Lesson Minimize the cost of an aquarium with the given volume
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<H2>Minimize the cost of an aquarium with the given volume</H2> <H3>Problem 1</H3>An aquarium that holds 40 cubic meters of water is to be made such that the length of its base is twice the width. If material for the base costs $20 per square meter, and the material for the sides costs $16 per square meter, find the cost of the materials for the cheapest such aquarium. <B>Solution</B> <pre> Let w be the width of the aquarium; then its length is 2w, according to the problem. If the height is h, then the volume is V = w*(2w)*h = 2w^2*h, so 2w^2*h = 40 cubic meters, or w^2*h = 20 cubic meters. It gives h = {{{20/w^2}}}. (1) The base area is w*(2w) = 2w^2; the base cost is 20*2w^2 = 40w^2 dollars. The lateral area is (w + 2w + w + 2w)*h = 6wh. The lateral sides cost is 16*6wh = 96wh dollars. The total cost is C = 40w^2 + 96wh = substitute h from (1) = {{{40w^2 + 96w*(20/w^2)}}} = {{{40w^2}}} + {{{(96*20)/w}}} = {{{40w^2}}} + {{{1920/w}}}. So, we want to minimize this function C(w) = {{{40w^2}}} + {{{1920/w}}}. (2) To find the minimum, take the derivative and equate it to zero. Doing it, you will get, step by step 80w = {{{1920/w^2}}} 80w^3 = 1920 w^3 = 1920/80 = 24 w = {{{root(3,24)}}} = 2.88. Thus the width is 2.885 m; the length is twice of it 2*2.885 = 5.77 m. the height is {{{20/w^2}}} = {{{20/2.88^2}}} = 2.41 m. The cheapest cost is C = formula (2) = {{{40*2.88^2}}} + {{{1920/2.88}}} = 998.43 dollars. </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>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/Maximize-the-area-of-a-trapezoid.lesson>Maximize the area of a trapezoid </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-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.
