5--120-72901992026-04-22T06:42:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-tout-dabord-nous-listons-tous-les-facteurs-de-60-la-factorisation-en-nombres-premiers-de-60-est--70819742026-03-12T06:44:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-tout-dabord-trouvez-la-factorisation-en-nombres-premiers-de-84--35881862026-03-12T06:44:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-pour-trouver-la-moyenne-additionnez-les-longueurs-et-divisez-par-2--68677912026-03-12T06:44:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve-the-system-of-equations-48658122026-04-24T12:48:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-with--a1--1--compute-each-term-step-by-step-3550352026-04-24T12:48:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--y--3x---10--into-the-first-equation-90633532026-04-24T12:49:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-the--t--intercept-occurs-when--e--0--substitute--e--0--into-the-equation-61520812026-04-24T12:49:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--b--a--3--into-the-equation-30885152026-04-24T12:49:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sum-of-the-cubes-of-the-first--n--positive-integers-is-given-by-the-formula-86938732026-04-24T12:49:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve-the-system-of-equations-from-a---b--1-express-a--b--1-substitute-into-3a--2b--18-85196772026-03-12T08:31:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute-x--3-into-both-functions-78404052026-03-12T08:32:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute-u--4-and-p--10-into-the-equation-33086012026-04-24T12:50:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-three-consecutive-odd-integers-be--x---2---x--and--x--2--their-sum-is-55077162026-03-12T08:32:15+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-equation-simplifies-to-87410392026-03-12T08:32:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--d--be-the-number-of-deliveries-the-net-profit-per-delivery-is-11563552026-03-12T08:32:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-the-square-is-95453332026-03-12T08:32:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-arithmetic-with-first-term--a1--3--common-difference--d--6--and-last-term--an--63--using-the-formula--an--a1--n-1d--17965422026-04-24T12:51:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-problem-reduces-to-permutations-of-a-multiset-there-are--7--5--12--total-shards-with-7-identical-peruvian-and-5-identical-mesoamerican-the-number-of-distinct-arrangements-is--73035912026-04-24T12:52:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-problem-is-equivalent-to-arranging-4-identical-tablets-and-3-identical-vessels-the-number-of-distinct-arrangements-is--97682102026-04-24T12:52:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-center-rewrite-the-equation-by-completing-the-square-group--x--terms-and--y--terms-54926762026-04-24T12:52:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-set-the-two-equations-equal-to-find-the-intersection-55898012026-04-24T12:52:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-a-functional-equation-80247702026-04-24T12:52:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-a-triangle-with-sides-a--13-b--14-and-c--15-the-circumradius-r-of-a-triangle-with-sides-a-b-c-is-given-by-the-formula-84771572026-04-22T07:03:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-side-length-of-the-original-equilateral-triangle-be-s-the-area-a-of-an-equilateral-triangle-is-73551322026-04-22T07:03:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-volume-of-a-sphere-with-radius-2r-is-30964932026-04-22T07:03:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-shortest-altitude-corresponds-to-the-longest-side-since-area-is-fixed-and-altitude-is-inversely-proportional-to-base-the-longest-side-is-21-km-so-we-compute-the-altitude-to-this-side-8817842026-04-22T07:03:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-anthropologist-can-only-compare-ritual-pairs-that-originate-from-neighboring-tribes-there-are-exactly-3-such-neighboring-pairs-since-each-comparison-is-between-a-unique-pair-of-rituals-and-no-pair-is-repeated-or-overlap-implied-the-total-number-of-valid-comparisons-is-simply-the-number-of-given-neighboring-pairs-5021682026-04-22T07:03:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-choosing-4-species-from-9-of-which-4-are-rainforest-dominants-and-5-are-not-we-want-exactly-2-rainforest-dominants-and-2-non-rainforest-dominants-the-number-of-favorable-outcomes-is-18156642026-04-22T07:03:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-shortest-altitude-in-a-triangle-with-side-lengths--a--13---b--14--and--c--15--we-first-compute-the-area-using-herons-formula-the-semi-perimeter--s--is-48159942026-04-24T12:56:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-rectangle-with-dimensions-5-cm-by-12-cm-is-inscribed-in-a-circle-so-its-diagonal-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-6325772026-04-24T12:56:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-a-right-triangle-with-legs--a---b--hypotenuse--z--25--and-inradius--c--5--we-use-the-formula-for-the-inradius-of-a-right-triangle-7543792026-04-24T12:56:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area--a--of-an-equilateral-triangle-with-side--s--is-96188272026-04-24T12:56:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-base-and-each-leg-have-length--x--then-the-perimeter-is--x--x--x--3x--36--so--x--12--89073212026-04-24T12:56:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-determine-the-prime-factorization-of-4800-begin-by-factoring-step-by-step-86076942026-03-12T12:20:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--v--6k--where--k--is-a-positive-integer-then-83397712026-03-12T12:20:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-simplify-the-expression-39738152026-03-12T13:09:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-looking-for-the-number-of-integers-from-1-to-100-that-are-congruent-to-3-modulo-7-these-numbers-form-an-arithmetic-sequence-27307562026-04-24T12:58:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-be-of-the-form-1000a--10b--7-ie-ends-in-7-we-want-the-smallest-such-number-divisible-by-13-66418902026-04-22T07:20:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-side-length-of-the-square-be--s--the-diagonal-of-a-square-divides-it-into-two-congruent-right-triangles-and-the-relationship-between-the-side-length-and-the-diagonal-is-given-by-the-pythagorean-theorem-7497082026-04-22T07:20:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-shortest-altitude-we-first-compute-the-area-using-herons-formula-let-the-sides-be--a--13---b--14---c--15--the-semi-perimeter-is-73754192026-04-22T07:20:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-radius-of-the-cylinder-be--r--and-its-height-be--h--2r--the-cylinder-is-inscribed-in-a-sphere-of-radius-10-so-the-diagonal-from-the-center-of-the-sphere-to-the-top-rim-of-the-cylinder-forms-a-right-triangle-with-legs--r--and--r--and-hypotenuse-equal-to-the-radius-of-the-sphere-10-cm-52384092026-04-22T07:20:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-the-number-of-6-digit-binary-numbers-with-exactly-one-11-block-we-analyze-each-possible-position-of-the-11-and-ensure-no-additional-consecutive-1s-the-11-can-occupy-positions-12-23-34-45-or-56-for-each-case-the-remaining-digits-must-avoid-forming-additional-11-blocks-2096102026-04-22T07:24:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-the-number-of-onto-functions-from-5-species-to-3-zones-calculated-using-inclusion-exclusion-21806432026-04-22T07:23:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-shoelace-formula-for-the-area-of-a-triangle-with-vertices--x1-y1---x2-y2---x3-y3--95603442026-04-22T07:24:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-two-complex-numbers--z1--x1--y1i--and--z2--x2--y2i--are-orthogonal-if-their-dot-product--x1x2--y1y2--0--5331372026-04-22T07:24:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-variance-of-a-dataset-is-defined-as-the-average-of-the-squared-deviations-from-the-mean-given-that-there-are-12-samples-and-the-sum-of-the-squares-of-the-deviations-is-144-we-compute-the-variance-as-follows-10037022026-04-22T07:24:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-function--pd--3d--7--and-we-are-to-compute-the-total-number-of-microplastic-particles-from-depth--d--1--to--d--10--this-requires-summing--pd--over-integer-values-from-1-to-10-84411302026-04-22T07:24:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-first-term-of-the-arithmetic-sequence-be--a--and-the-common-difference-be--d--the-sequence-has-8-terms-so-the-sum--s8--is-35600622026-04-22T07:24:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-asked-to-count-the-number-of-distinct-sequences-in-which-the-drone-can-visit-5-fields-2-wheat-w-2-corn-c-and-1-soybean-s-where-fields-of-the-same-crop-are-indistinguishable-this-is-a-permutation-of-a-multiset-48636492026-04-24T13:00:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-9-nucleotides-4-as-3-cs-and-2-gs-where-nucleotides-of-the-same-type-are-indistinguishable-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-74344892026-04-24T13:00:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-assigning-6-days-to-cultivation-tasks-3-wheat-w-2-corn-c-and-1-soybean-s-where-the-order-matters-but-tractors-are-indistinct-the-number-of-distinct-sequences-is-the-multinomial-coefficient-60672542026-04-24T13:00:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-5-proteins-2-as-2-gs-and-1-f-with-indistinguishable-items-within-each-type-the-number-of-distinct-sequences-is-40568742026-04-24T13:00:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-asked-to-count-the-number-of-distinct-permutations-of-a-multiset-the-total-number-of-icons-is-4--3--2--9-with-repetitions-of-each-type-the-number-of-distinct-arrangements-is-given-by-the-multinomial-coefficient-82465052026-04-24T13:00:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-8-individuals-with-repetitions-3-chimpanzees-2-gorillas-and-3-orangutans-the-number-of-distinct-arrangements-is-3135942026-04-24T13:00:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-10-items-with-repetitions-5-of-one-type-3-of-another-and-2-of-a-third-the-number-of-distinct-sequences-is-15688972026-04-24T13:00:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-9-items-with-repetitions-4-males-3-females-2-juveniles-the-number-of-distinct-sequences-is-79798082026-04-22T07:24:58+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-we-add-the-three-expressions-and-divide-by-3-46117422026-04-24T13:01:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-regular-icosahedron-all-edges-are-of-equal-length-first-compute-the-distance-between-the-given-points-27074052026-04-24T13:01:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-solve-this-we-use-combinations-to-count-favorable-outcomes-over-total-outcomes-31390382026-04-24T13:01:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-each-altitude-has-one-sensor-selecting-3-sensors-with-no-two-from-the-same-altitude-simply-means-choosing-any-3-distinct-sensors-which-is-guaranteed-as-all-sensors-are-at-different-altitudes-37826202026-04-24T13:01:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-robot-arms-are-identical-the-problem-reduces-to-choosing-4-distinct-arms-out-of-6-to-place-the-sensors-however-because-the-arms-are-indistinct-the-order-of-placement-does-not-matter-thus-the-number-of-ways-is-the-number-of-combinations-of-6-arms-taken-4-at-a-time-44524332026-04-24T13:01:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-equivalent-to-partitioning-7-distinct-calculators-into-4-non-empty-identical-subsets-the-number-of-ways-is-given-by-the-stirling-numbers-of-the-second-kind--s7-4--using-the-formula-or-recurrence-relation-16980422026-04-24T13:01:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-four-terms-of-the-arithmetic-sequence-be-a---3d-a---d-a--d-and-a--3d-where-a-is-the-middle-term-and-d-is-the-common-difference-974292026-04-22T07:37:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-diagonal-of-the-rectangle-is-the-hypotenuse-of-a-right-triangle-with-legs-5-cm-and-12-cm-using-the-pythagorean-theorem-40366752026-04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rian-2-nomadic-and-2-maritime-13124252026-04-24T13:19:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-assigning-12-distinguishable-job-roles-into-3-categories-with-fixed-counts-5-ai-driven-4-enhanced-by-ai-and-3-unaffected-29466362026-04-24T13:19:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-8-events-where-3-are-strong-3-are-moderate-and-2-are-weak-since-events-of-the-same-type-are-indistinguishable-the-number-of-sequences-is-85007362026-04-24T13:19:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-assigning-9-distinguishable-artifacts-into-3-time-periods-with-fixed-counts-4-ancient-3-classical-and-2-modern-65095372026-04-24T13:19:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--m--be-the-midpoint-of--ab--and--n--be-the-midpoint-of--ac--then-29137492026-04-24T13:19:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-partial-fraction-decomposition-70477392026-04-22T08:53:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-difference--fx1---fx--is-linear--fx--must-be-a-quadratic-29547312026-04-22T08:53:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-radius--r--of-the-circumscribed-circle-of-a-triangle-with-sides--a--5---b--6--and--c--7--we-use-the-formula-21601782026-04-22T08:54:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area--a--of-a-triangle-can-be-calculated-using-the-base-and-height-76582122026-04-22T08:53:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-square-inscribed-in-a-circle-the-diagonal-of-the-square-is-equal-to-the-diameter-of-the-circle-let--s--be-the-side-length-of-the-square-then-the-diagonal--d--is-34713412026-04-22T08:53:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-shortest-altitude-corresponds-to-the-longest-side-which-is--13--cm-first-compute-the-area--k--using-herons-formula-the-semi-perimeter--s--is-34878322026-04-22T08:53:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-30-of-120-calculate-75523142026-04-22T08:54:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-find-the-daily-energy-production-47892302026-04-22T08:54:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-multiply-the-total-number-by-the-fraction-51607412026-04-22T08:54:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-multiply-the-population-by-the-fraction-59829252026-04-22T08:54:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-a-right-triangle-with-legs--a---b--and-hypotenuse--c--the-area--a--is-38231112026-04-22T08:57:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--n--5--into-the-sequence-formula-71774752026-04-22T09:01:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-square-both-sides-of--x--32222022026-04-22T09:01:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-longest-side-is-15-cm-to-find-the-altitude-to-this-side-first-calculate-the-area-using-herons-formula-the-semi-perimeter-is-30316322026-04-22T09:02:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal-d-is-82879102026-04-22T09:09:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-sphere-with-radius-3x-is-35715592026-04-22T09:09:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-from-trigonometric-identities-or-standard-values-36662622026-04-22T09:09:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-a-of-an-equilateral-triangle-with-side-length-s-is-6143912026-04-22T09:09:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-from-1950-to-2050-is-100-years-or-10-decades-7800092026-03-13T04:22:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-an-arithmetic-sequence-with-first-term-a--2-and-common-difference-d--3-17072652026-03-13T04:22:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-rise-is-given-by-11110402026-03-13T04:21:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-find-how-many-are-related-to-renewable-energy-72194682026-03-13T04:22:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-an-arithmetic-sequence-with-a--410-d--2-23027392026-03-13T04:22:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-speed-grows-exponentially-59019062026-03-13T04:22:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-length-we-add-the-lengths-of-the-two-stems-and-divide-by-2-46323582026-03-13T04:22:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-we-sum-the-data-usage-values-and-divide-by-the-number-of-departments-59828772026-03-13T04:22:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-difference-in-width-subtract-the-smaller-width-from-the-larger-66066752026-04-22T09:18:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-calculate-the-average-sum-the-times-and-divide-by-the-number-of-measurements-4988762026-04-22T09:18:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-width-be-w-then-the-length-is-2w-the-perimeter-formula-for-a-rectangle-is-37917512026-04-22T09:18:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-total-height-we-sum-all-the-individual-heights-90512062026-04-22T09:18:58+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-of-devices-follows-an-arithmetic-sequence-where-the-first-term-is-13-and-the-common-difference-is-7-the-formula-for-the-nth-term-is-9117282026-04-22T09:18:58+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-maximum-number-of-schools-divide-the-total-kits-by-the-minimum-per-school-39569602026-04-22T09:18:58+00:00daily0.8https://pg.groshi247.com/highlight/solution-the--y--intercept-of-a-linear-equation--ex--mx--b--is-the-value-of--ex--when--x--0--substituting--x--0--into--ex--35x--12--we-get--e0--350--12--12--73213942026-04-22T09:20:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area--a--of-an-equilateral-triangle-with-side-length--s--is-given-by-63664612026-04-22T09:20:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-five-numbers-form-an-arithmetic-sequence-we-write-them-as-4187832026-04-22T09:21:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-by-expanding-both-sides-18565192026-04-22T09:21:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-add-the-two-values-and-divide-by-2-42324592026-04-22T09:22:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-by-expanding-the-left-side-99956802026-04-22T09:32:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-length-is-found-by-adding-the-lengths-and-dividing-by-2-18565192026-04-22T09:32:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-regular-tetrahedron-all-edges-are-equal-compute-the-distance-between-given-points-16502352026-04-22T09:32:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-looking-for-the-smallest-three-digit-number-n-such-that-44373392026-04-22T09:33:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-point-on-the-line--y--14371332026-04-22T09:40:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-three-given-vertices-be--a100---b010---c001--and-let--dxyz--be-the-fourth-vertex-of-the-regular-tetrahedron-with-integer-coordinates-in-a-regular-tetrahedron-all-edges-are-equal-79094862026-04-24T13:38:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-legs-of-the-right-triangle-be--a--and--b--and-the-hypotenuse--c--13--the-radius--r--of-the-incircle-of-a-right-triangle-is-given-by-58134372026-04-22T09:42:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-5-character-strings-using-the-letters-a-b-c-d-4-letters-such-that-each-letter-appears-at-least-once-this-is-a-classic-inclusion-exclusion-problem-17012832026-04-22T09:43:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-diagonal-of-a-rectangle-forms-a-right-triangle-with-its-length-and-width-using-the-pythagorean-theorem-the-diagonal-d-is-calculated-as-31749852026-04-22T09:43:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-regular-hexagon-inscribed-in-a-circle-the-radius-equals-the-side-length-of-the-hexagon-this-is-because-the-hexagon-can-be-divided-into-six-equilateral-triangles-thus-the-radius-is-equal-to-the-side-length-79647872026-04-22T09:43:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-center-complete-the-square-for-both--x--and--y--31748072026-04-22T09:43:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-two-vectors-to-be-perpendicular-their-dot-product-must-be-zero-62639272026-04-24T13:41:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area--a--of-a-regular-hexagon-with-side-length--s--is-given-by-33867992026-04-24T13:41:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-the-formula-for-the-inradius-68097112026-04-24T13:41:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-selecting-a-team-of-4-experts-from-7-mathematicians-5-physicists-and-6-computer-scientists-with-the-restriction-that-at-least-one-expert-must-come-from-each-discipline-91022982026-04-24T13:42:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-5-distinct-people-around-a-circular-table-where-rotations-are-considered-identical-in-circular-permutations-the-number-of-arrangements-of--n--distinct-people-is--n-1--40321202026-04-24T13:42:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-forming-a-6-digit-code-with-the-following-constraints-7372052026-04-24T13:42:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-selecting-a-6-member-committee-from-10-scientists-including-3-ai-experts-and-8-engineers-with-83380822026-04-22T09:53:31+00:00dai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ot-product-is-zero-compute-the-dot-product-44895882026-04-22T12:12:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-closest-point-on-a-line-to-a-given-point-is-the-perpendicular-projection-of-that-point-onto-the-line-40836012026-04-22T12:12:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-width-be--x--cm-then-the-length-is--x--4--cm-the-perimeter-is-given-by--2x--x--4--80--which-simplifies-to-99081042026-03-14T00:37:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-width-be--x--inches-then-the-length-is--2x--inches-the-perimeter-is-84682972026-03-14T00:37:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-functional-equation-known-as-cauchys-functional-equation-in-three-variables-we-analyze-it-step-by-step-3715812026-03-14T00:37:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-calculate-the-average-sum-the-two-values-and-divide-by-2-10774312026-04-22T12:17:46+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-arithmetic-with-first-term--a1--5--common-difference--d--4--and-last-term--an--41--83333792026-04-22T12:17:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-set-up-the-equation-for-the-average-of-the-two-expressions-66048612026-04-24T14:23:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-number-of-migration-events-divide-the-total-days-by-the-cycle-length-50642592026-04-24T14:23:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-least-common-multiple-lcm-of-8-and-12-determines-when-both-events-align-prime-factorizations-88915332026-04-24T14:23:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-lcm-of-6-and-9-is-calculated-as-follows-20111992026-04-24T14:23:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-desired-integer-be--x--96914022026-04-22T12:23:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-blue-markers-be-x-the-ratio-54-implies-75505002026-04-24T14:25:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-mean-is-calculated-by-summing-the-values-and-dividing-by-2-98868692026-04-24T14:25:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-reactions-are-independent-the-probability-that-all-three-succeed-is-69191692026-04-24T14:25:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-n--4-so-the-deck-has-3n--12-cards-labeled-from-1-to-12-we-want-the-probability-that-the-sum-of-3-randomly-drawn-distinct-cards-is-divisible-by-3-76895622026-04-24T14:25:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--1--into-both-functions-91551862026-04-22T12:28:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-set-the-equations-equal-at--x--3--67660262026-04-22T12:28:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-multinomial-coefficient-the-number-of-ways-is--48642172026-04-22T12:29:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume--v--of-a-prism-is-given-by-the-base-area--a--multiplied-by-the-height--h--thus--v-88745412026-04-24T14:29:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-is-calculated-as--7389892026-04-24T14:30:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-find-each-number-modulo-13-89507132026-04-24T14:30:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-first-team-has-4-prototypes-each-with-a-distinct-rank-from-1-to-4-the-third-team-has-3-prototypes-ranked-1-to-3-since-one-prototype-is-selected-uniformly-at-random-from-each-team-the-selection-within-each-team-is-independent-91633682026-04-22T12:45:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-a-total-of--5--2--1--8--atoms-where-5-are-identical-hydrogen-h-2-are-identical-oxygen-o-and-1-is-nitrogen-n-the-number-of-distinct-permutations-is-given-by-the-multinomial-coefficient-62261632026-04-22T12:45:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-greatest-common-factor-gcf-of-78-and-104-we-start-by-finding-their-prime-factorizations-18696612026-04-24T14:31:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-of-three-expressions-is-given-by-54621132026-04-24T14:31:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--t--15--into-the-equation-52451322026-04-22T12:52:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-lengths-be--3x---5x--and--7x--the-sum-is--3x--5x--7x--15x--45--solving-for--x--41538352026-04-22T12:52:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-rectangular-trench-is-inscribed-in-the-circular-foundation-the-diagonal-of-the-rectangle-is-equal-to-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal-d-is-49551632026-04-22T12:52:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-rectangle-is-inscribed-in-the-circle-so-its-diagonal-is-the-diameter-the-diagonal-d-is-39587522026-04-22T12:52:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-simplify-the-expression--fx--19907552026-04-22T13:03:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-arithmetic-mean-sum-all-the-measurements-and-divide-by-the-number-of-days-96723772026-03-14T06:02:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-mixture--15--35--50-liters-fraction-of-rainwater---29542892026-03-14T06:03:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-there-are-only-two-types-high-income-and-low-income-it-is-impossible-to-select-3-zones-of-different-types-without-including-at-least-two-of-the-same-type-therefore-the-event-of-selecting-3-zones-all-of-different-types-is-impossible-42167572026-03-14T06:03:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-intersection-point-solve-the-system-of-equations-61940232026-03-14T06:03:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-nous-utilisons-la-formule-daddition-pour-la-tangente--45682772026-04-22T13:08:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-distributive-property-to-expand-73245662026-04-22T13:11:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-area-of-a-triangle-with-sides-a--7-b--10-and-c--13-we-use-herons-formula-254422026-03-14T07:04:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-vectors-to-be-orthogonal-their-dot-product-must-be-zero-compute-3-2--x5--0-this-simplifies-to--6--5x--0-solving-gives-x--89258012026-04-22T13:25:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-chips-is-5--7--3--15-the-probability-the-first-chip-is-amber-is--14120062026-04-22T13:26:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-shortest-altitude-corresponds-to-the-longest-side-8-cm-compute-area-using-herons-formula--s--39336192026-04-22T13:26:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-factor-1080-we-have-5737422026-04-24T14:43:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-looking-for-the-largest-number-n--200-such-that-14853442026-04-24T14:43:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-area-using-herons-formula-the-semiperimeter-is-16776942026-04-24T14:45:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-given-hypotenuse--25--one-leg--24--the-other-leg-is-59279322026-04-24T14:45:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-problem-involves-distributing-4-distinct-objects-into-2-identical-boxes-this-corresponds-to-the-sum-of-stirling-numbers-of-the-second-kind-for-2-boxes-considering-non-empty-subsets-the-formula-is--s41--s42--here--s41--1--all-groups-in-one-zone-and--s42--7--split-into-two-non-empty-groups-thus-the-total-number-of-ways-is--1--7--8--47077652026-04-22T13:37:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-circular-arrangements-of-n-distinguishable-people-the-number-of-distinct-arrangements-is-n-1-here-there-are-8--5--13-individuals-thus-the-number-of-distinct-seating-arrangements-is-13-1--12-29507802026-04-22T13:38:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-equivalent-to-finding-the-number-of-ways-to-partition-6-distinguishable-objects-into-exactly-4-non-empty-indistinguishable-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind-s6-4-from-known-values-s6-4--65-since-the-cabinets-are-indistinguishable-we-do-not-multiply-by-permutations-8924342026-04-22T13:38:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-an-arithmetic-sequence-where-the-first-term--a--3--the-common-difference--d--4--and-the-last-term--l--99--to-find-the-number-of-terms--n--we-use-the-formula-for-the--n--th-term-of-an-arithmetic-sequence-60287692026-04-22T13:38:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-smallest-4-digit-number-is-1000-and-the-largest-is-9999-to-find-how-many-of-these-are-divisible-by-11-we-find-the-smallest-and-largest-multiples-of-11-in-this-range-12344162026-04-22T13:38:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-count-the-positive-divisors-of-60-the-prime-factorization-of-60-is-84600472026-04-22T13:38:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-center-rewrite-the-equation-by-completing-the-square-group-x-and-y-terms-39833122026-04-22T13:38:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-a-system-of-congruences-42372382026-03-14T10:18:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-3-toucans-t-2-tanagers-a-1-macaw-m-and-1-hummingbird-h-and-we-want-to-count-the-number-of-distinct-sequences-of-length-7-using-these-indistinguishable-items-the-total-number-of-distinct-permutations-of-a-multiset-is-given-by-45734832026-04-22T13:43:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-assigning-4-identical-robots-r-3-identical-detectors-d-and-2-identical-sensors-s-to-9-distinct-stations-with-one-device-per-station-this-is-a-multinomial-coefficient-problem-65878292026-04-22T13:43:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-choose-2-different-flavors-from-8-the-number-of-combinations-is-39330732026-04-22T13:43:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-2-jaguars-j-2-capybaras-c-and-2-sloths-s-all-indistinguishable-within-species-the-number-of-distinct-permutations-is-65526862026-04-22T13:43:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-distributing-5-distinguishable-objects-wires-into-3-indistinguishable-non-empty-boxes-this-is-equivalent-to-finding-the-stirling-numbers-of-the-second-kind-s53-which-counts-the-number-of-ways-to-partition-5-labeled-items-into-3-non-empty-unlabeled-subsets-from-known-values-49439392026-04-22T13:43:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-an-equilateral-triangle-with-side-length-6-km-and-we-are-to-find-the-perpendicular-distance-from-a-vertex-to-the-opposite-sidethis-is-the-altitude-of-the-equilateral-triangle-61324192026-04-22T13:43:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-triangle-has-side-lengths-a--13-b--14-c--15-the-longest-side-is-15-cm-so-the-shortest-altitude-corresponds-to-this-side-32075952026-04-22T13:43:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-the-sphere-is-77336632026-03-14T10:53:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-verify-the-triangle-is-right-angled-96091672026-03-14T10:53:44+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-convert--3427--to-base-ten-we-expand-it-using-powers-of-7-63543492026-04-22T13:53:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-7-and-11-are-coprime-their-least-common-multiple-is-3829842026-04-24T14:50:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-length-is-calculated-by-summing-the-lengths-and-dividing-by-2-77779682026-04-22T14:00:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-find-the-cost-of-one-sensor-13551692026-04-22T14:00:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-numerical-values-2804432026-04-22T14:00:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-total-cost-of-three-smartwatches-multiply-the-cost-of-one-by-3-34169162026-04-22T14:08:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-add-the-three-expressions-and-divide-by-3-26342502026-04-22T14:08:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-red-marbles-be-4x-and-blue-marbles-be-7x-since-4x--28-solve-for-x-43750532026-04-22T14:08:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-amount-each-friend-pays-divide-the-total-cost-by-4-66478102026-04-22T14:08:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-yellow-candies-be-5x-and-orange-candies-be-8x-since-8x--40-solve-for-x-12169202026-04-22T14:08:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-a-multiset-of-12-soil-cores-3-from-plot-a-2-from-plot-b-and-1-each-from-plots-c-d-e-and-f-the-total-number-of-distinct-sequences-arrangements-in-which-these-indistinct-items-can-be-arranged-is-a-multinomial-coefficient-9226082026-04-22T14:08:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-distinct-sequences-of-12-blocks-where-there-are-5-red-r-4-blue-b-and-3-green-g-and-blocks-of-the-same-color-are-indistinguishable-this-is-a-permutation-of-a-multiset-27187472026-04-22T14:08:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-volume-of-a-sphere-with-radius--r--2y--15159732026-04-24T14:55:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-shortest-distance-from-a-point-x0-y0-to-a-line-defined-by-ax--by--c--0-is-given-by-the-formula-77146042026-04-22T14:14:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-two-area-functions-33112252026-04-22T14:14:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-first-term-be-a-and-the-common-difference-be-d-8022282026-04-22T14:14:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-the-number-of-ways-to-choose-3-compounds-from-8-and-2-delivery-mechanisms-from-5-we-use-combinations-44548412026-04-22T14:15:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-oak-pollen-grains-be--x--the-ratio--75--implies-89516152026-04-22T14:32:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-white-flowered-plants-be--y--the-ratio--53--implies-27532552026-04-22T14:32:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-dominant-monkeys-be--z--the-ratio--411--implies-26879782026-04-22T14:32:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-an-arithmetic-progression-3869732026-04-22T14:32:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-seating-is-circular-and-the-2-newtonian-experts-must-sit-together-we-treat-them-as-a-single-block-this-reduces-the-problem-to-arranging-6-units-the-block-and-5-historians-in-a-circle-the-number-of-circular-permutations-of-n-objects-is-n-1-so-we-fix-the-block-and-arrange-the-5-historians-around-it-giving-6---1--5-arrangements-within-the-block-the-2-newtonian-experts-can-be-arranged-in-2-ways-thus-the-total-number-of-arrangements-is-89380462026-04-22T14:32:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--c--be-coding-rate-and--t--testing-rate-the-system-is-2788922026-04-24T15:01:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-y-intercept-set--h--0--71807292026-04-24T15:01:58+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-expression-simplifies-to-29360672026-04-22T14:42:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-given-the-average-detection-over-three-locations-is-20-ppm-we-set-up-the-equation-for-the-average-582902026-04-22T14:46:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-processing-rate-is-given-by-28769842026-04-22T14:46:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-pressure-change-is-given-by-66448012026-04-22T14:46:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio-of-kale-to-spinach-is-75-meaning-for-every-7-kale-plants-there-are-5-spinach-plants-since-there-are-35-kale-plants-we-determine-how-many-groups-of-7-are-in-35-30246072026-04-24T15:05:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio-of-basil-to-mint-channels-is-43-so-for-every-4-basil-channels-there-are-3-mint-channels-since-there-are-28-basil-channels-we-calculate-the-number-of-groups-of-4-in-28-35514212026-04-24T15:05:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-least-common-multiple-lcm-of-1660-and-1666-factorize-both-numbers-13769292026-04-24T15:05:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio-74-means-7-parts-wheat-correspond-to-4-parts-barley-34686602026-03-14T19:29:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-pour-trouver-la-valeur-minimale-de--6071372026-03-14T19:39:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-the-path-is-the-difference-between-the-area-of-the-larger-circle-garden--path-and-the-garden-itself-the-radius-of-the-larger-circle-is--r--w--thus-the-area-of-the-path-is-95532502026-04-22T14:52:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-set-the-expression-equal-to-5-and-solve-for--x--48777542026-04-22T14:52:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--1--into-the-polynomial-95768942026-04-22T14:52:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--3--into-the-quadratic-equation-38561652026-04-22T14:52:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--6--into-the-function-3793282026-04-22T14:52:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--0--into-the-function-51011002026-04-22T14:52:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--4--into-the-function-96840022026-04-22T14:52:57+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-toucans-be--t--and-the-number-of-parrots-be--p--from-the-given-ratio-we-have-89328432026-04-22T15:03:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-the-formula-for-the-area-of-a-triangle-with-coordinates-15956642026-04-22T15:06:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-selecting-3-applications-at-random-without-replacement-from-120-total-and-we-want-the-probability-that-each-application-is-from-a-distinct-fieldone-from-cancer-one-from-neuroscience-and-one-from-infectious-diseases-23669662026-04-24T15:10:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-greatest-common-divisor-gcd-of-144-and-60-we-first-perform-the-prime-factorization-of-each-number-54520112026-04-22T15:07:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-find-the-cost-of-one-sensor-by-dividing-the-total-cost-by-the-number-of-sensors-30736592026-04-22T15:07:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-smallest-5-digit-number-is-10000-and-the-largest-is-99999-23723902026-04-22T15:07:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-area-of-the-square-10006082026-04-22T15:07:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-arithmetic-with-first-term--a1--2--common-difference--d--3--and-last-term--an--50--using-the-formula-for-the--n--th-term-of-an-arithmetic-sequence-62214762026-03-14T21:08:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-energy-after--n--months-is-the-sum-of-the-arithmetic-series-29704362026-03-14T21:08:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--2--and--y---1--into-the-expression-for--e--18891162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sum-is-62322062026-04-22T15:32:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-find-the-least-common-multiple-of-18-and-24-first-factor-both-31179632026-04-22T15:32:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-triangle-is-a-right-triangle-with-base-along-the-x-axis-from--a--to--b--and-height-from--a--to--c--57526442026-04-22T15:41:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-smallest-circle-enclosing-an-equilateral-triangle-is-its-circumcircle-44763972026-04-24T15:20:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-hypotenuse-of-the-triangle-is-the-diameter-of-the-circumscribed-circle-3748942026-04-24T15:20:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-an-isosceles-triangle-with-equal-sides-of-10-cm-and-base-12-cm-48842632026-04-22T15:41:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-verify-its-a-right-triangle-92110352026-04-22T15:41:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-remainder-theorem-substitute--v--1--into-the-polynomial-68217812026-04-22T15:55:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-greatest-common-divisor-gcd-of--78--and--130--we-use-the-euclidean-algorithm-25193482026-04-22T15:59:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-an-arithmetic-sequence-where-the-first-term-is--a--3--the-common-difference-is--d--4--and-the-last-term-is--l--99--34443062026-04-22T15:59:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-subtract-the-amount-used-from-the-original-amount-67195332026-04-22T15:59:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-of-two-numbers-is-the-sum-divided-by-2-54247062026-04-22T15:59:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-find-the-centroid-g-of-the-triangle-with-vertices-a12-b46-c72-the-centroid-is-85937662026-04-22T15:59:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-horizontal-asymptote-of-the-rational-function--tm--1345372026-04-22T16:02:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-test-divisibility-starting-from-the-smallest-primes-16674932026-04-24T15:26:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-arithmetic-with-first-term--a--3--common-difference--d--5--and-last-term--l--198--the-formula-for-the--n-th-term-of-an-arithmetic-sequence-is-20193802026-04-24T15:26:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-of-number-of-blocks-per-layer-forms-a-sequence-where-the-first-term-is-2-layer-1-and-the-differences-between-terms-increase-by-3-each-time-6713492026-03-15T01:15:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio-of-red-to-blue-pixels-is--74--given-28-red-pixels-set-up-a-proportion-50852512026-04-24T15:26:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-an-arithmetic-sequence-with--a--5000---d--500---l--9500--use-80285002026-04-24T15:26:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-starts-at-1-and-the-differences-increase-by-3-86195562026-04-24T15:26:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-ratio--58--means-for-every-5-males-there-are-8-females-given-40-males-scale-the-ratio-50822552026-04-24T15:26:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-an-arithmetic-sequence-with--a--20---d--4---n--10--use-4771972026-04-24T15:26:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-factor-out-the-greatest-common-factor-which-is-2-73355492026-04-22T16:03:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-an-arithmetic-sequence-with-first-term--a--2--common-difference--d--4--and-last-term--l--202--50959692026-04-22T16:03:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-evaluate--f-2--44384672026-03-15T01:22:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-distributive-property-to-expand-57412352026-03-15T01:22:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-efficiency-over-25-days-at-72-2590762026-03-15T01:21:58+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-convert--5728--to-base-ten-expand-using-powers-of-8-59916652026-04-22T16:12:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-semicircle-any-triangle-inscribed-with-the-hypotenuse-as-the-diameter-is-a-right-triangle-since-the-triangle-is-isosceles-and-right-its-legs-are-equal-and-the-hypotenuse-is-the-diameter-10305272026-03-15T02:03:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-a-sector-is-given-by-13056362026-03-15T02:03:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-distance-between-opposite-vertices-of-a-regular-hexagon-is-twice-the-side-length--s--so-65165902026-03-15T02:03:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume--v--of-a-cone-is-38040152026-03-15T02:03:55+00:00daily0.8https://pg.groshi247.com/high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ir-least-common-multiple-since-7-and-13-are-both-prime-their-least-common-multiple-is-91890592026-04-22T18:02:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-radius--r--of-the-inscribed-circle-in-a-right-triangle-with-legs--a--and--b--and-hypotenuse--c--is-given-by-the-formula-3782572026-04-22T18:02:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-circle-is-inscribed-in-an-equilateral-triangle-and-has-diameter-10-cm-its-radius-is-66983242026-04-22T18:02:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-rectangle-have-width--2x--and-height--y--inscribed-in-a-semicircle-of-radius-5-with-its-base-on-the-diameter-the-top-corners-lie-on-the-semicircle-so-by-the-pythagorean-theorem-10275992026-04-22T18:02:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume--v--of-a-sphere-of-radius--r--is-33312202026-04-22T18:02:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-calculate-the-probability-that-the-sum-of-four-6-sided-dice-equals-14-we-begin-by-determining-the-total-number-of-possible-outcomes-33377812026-04-22T18:03:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-classic-hypergeometric-probability-problem-we-are-selecting-15-fish-from-a-population-of-200-of-which-10-are-tagged-we-want-the-probability-that-exactly-4-of-the-15-sampled-fish-are-tagged-67210082026-04-22T18:03:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-dividing-12-distinct-volunteers-into-3-unlabeled-teams-of-4-each-96849582026-04-22T18:03:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sum-of-three-numbers-is-even-if-52360852026-04-22T18:03:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-probability-that-at-least-2-of-the-4-selected-compounds-are-placebos-we-calculate-the-complement-of-the-probability-that-fewer-than-2-are-placebos-ie-0-or-1-placebo-10438992026-04-22T18:03:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-choosing-5-practices-from-9-total-with-4-coastal-and-5-non-coastal-35767062026-04-22T18:04:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-compute-the-total-profit-after-3-years-we-evaluate-both-profit-functions-at--x--3--and-sum-the-results-32080352026-04-24T15:58:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-slope--m--74067072026-04-24T15:58:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-how-many-drones-remain-undistributed-we-divide-120-by-8-and-find-the-remainder-41580232026-04-22T18:04:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-temperature-drop-we-add-the-two-drops-and-divide-by-2-455212026-04-24T15:58:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-smallest-number-of-display-cases-we-determine-the-greatest-common-divisor-gcd-of-72-and-108-to-find-the-maximum-number-of-items-per-case-674682026-04-24T15:58:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-maximum-number-of-swarm-units-we-compute-the-greatest-common-divisor-gcd-of-168-and-210-50604592026-04-24T15:58:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find--m--substitute--x--3--into-the-function-3480062026-04-22T18:04:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-greatest-common-factor-gcf-of-48-and-64-factor-both-numbers-91639162026-04-24T15:59:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-arithmetic-with-first-term-a1--10-common-difference-d--5-and-last-term-an--50-51370862026-04-24T15:59:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-factors-of-30-are-1-2-3-5-6-10-15-30-totaling-8-there-are-30-possible-integers-499782026-04-24T15:59:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-orthogonality-their-dot-product-must-be-zero-2x--3x---14--0-simplifying-5x---4--0-so-x--97599632026-04-24T15:59:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-length-we-sum-the-lengths-and-divide-by-the-number-of-fish-50673152026-04-24T15:59:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-of-pollinators-per-day-is-an-arithmetic-sequence-with-first-term--a--1--and-common-difference--d--3--the-formula-for-the--n--th-term-of-an-arithmetic-sequence-is-60131912026-04-24T15:59:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-minimize-the-number-of-rectangles-we-should-maximize-the-area-of-each-rectangle-without-exceeding-15-square-inches-the-largest-area-per-rectangle-is-14-square-inches-a-possible-value-in-the-allowed-range-then-we-divide-the-total-area-by-this-value-27670432026-04-24T15:59:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-list-all-positive-divisors-of-60-the-prime-factorization-of-60-is-18720652026-03-15T07:48:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-smallest-prime-factor-of-143-we-test-divisibility-by-small-primes-93119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r-of-initially-infected-individuals-by-the-average-number-of-new-infections-per-person-3914632026-04-22T18:34:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-length-of-the-two-segments-we-add-the-two-lengths-together-and-then-divide-by-2-67499112026-04-22T18:34:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-arithmetic-mean-is-found-by-summing-the-values-and-dividing-by-the-number-of-values-89055832026-04-22T18:34:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-radius-be--r--and-the-height-be--h--65267892026-04-24T16:06:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-herons-formula-to-compute-the-area-95024802026-04-24T16:06:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-a-triangle-with-an-incircle-is-related-to-the-semiperimeter-and-inradius-by-58777792026-04-24T16:06:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-the--y--intercept-occurs-where--x--0--first-compute-the-slope--m--of-the-line-51538422026-04-24T16:06:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve-the-system-of-equations-from-the-second-equation--x--y--2--substitute-into-the-first-54963832026-04-24T16:06:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--with--2x---1--in-the-original-function--fx--5577912026-04-22T18:37:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-square-both-sides-of-the-equation-32529632026-04-22T18:41:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-be--x--then-44091322026-04-22T18:41:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--4---y--21--into-the-equation-10420522026-04-22T18:41:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-selecting-4-samples-out-of-7-where-3-are-anomalous-call-them-type-a-and-4-are-normal-type-n-we-want-the-probability-that-at-least-2-of-the-4-selected-are-anomalous-17021082026-04-22T18:41:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-number-of-ways-to-choose-5-modules-from-8-74869672026-04-22T18:41:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-number-of-ways-to-assign-6-distinct-drugs-to-6-patients-is-42461552026-04-22T18:41:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-number-of-ordered-sequences-of-3-distinct-layers-from-5-is-22202622026-04-22T18:41:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-net-profit-per-delivery-is--12---150--1050--let--x--be-the-number-of-deliveries-41471842026-04-22T18:44:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-parameter-t-satisfy-90106302026-04-22T18:47:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-a-surface-in-spherical-coordinates-89678182026-04-24T16:09:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-sum-to-product-identity-72449282026-04-24T16:09:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-identity-for-tangent-addition-note-that-3747892026-04-24T16:09:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-distinct-sequences-is-the-multinomial-coefficient-accounting-for-indistinguishable-items-70827422026-03-15T08:32:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-load-we-sum-the-three-expressions-and-divide-by-3-65832612026-04-24T16:09:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-mean-sum-the-three-expressions-and-divide-by-3-11676022026-04-24T16:09:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-an-arithmetic-sequence-with-first-term--a1--3--common-difference--d--5--and-last-term--an--48--58942952026-04-22T18:54:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-documents-is-4--3--2--9-since-the-documents-within-each-category-are-indistinguishable-the-number-of-distinct-orders-is-given-by-the-multinomial-coefficient-41115722026-04-22T18:54:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-samples-is-3--5--2--10-since-samples-within-each-category-are-indistinguishable-the-number-of-distinct-sequences-is-79298732026-04-22T18:54:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-individuals-is-4--5--3--12-since-individuals-within-each-status-category-are-indistinguishable-the-number-of-distinct-sequences-is-12988772026-04-22T18:54:46+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-the-euclidean-algorithm-74291912026-04-22T18:57:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--4--into-the-equation-89168982026-04-22T18:57:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x--y--10--and--15x--20y--10x--25y--350--simplify-68434312026-04-22T18:57:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-half-life-we-add-the-two-half-lives-and-divide-by-2-69290532026-04-22T18:58:23+00:00daily0.8ht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-hypotenuse-is-59593152026-04-24T16:59:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area--a--of-an-equilateral-triangle-with-side-length--s-65360052026-03-15T12:23:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-equation-in-spherical-coordinates-89830572026-04-24T17:00:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-arithmetic-sequence-be-83173372026-04-24T17:00:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-time-per-instrument-is--x--x--10--2x--10--for-120-instruments-the-total-time-is--1202x--10--3300--expanding-52764612026-04-24T17:00:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-five-sensor-readings-in-arithmetic-progression-be--a---2d-a---d-a-a--d-a--2d--given-that-the-sum-of-the-first-and-fifth-readings-is-40-32039362026-04-22T22:50:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-given--e--aa--b--ba--b--factor-out--a--b--63848462026-04-22T22:50:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-convert-the-base-eight-number--3218--to-base-ten-we-use-the-place-value-expansion-2951162026-04-24T17:00:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-formula-for-the-sum-of-the-first--n--squares-is-98956002026-04-24T17:00:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-lets-define--an--as-the-number-of-valid-forecasts-for--n--days-where-no-two-consecutive-days-are-rainy-85068102026-04-22T22:55:06+00:00daily0.8https://pg.groshi247.com/highlight/solution-total-animals--4--3--7--all-are-distinguishable-27086522026-04-24T17:02:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume--v--of-a-regular-tetrahedron-with-edge-length--a--is-given-by-39612252026-03-15T12:34:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-shortest-altitude-corresponds-to-the-longest-side-which-is-15-cm-first-compute-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter--s--is-80050572026-03-15T12:36:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-9-patent-applications-4-from-company-a-indistinguishable-among-themselves-3-from-company-b-and-2-from-company-c-we-are-to-count-the-number-of-distinct-sequences-in-which-these-can-be-reviewed-where-only-the-company-of-origin-matters-not-individual-applications-and-indistinguishability-within-companies-94014262026-03-15T12:36:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-distinct-sequences-of-length-6-where-there-are-2-low-2-medium-and-2-high-transmission-days-since-the-days-are-ordered-over-time-and-the-counts-per-category-are-fixed-this-is-a-permutation-of-a-multiset-72842052026-03-15T12:36:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-the-remainder-theorem-again-we-evaluate--f-1--since-the-divisor-is--u--1--u----1--92548122026-03-15T12:40:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-that--f3--g3--first-compute--f3--16701382026-04-24T17:05:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-first-term-be--a--and-the-common-difference-be--d--84973902026-03-15T12:56:58+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-value-of--m--such-that--fx--and--gx--intersect-at-exactly-one-point-set--fx--gx--27649712026-04-24T17:07:55+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find--k--such-that--px--qx--has-exactly-one-solution-set-31284712026-04-24T17:08:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-given-functional-equation-is-57138282026-04-22T23:23:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-permutations-of-6-chimpanzees-is-6--720-to-count-the-number-of-arrangements-where-kali-and-momo-are-adjacent-treat-them-as-a-single-entity-this-gives-5-arrangements-and-kali-and-momo-can-be-ordered-in-2-ways-so-2-imes-5--240-subtracting-from-the-total-720---240--480-72075792026-04-22T23:25:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-permutations-of-5-monkeys-is-5--120-in-half-of-these-aki-precedes-bongo-in-the-other-half-bongo-precedes-aki-so-the-number-of-sequences-where-aki-does-not-precede-bongo-is--17360022026-04-22T23:25:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-herons-formula-to-find-the-area-of-the-triangle-first-compute-the-semi-perimeter-78128362026-04-22T23:27:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-formula-for-the-area-of-a-triangle-in-terms-of-its-inradius--r--and-semi-perimeter--s--71599742026-04-22T23:27:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-area-formula-for-an-equilateral-triangle-86298992026-04-24T17:10:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-factor-both-numbers-into-primes-75867532026-04-22T23:32:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-geometric-series-with-first-term--a--200--common-ratio--r--085--and--n--4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mber-of-distinct-sequences-is-45149442026-04-23T02:24:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-total-number-of-ways-to-choose-4-insects-from-10-94290062026-04-23T02:24:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-total-number-of-fossils-is-5--4--3--12-and-we-are-arranging-a-sequence-of-12-items-where-5-are-of-one-type-dinosaur-4-of-a-second-type-reptile-and-3-of-a-third-type-insect-with-indistinguishability-within-each-type-the-number-of-distinct-arrangements-is-given-by-the-multinomial-coefficient-33263802026-04-23T02:24:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-find-the-area-enclosed-by-the-curve-49164972026-03-15T16:07:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-arithmetic-mean-we-sum-the-integers-and-divide-by-the-number-of-integers-42408292026-03-15T16:08:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-circular-arrangement-fix-one-87990952026-04-23T02:39:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-when-all-three-servers-synchronize-compute-the-least-common-multiple-lcm-of-18-24-and-30-80281422026-04-23T02:51:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-factor-out-the-greatest-common-divisor-gcd-of-the-coefficients-which-is-3-31206462026-04-24T18:11:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-set-the-equations-equal-to-solve-for-x-14207682026-04-24T18:11:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-y-intercept-occurs-when-x--0-substituting-51638192026-04-24T18:11:17+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-probability-follows-a-binomial-distribution-with-parameters--n--6--and--p--56111112026-04-23T03:04:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-the-binomial-probability-formula-with--n--7---p--16695122026-04-23T03:04:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-since--x--is-a-positive-multiple-of-6-we-can-write--x--6k--where--k--is-a-positive-45677342026-04-23T03:04:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-sum-the-budgets-and-divide-by-the-number-of-grants-24970622026-04-24T18:13:15+00:00daily0.8https://pg.groshi247.com/highlight/solution-sum-the-depths-and-divide-by-3-68942732026-04-23T03:10:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-set-up-the-equation-for-the-average-9986582026-04-23T03:10:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-seed-cones-be-x-the-ratio-53-implies--26468332026-04-23T03:15:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-sum--14--18--22--16--20--90-mean---75654952026-04-23T03:15:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-sum-the-expressions-4u--1--2u--5--6u---3--12u--3-divide-by-3--50207672026-04-23T03:15:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-sum-the-funding-amounts-and-divide-by-the-number-of-projects-80239642026-04-24T18:19:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-total-trees-28--36--42--54--160-the-smallest-prime-factor-of-160-is-2-as-160-is-even-50876632026-04-24T18:19:22+00:00daily0.8https://pg.groshi247.com/highlight/solution-since--v--is-a-multiple-of-6-lets-try-successive-multiples-of-6-and-cube-them-40427312026-04-24T18:19:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-centroid-g-of-triangle-abc-288242026-04-24T18:20:43+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-the-number-of-distinct-soundscapes-we-calculate-the-number-of-ways-to-choose-3-ambient-layers-from-8-and-2-effects-from-5-since-the-order-does-not-matter-we-use-combinations-66168452026-04-23T03:50:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-of-ways-to-choose-4-proteins-from-12-is-given-by-29969592026-04-23T03:50:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-of-ways-to-choose-2-interpretations-from-7-is-97171752026-04-23T03:50:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-order-of-the-effects-matters-and-no-repetition-is-allowed-we-are-looking-for-the-number-of-permutations-of-10-effects-taken-4-at-a-time-97453562026-04-23T03:50:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-of-ways-to-choose-3-drugs-from-9-is-17403352026-04-23T03:50:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-choose-5-participants-from-10-66486042026-04-23T03:50:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-shortest-distance-from-a-point--x0-y0--to-a-line--ax--by--c--0--is-given-by-the-formula-49367542026-04-23T03:52:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-recall-that-in-spherical-coordinates-62908402026-04-23T03:53:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-shortest-altitude-we-first-note-that-the-shortest-altitude-corresponds-to-the-longest-side-which-is-15-cm-we-begin-by-computing-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter-is-4594982026-04-23T03:52:58+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-side-length-of-the-original-equilateral-triangle-be--s--the-area-of-an-equilateral-triangle-is-40366552026-04-23T03:53:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-triangle-have-legs--a---b--and-hypotenuse--c--25--the-inradius--r--5--in-a-right-triangle-the-inradius-is-given-by-33780332026-04-23T03:53:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-triangle-be-isosceles-with-sides--a--10---b--10---c--12--the-semi-perimeter-is-27562352026-04-23T03:53:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-overall-average-word-length-we-compute-the-weighted-average-based-on-the-number-of-words-in-each-manuscript-97067922026-04-23T03:53:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-the-square-region-is-18791062026-04-23T03:53:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-list-all-prime-numbers-less-than-20-99091472026-04-23T03:53:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-given-complex-number-is-in-polar-form-48808202026-04-23T03:53:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-diameter-add-the-two-measurements-and-divide-by-2-1878302026-04-24T18:26:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-find-the-cost-of-one-airplane-76346372026-04-24T18:26:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-treat-the-3-gemstones-as-a-single-super-object-this-gives-6--4--1--11-items-to-arrange-with-the-6-pottery-shards-and-4-statues-being-distinct-the-number-of-arrangements-is--14384672026-04-23T03:56:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-fraction-calibrated-is--78271402026-03-15T17:32:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-rational-functions-the-horizontal-asymptote-depends-on-degrees-of-numerator-and-denominator-here-both-are-degree-1-so-the-asymptote-is-y--85006152026-03-15T17:32:47+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-minimum-distance-from-the-origin-to-the-surface-occurs-when--10895592026-04-23T03:56:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-choosing-3-viral-strains-from-7-with-4-enveloped-e-and-3-non-enveloped-n-we-want-the-probability-that-exactly-2-are-enveloped-54960762026-03-15T17:34:18+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-5-animals-where-there-are-2-identical-males-m-2-identical-females-f-and-1-juvenile-j-the-number-of-distinct-permutations-of-a-multiset-is-given-by-43033302026-03-15T17:34:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-lets-define--an--as-the-number-of-valid-sequences-of-length--n--using-symbols-l-m-h-with-no-two-consecutive-hs-5585832026-03-15T17:34:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-9-rna-segments-where-4-are-identical-of-type-a-3-of-type-b-and-2-of-type-c-the-number-of-distinct-permutations-is-2486942026-03-15T17:34:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-6-events-3-identical-grooming-g-2-identical-play-p-and-1-aggression-a-the-number-of-distinct-permutations-is-68053372026-03-15T17:34:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--an--be-the-number-of-valid-sequences-of-length--n--with-no-two-consecutive-high-h-readings-4000432026-03-15T17:34:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-ensure-diversification-the-vectors-must-be-orthogonal-so-their-dot-product-is-zero-79873542026-04-24T18:27:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-by-expanding-both-terms-68297592026-04-24T18:27:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-sum-of-the-numbers-68736262026-04-23T04:08:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-four-consecutive-subgroup-sizes-be--2n1-2n3-2n5-2n7--these-are-four-consecutive-odd-numbers-84745032026-04-23T04:08:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-the-inscribed-circle-is-32416042026-04-23T04:08:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-note-that--1---x--y--z---1---y--x--z---1---z--x--y--so-the-expression-becomes-27975632026-04-23T04:08:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-set--x--0--in--2x--3y--6--95496622026-04-23T04:17:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-greatest-common-factor-gcf-of-72-and-168-we-first-perform-prime-factorization-71355182026-04-23T04:19:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-count-the-total-number-of-positive-integers-less-than-or-equal-to-60-which-is-60-28916952026-04-23T04:19:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-equivalent-to-finding-the-greatest-common-factor-of-24-and-36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ps-are-indistinguishable-and-each-must-receive-at-least-one-project-the-problem-reduces-to-partitioning-6-distinct-projects-into-4-non-empty-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind--s6-4--the-formula-for--sn-k--is-34813702026-04-23T05:10:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-selecting-3-modules-where-7-are-distinguishable-prototypes-and-5-are-identical-backup-units-since-the-prototype-units-are-unique-and-the-backup-units-are-indistinguishable-we-consider-combinations-based-on-how-many-prototypes-are-selected-from-0-to-3-and-how-many-backup-units-are-used-up-to-3-but-limited-by-availability-and-total-draw-87513062026-04-23T05:14:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-selecting-one-indistinguishable-stem-cell-line-from-4-identical-types-so-only-one-choice-for-the-stem-cell-category-and-three-distinct-neural-templates-from-6-distinguishable-ones-since-the-stem-cell-line-is-identical-across-all-selecting-any-one-gives-one-unique-type-1269802026-04-23T05:14:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-since--fx--is-a-cubic-polynomial-its-third-differences-are-constant-we-compute-the-values-92813152026-04-23T05:17:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-using-the-binomial-probability-formula-with--n--6---p--03--and--k--2--8201332026-04-23T05:20:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-least-common-multiple-lcm-of-12-15-and-20-we-begin-by-factoring-each-number-into-its-prime-components-9682912026-04-23T05:20:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-system-of-congruences-36078012026-04-23T05:20:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-looking-for-the-smallest-integer-greater-than-100-that-is-a-common-multiple-of-18-and-24-but-not-a-multiple-of-30-first-compute-the-least-common-multiple-lcm-of-18-and-24-13735522026-04-23T05:20:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-when-both-cycles-align-we-compute-the-least-common-multiple-lcm-of-36-and-60-66485042026-04-23T05:20:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-compute-the-probability-that-exactly-2-out-of-5-students-receive-their-originally-assigned-experiment-a-fixed-set-of-permutations-with-the-remaining-3-elements-forming-a-derangement-with-no-fixed-points-among-the-other-students-43725652026-04-23T05:20:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-maximum-population-we-first-compute-the-derivative-of--pt--66888822026-04-23T05:24:52+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-progression-rate-is--ln--29601122026-04-23T05:24:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--h--3--into-the-function-72331682026-04-23T05:24:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--t--5--into-the-function-66070602026-04-23T05:24:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-recognize-this-as-a-perfect-square-trinomial-92723572026-04-23T05:24:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area-of-a-circle-is-proportional-to-the-square-of-its-radius-55664922026-04-23T05:24:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area--a--of-a-right-triangle-can-be-written-as-14953932026-03-15T18:59:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-a-combinatorics-problem-with-restrictions-based-on-categories-the-investor-must-choose-3-startups-each-from-a-different-research-area-the-available-areas-are-64735472026-03-15T19:04:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-distinct-permutations-of-5-projects-where-72570152026-03-15T19:04:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-6-initiatives-with-indistinguishability-within-categories-47193152026-03-15T19:04:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-sum-the-values-and-divide-by-the-number-of-phases-6461602026-04-23T05:31:23+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-assigning-12-distinct-zones-to-12-stages-4-of-one-type-high-pressure-3-of-another-injectant-and-5-of-a-third-monitoring-since-the-zones-are-distinguishable-but-stages-of-the-same-type-are-indistinct-in-labeling-the-number-of-distinct-arrangements-is-the-multinomial-coefficient-5906412026-04-24T18:51:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-treat-the-2-specific-concepts-that-must-be-adjacent-as-a-single-block-this-reduces-the-problem-to-arranging-6-items-the-5-other-concepts-and-the-1-block-these-6-items-can-be-arranged-in-6-ways-within-the-block-the-2-concepts-can-be-arranged-in-2-ways-therefore-the-total-number-of-arrangements-is-32217462026-04-24T18:52:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-sphere-of-radius-r-is-98663782026-04-23T05:33:48+00:00daily0.8https://pg.groshi24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lution-this-is-a-permutation-of-a-multiset-9-total-items-with-repetitions-the-number-of-distinct-sequences-is-76347682026-04-23T06:17:20+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-five-distances-in-arithmetic-sequence-be--a---2d-a---d-a-a--d-90213152026-04-23T06:17:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-equation-in-cylindrical-coordinates-79153282026-04-23T06:20:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-sum-of-the-four-numbers-55417032026-04-23T06:20:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-four-consecutive-odd-integers-are--11071-11073-11075-11077--their-sum-is-58831982026-04-23T06:20:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-four-consecutive-odd-integers-be--n-n2-n4-n6--where--n--is-odd-the-product-is-81008422026-04-23T06:21:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-convert--27538--to-base-ten-expand-using-powers-of-8-50491102026-04-23T06:21:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-intersection-set-the-two-equations-equal-40201292026-04-23T06:30:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-evaluate--f3--and--g3--15335512026-04-23T06:30:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-solve-for--x--in-the-equation--2x--3--11--72745812026-04-23T06:30:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-y-intercept-occurs-when--x--0--substitute--x--0--into-the-equation-72513492026-04-23T06:30:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-set-the-two-equations-equal-47240432026-04-23T06:30:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-distributing-6-distinguishable-fish-species-into-4-indistinguishable-zones-with-each-zone-receiving-at-least-one-species-this-is-equivalent-to-finding-the-number-of-ways-to-partition-a-set-of-6-labeled-elements-into-exactly-4-non-empty-unlabeled-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind-denoted--sn-k--here--n--6---k--4--37258552026-04-23T06:34:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-sequences-of-length-7-using-symbols-w-m-c-such-that-47289072026-04-23T06:34:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-hypergeometric-probability-problem-we-have-39158192026-04-23T06:34:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-ways-to-distribute-8-distinguishable-fish-into-3-distinguishable-zones-a-b-c-with-22674462026-04-23T06:34:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-5-distinct-transitions-occurring-at-unique-seconds-in-a-3600-second-interval-with-all-orderings-of-transition-times-uniformly-random-we-are-to-compute-the-probability-that-two-specific-transitions-say-t1-and-t2-occur-at-consecutive-seconds-in-the-sequence--that-is-one-immediately-follows-the-other-either-t1-then-t2-or-t2-then-t1-49782782026-04-23T06:34:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-distinct-circular-arrangements-of-12-pollen-grains--5-oak-o-4-pine-p-and-3-birch-b--such-that-no-two-birch-grains-are-adjacent-all-grains-of-the-same-type-are-indistinguishable-arrangements-are-considered-up-to-rotation-circular-symmetry-90548972026-04-23T06:34:30+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-compute-the-probability-that-in-a-random-assignment-of-difficulty-levels-e-m-h-to-7-tasks-exactly-2-are-hard-and-no-two-hard-tasks-are-adjacent-36217922026-04-24T19:06:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-following-system-of-congruences-24093252026-04-23T06:36:25+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-compute-the-difference-of-squares-65709062026-04-23T06:54:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-square-root-of-a-square-gives-the-absolute-value-96284072026-04-23T06:54:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-cost-per-unit-is-the-total-cost-divided-by-the-number-of-units-35898552026-04-23T06:54:10+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-mutation-appears-every-15th-strand-so-it-occurs-at-multiples-of-15-32832872026-04-23T06:54:04+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-fourth-vertex-be--d--x-y-z--and-denote--a--000---b--220---c--202--in-a-regular-tetrahedron-all-edges-have-equal-length-31216652026-04-23T06:55:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-sequence-be--a---3d-a---d-a--d-a--3d--56675182026-04-24T19:12:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-three-given-vertices-be-a--100-b--010-c--001-and-let-d--xyz-be-the-fourth-vertex-of-a-regular-tetrahedron-so-that-all-edges-have-equal-length-21387972026-03-15T20:44:45+00:00daily0.8https://pg.groshi247.com/highlight/solution-square-both-sides-to-eliminate-the-square-root-84303592026-04-23T07:14:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-horizontal-asymptote-compare-the-degrees-of-the-numerator-and-denominator-both-are-degree-1-so-the-horizontal-asymptote-is-the-ratio-of-the-leading-coefficients-66142272026-04-23T07:14:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--2--into-both-functions-90863102026-04-23T07:31:36+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--x--5--into--fx--22602102026-04-23T07:31:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-speed-is-the-magnitude-of-the-velocity-vector-calculated-as-56882402026-04-23T07:31:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-legs-of-the-right-triangle-be--a--and--b--and-hypotenuse--z--for-any-right-triangle-the-inradius--c--is-given-by-99660712026-04-23T07:31:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-rectangle-is-inscribed-in-the-circle-so-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-63054902026-04-23T07:31:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-asked-to-count-the-number-of-distinct-ways-to-assign-responses-to-100-participants-such-that-40-say-yes-35-say-no-and-25-say-unsure-since-the-participants-are-distinguishable-but-we-are-only-counting-the-number-of-distinct-groupings-ie-how-many-ways-to-divide-the-100-people-into-three-labeled-groups-of-fixed-sizes-this-is-a-multinomial-coefficient-problem-the-number-of-such-response-patterns-is-given-by-20494272026-04-23T07:32:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-assigning-one-of-3-priority-levels-say-a-b-c-to-each-of-7-distinct-signals-in-sequence-with-the-restriction-that-no-two-adjacent-signals-have-the-same-priority-the-first-signal-has-3-choices-for-each-subsequent-signal-since-it-cannot-have-the-same-priority-as-the-previous-one-there-are-2-choices-therefore-the-total-number-of-valid-assignments-is-68005072026-04-23T07:32:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-a-sequence-of-8-video-segments-with-fixed-counts-3-animations-a-2-simulations-s-2-data-overlays-d-and-1-live-action-l-since-the-segments-of-the-same-type-are-indistinct-in-effect-though-filmed-separately-we-are-counting-the-number-of-distinct-permutations-of-a-multiset-the-number-of-distinct-editing-orders-is-96103532026-04-23T07:32:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-regular-hexagon-inscribed-in-a-circle-the-side-length-equals-the-radius-of-the-circle-16352472026-04-23T07:32:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-least-common-multiple-lcm-of-18-24-and-36-we-first-perform-prime-factorization-11855832026-04-23T07:34:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-the-probability-we-first-count-the-positive-divisors-of-120-that-are-less-than-or-equal-to-60-the-prime-factorization-of-120-is-47337512026-04-23T07:34:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-factor-numerator-and-denominator-26830552026-03-15T21:15:39+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-local-minima-first-compute-the-derivative-99065662026-04-23T07:36:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-polynomial-99147032026-04-23T07:36:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-multiply-each-term-in-the-first-polynomial-by-each-term-in-the-second-88599862026-04-23T07:38:27+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-mass-of-one-astrolabe-be--m--kilograms-since-there-are-three-identical-astrolabes-their-total-mass-is--3m--this-is-given-as-27-kilograms-72300022026-04-23T07:41:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-arithmetic-sequence-be--a-a--d-a--2d-a--3d-a--4d--7803882026-04-23T07:43:59+00:00daily0.8https://pg.groshi247.com/highlight/solution-tout-dabord-convertissons-3-livres-en-kilogrammes--34240562026-04-23T07:56:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-volume-formula--v--37402532026-04-23T07:56:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-five-weights-be--14972042026-04-23T07:56:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-largest-square-plot-size-we-compute-the-greatest-common-divisor-gcd-of-72-and-48-36959312026-04-24T19:27:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-an-arithmetic-sequence-with-first-term--a--3--common-difference--d--5--and-last-term--l--198--90510932026-04-23T07:59:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-multiply-both-sides-by--x---1x--1--to-eliminate-denominators-87454442026-04-23T08:00:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-composition--fgx--is-defined-when--gx--is-defined-and-when--gx--lies-in-the-domain-of--f--since--fx--is-a-polynomial-its-defined-for-all-real-numbers-so-we-only-need--gx--to-be-defined-4397622026-04-23T08:00:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-the-expression-under-the-square-root-must-be-non-negative-65618112026-04-23T08:00:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-given--x--y--z--1--note-that--y--z--1---x---z--x--1---y---x--y--1---z--so-the-expression-becomes-66051112026-04-23T08:00:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-solve-this-first-compute-each-square-and-sum-them-20261902026-04-23T08:37:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-find-the-least-common-multiple-of-12-and-18-68270362026-04-23T08:37:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-treat-each-species-as-a-single-block-there-are-3-ways-to-arrange-the-3-blocks-chimpanzees-gorillas-orangutans-within-each-block-the-animals-can-be-permuted-among-themselves-5-for-chimpanzees-3-for-gorillas-and-4-for-orangutans-the-total-number-of-arrangements-is-48653182026-04-23T08:37:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-multinomial-problem-where-we-count-the-number-of-permutations-of-a-multiset-the-total-number-of-sequences-is-12584492026-04-23T08:37:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-two-digit-number-be--n--we-are-given-31421602026-04-23T08:40:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--n--be-the-two-digit-number-such-that-71940302026-04-23T08:40:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-no-two-adjacent-blocks-can-be-the-same-color-and-we-have-equal-numbers-of-red-and-blue-blocks-the-only-possible-alternating-patterns-are-rbrbrb-and-brbrbr-these-are-the-only-two-sequences-that-satisfy-the-condition-and-use-3-red-and-3-blue-blocks-no-other-arrangement-avoids-adjacent-duplicates-for-example-starting-with-b-leads-to-brbrbr-and-starting-with-r-leads-to-rbrbrb-no-other-starting-point-avoids-conflict-thus-there-are-exactly-2-valid-arrangements-67497842026-04-23T08:42:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-area--a--of-a-right-triangle-is-given-by--a--18963812026-04-23T08:42:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume-of-a-hemisphere-of-radius--r--is-67234522026-04-23T08:42:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-semiperimeter--s--of-the-triangle-is-10047312026-04-23T08:42:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-volume--v--of-a-cone-is--33099882026-04-23T08:42:53+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-length-and-width-of-the-rectangle-be--l--and--w--the-perimeter-is-35744982026-04-23T08:42:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-looking-for-the-smallest-positive-integer--n--such-that-27312472026-04-23T08:44:37+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-a-system-of-three-dot-product-equations-77248822026-04-24T19:42:07+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-side-length-of-the-original-equilateral-triangle-be--s--the-area--a--of-an-equilateral-triangle-is-given-by-78345872026-04-23T09:02:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-right-triangle-have-legs--a---b--and-hypotenuse--z--the-area--a--of-the-triangle-is-38645322026-04-23T09:02:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-average-ph-we-sum-the-three-expressions-and-divide-by-3-81338042026-03-15T22:45:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-is-the-sum-divided-by-3-87236132026-03-15T22:45:31+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-average-is-the-sum-divided-by-2-35472722026-03-15T22:45:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-us-define-a-substitution-to-simplify-the-expression-set-46293382026-04-23T09:08:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-an-equilateral-triangle-the-altitude-from-one-vertex-to-the-opposite-side-splits-the-triangle-into-two-congruent-right-triangles-using-the-pythagorean-theorem-the-altitude--h--satisfies-5524242026-04-23T09:08:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-right-triangle-the-radius--r--of-the-circumscribed-circle-is-half-the-length-of-the-hypotenuse-first-compute-the-hypotenuse--c--78499972026-04-23T09:08:12+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-closest-point-on-the-line--y--2x--1--to--3--2--is-found-by-minimizing-the-distance-squared-82690322026-04-23T09:10:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-12-simulations-where-each-of-the-4-scenarios-appears-exactly-3-times-this-is-a-multinomial-coefficient-problem-the-number-of-distinct-sequences-is-the-multinomial-coefficient-63737902026-04-23T09:11:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-fx-is-a-cubic-polynomial-its-third-differences-are-constant-compute-the-values-8877522026-04-23T09:11:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-v3-integrate-vt-91938232026-04-23T09:11:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-particle-is-at-rest-when-vt--st--0-56248802026-04-24T19:46:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-cauchy-schwarz-inequality-in-the-engel-form-titus-lemma-48466072026-04-24T19:46:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-number-of-distinct-arrangements-is-calculated-by-dividing-the-factorial-of-the-total-number-of-items-by-the-factorial-of-the-number-of-identical-items-here-there-are-5-total-permutations-but-since-2-charts-are-identical-we-divide-by-2-thus-the-number-of-distinct-arrangements-is--38678142026-04-24T19:48:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-convert-the-base-5-number-32145-to-base-10-expand-it-using-powers-of-5-35036682026-04-24T19:48:08+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-first-time-both-signals-align-is-the-least-common-multiple-lcm-of-18-and-30-factor-70427662026-03-15T22:59:14+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-number-of-clay-pots-be--7k--and-the-number-of-wooden-tools-be--4k--given-that--7k--28--solve-for--k--16088452026-04-23T09:16:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-first-six-positive-even-integers-are-1321662026-04-23T09:16:11+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-constraints-on-the-sequence-of-5-moves-32558432026-04-23T09:21:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-arranging-9-artifacts-where-84175322026-04-23T09:21:48+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-identify-the-prime-numbers-between-1-and-8-53348512026-04-23T09:21:49+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-x-be-the-integer-then-78186362026-04-24T19:49:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-smallest-four-digit-number-divisible-by-both-12-and-18-compute-the-least-common-multiple-lcm-of-12-and-18-51640172026-04-24T19:49:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-the-pressure-function-95342762026-04-24T19:50:24+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-each-hour-mia-tutors-her-net-earnings-are-41661022026-04-23T09:32:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-by-expanding-the-left-hand-side-99296062026-04-23T09:32:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-sequence-is-arithmetic-with-first-term--a--3--common-difference--d--4--and-last-term--l--43--the-formula-for-the--n--th-term-is--an--a--n-1d--setting--an--43--25533322026-04-23T09:47:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-subtract--2x--from-both-sides-53878292026-04-23T09:47:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-test-divisibility-by-primes-221-is-not-divisible-by-2-3-or-5-dividing-by-13-30551992026-04-23T09:47:41+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-the-order-of-addition-matters-we-are-looking-for-the-number-of-permutations-of-5-catalysts-taken-2-at-a-time-this-is-given-by-60536312026-04-23T09:47:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-classic-surjection-problem-distributing-6-distinguishable-participants-to-4-distinguishable-designs-such-that-no-design-is-left-untested-the-number-of-such-surjective-functions-is-given-by-19871222026-04-23T09:47:40+00:00daily0.8https://pg.groshi247.com/highlight/solution-in-a-circular-arrangement-with-alternating-roles-we-fix-one-persons-position-to-account-for-rotational-symmetry-fix-a-government-official-at-one-seat-then-the-seats-alternate-g-i-g-i-g-i-the-remaining-2-government-officials-can-be-arranged-in-2-ways-and-the-3-industry-leaders-in-3-ways-so-the-total-number-of-distinct-arrangements-is-24359652026-04-23T09:47:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-algorithm-evaluates-configurations-based-on-departure-times-and-traffic-conditions-since-the-choices-are-independent-the-total-number-of-configurations-is-the-product-of-the-number-of-departure-times-and-traffic-conditions-22150252026-04-24T19:56:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-problem-reduces-to-arranging-a-multiset-of-elements-there-are-a-total-of-2--3--1--6-transmission-events-with-2-indistinct-water-based-3-indistinct-contact-based-and-1-indistinct-airborne-method-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-19862762026-04-24T19:56:02+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-a-permutation-of-a-multiset-there-are-4--3--2--9-fish-total-with-4-indistinct-river-a-3-indistinct-river-b-and-2-indistinct-river-c-tags-the-number-of-distinct-sequences-is-16329852026-04-24T19:56:00+00:00daily0.8https://pg.groshi247.com/highlight/solution-this-is-equivalent-to-arranging-a-multiset-of-4-urgent-u-3-routine-r-and-2-follow-up-f-visits-the-total-number-of-distinct-schedules-is-61270892026-04-24T19:56:03+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-angle-between-two-unit-vectors-is-given-by-the-absolute-difference-of-their-direction-angles-15682612026-03-15T23:45:28+00:00daily0.8https://pg.groshi247.com/highlight/solution-since--gx--is-a-cubic-polynomial-we-assume-73942592026-04-23T10:01:38+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-two-functions-87069162026-04-24T20:02:19+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--s--e1--e2---p--e1-e2--from-the-equations-32491552026-04-23T10:19:13+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-greatest-common-factor-gcf-of-36-and-48-factor-both-numbers-42828002026-04-23T10:42:26+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-analyze-the-given-equation-in-spherical-coordinates-1326672026-04-23T10:45:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-substitute--d--120---vp--6--and--vs--4--into-the-formula-52365722026-04-23T11:04:51+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--x--be-the-successful-outcomes-in-the-additional-10-hours-total-successful-outcomes--065-imes-40--x--26--x--total-hours--50--the-new-success-rate--4816132026-03-16T00:53:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--r--be-the-pre-update-retention-rate-after-the-update-retention-is--r--012--given--1000-imes-r--012--850--solve-31188532026-03-16T00:53:16+00:00daily0.8https://pg.groshi247.com/highlight/solution-let--p---a--and--i--represent-the-cost-of-protein-algae-paste-and-insects-in-cents-the-system-is-82136652026-04-23T11:06:29+00:00daily0.8https://pg.groshi247.com/highlight/solution-start-by-squaring-both-sides-of-the-given-equation-15137302026-04-23T11:10:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-quadratic-formula-7542862026-04-23T11:10:33+00:00daily0.8https://pg.groshi247.com/highlight/solution-first-list-all-prime-numbers-less-than-30-55411492026-04-24T20:17:42+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-arithmetic-progression-be--a---2d-a---d-a-a--d-a--2d-a--3d--where--a--is-the-fourth-term-and--d--is-the-common-difference-84312642026-04-23T11:16:01+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-given-summation-can-be-simplified-using-partial-fraction-decomposition-7705872026-04-23T11:20:50+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-formula-for-the-magnitude-of-the-sum-of-two-vectors-59587572026-04-23T11:23:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-determine-the-total-number-of-distinct-identity-markers-we-apply-the-multiplication-principle-each-marker-consists-of-one-triangle-one-circle-and-one-square-since-the-choices-are-independent-52492152026-04-24T20:20:21+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-to-count-the-number-of-sequences-of-length-4-using-3-types-of-pulses-pressure-temperature-vibration-such-that-each-sequence-contains-at-least-one-of-each-type-39531902026-03-16T01:13:32+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-distributing-3-distinct-crafts-to-12-distinguishable-families-with-fixed-group-sizes-5-to-pottery-4-to-weaving-and-3-to-carving-74855372026-03-16T01:13:35+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-original-side-length-be--s--the-area--a--of-an-equilateral-triangle-is-given-by-the-formula-28970092026-04-24T20:21:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-legs-of-the-right-triangle-be--a--and--b--and-the-hypotenuse--z--the-area--a--of-the-triangle-is-85473532026-04-24T20:21:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-legs-be--a--9---b--12--and-hypotenuse--c--first-compute--c--93526432026-04-24T20:21:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-triangle-have-legs--a--15---b--and-hypotenuse--h--the-inradius--r--of-a-right-triangle-is-12766272026-04-24T20:21:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-for-an-equilateral-triangle-circumscribed-about-a-circle-ie-the-incircle-has-radius--r--6--the-relationship-between-the-side--s--and-the-inradius-is-54589942026-04-24T20:21:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-compute-the-diagonal-using-the-pythagorean-theorem-7818772026-04-24T20:21:09+00:00daily0.8https://pg.groshi247.com/highlight/solution-let-the-legs-be--a--5---b--and--c--13--use-pythagoras-to-find--b--29295292026-04-24T20:21:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-since-c-divides-ab-in-the-ratio-23-the-position-vector-of-c-is-given-by-the-section-formula-73057142026-04-24T20:21:05+00:00daily0.8https://pg.groshi247.com/highlight/solution-the-shortest-distance-from-a-point-to-a-line-is-along-the-perpendicular-the-given-line-is-y--2x--1-so-slope-m--2-the-perpendicular-slope-is---4569142026-04-24T20:21:18+00:00daily0.8https://pg.groshi247.com/highligh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solution-we-are-looking-for-the-least-common-multiple-lcm-of-18-and-24-68942502026-03-16T03:43:56+00:00daily0.8https://pg.groshi247.com/highlight/solution-use-the-formula-for-the-magnitude-of-a-vector-871182026-03-16T03:43:54+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-that-x-y-z--0-and-x--y--z--1-and-we-are-to-minimize-the-expression-72658492026-04-23T11:59:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-we-are-given-a-recurrence-type-functional-equation-71482922026-04-23T11:59:34+00:00daily0.8https://pg.groshi247.com/highlight/solution-to-find-the-intersection-we-solve-the-system-of-equations-5114352026-04-24T20:37:24+00:00daily0.8