Lesson Maximize the area of a trapezoid
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<H2>Maximize the area of a trapezoid</H2> <H3>Problem 1</H3>Three sides of a trapezoid are each 10 cm long. How long is the 4th side, when the area of the trapezoid has the greatest value? <B>Solution</B> <pre> It is clear, that under given conditions, the trapezoid is isosceles with the shorter base of 10 cm long and lateral sides of 10 cm long, each. See my sketch below. The trapezoid is isosceles. The upper base is 10 cm long; the lower base is 10+2x cm long. 10 +------------------+ /| \ / | \ 10 / | \ 10 / | \ / | \ +------------------------------+ <- x -> The mid-line is {{{(10+(10+2x))/2}}} = 10+x cm long. The height is {{{sqrt(10^2-x^2)}}}. The area of the trapezoid is A(x) = {{{(10+x)*sqrt(100-x^2)}}}. To find the maximum of A(x), take the derivative {{{(dA)/(dx)}}} = {{{sqrt(100-x^2)}}} - {{{(1/2)*((10+x)/sqrt(100-x^2))*(-2x)}}} {{{(dA)/(dx)}}} = {{{sqrt(100-x^2)}}} - {{{((10+x)*x)/sqrt(100-x^2)}}} Equate it to zero {{{sqrt(100-x^2)}}} - {{{((10+x)*x)/sqrt(100-x^2)}}} = 0. Simplify and find x {{{sqrt(100-x^2)}}} = {{{((10+x)*x)/sqrt(100-x^2)}}} {{{100-x^2)}}} = (10+x)*x {{{100-x^2}}} = {{{10x + x^2}}} {{{2x^2 + 10x - 100}}} = 0 {{{x^2 + 5x - 50}}} = 0 (x-5)*(x+10) = 0 Of these two roots, x= 5 and x= -10, take the positive one, which gives the maximum to the trapezoid's area. <U>ANSWER</U>. The area of trapezoid is maximum when the 4-th side is 10+2*5 = 20 cm long. </pre> My other lessons on Calculus word problems at this site are - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-ladder-foot-slides-on-the-ground-txt.lesson>A ladder foot slides on the ground</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-rate-of-change-of-some-processes.lesson>Finding rate of change of some processes</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-defined-by-complicated-expression.lesson>Find the derivative of a function defined by complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Taking-derivative-of-a-function-which-is-defined-implicitly.lesson>Taking derivative of a function, which is defined implicitly</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-satisfying-given-functional-equation.lesson>Find the derivative for a function satisfying given functional equation</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/09-add-Find-the-Range-of-f%28x%29-=-%285cos%28x%29%29-div-%28x-%2B-1%29%29cos%28x%29.lesson>Find the range of f(x) = (5*cos(x))/(x + 1)), x >=0</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-tricky-Calculus-problem-on-derivative-and-anti-derivative.lesson>A tricky Calculus problem on derivative and anti-derivative</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Couple-of-non-standard-Calculus-problems.lesson#google_vignette>Couple of non-standard Calculus problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-the-minimum-of-a-function-defined-on-a-curve-in-the-coordinate-plane.lesson>Finding the minimum of a function defined on a curve in a coordinate plane</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Tricky-solution-to-find-the-maximum-of-a-function-defined-by-a-complicated-expression.lesson>Tricky solution to find the maximum of a function defined by a complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximmize-the-volume-of-an-open-box.lesson>Maximize the volume of an open box</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-surface-area-of-a-rectangular-prizm-with-the-given-volume.lesson>Minimize surface area of a rectangular box with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-the-cost-of-an-aquarium-with-the-given-volume.lesson>Minimize the cost of an aquarium with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-surface-area-of-a-conic-paper-cup-with-the-given-volume.lesson>Minimize surface area of a conical paper cup with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-volume-of-a-solid-body-obtained-by-rotation-the-area-about-an-axis.lesson>Find the volume of a solid obtained by rotation of some plane shape about an axis</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-the-volume-of-a-solid-body-mentally.lesson>Finding the volume of a solid body mentally</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/OVERVIEW-of-my-lessons-on-Calculus-word-problems.lesson>OVERVIEW of my lessons on Calculus word problems</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
