Lesson Minimize surface area of a rectangular box with the given volume
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<H2>Minimize surface area of a rectangular box with the given volume</H2> <H3>Problem 1</H3>A candy company wants to reduce the amount of packaging they use, by minimizing the surface area. If they still want the candy package to hold a volume of 250 cm3, what is the minimum surface area? Assume the candy comes packaged in rectangular prisms. <B>Solution</B> <pre> Let x, y and z be the dimensions of a rectangular prism. Then its volume is xyz = 250 cm^3, (1) a fixed value, and the problem wants we minimize the surface area 2xy + 2yz + 2xz under this restriction (1). It is the same as to minimize the function of 3 independent variables F(x,y,z) = xy + yz + xz under restriction (1). Using the restriction, we can reduce function F(x,y,z) to function of two independent variables f(x,y) = xy + {{{250/x}}} + {{{250/y}}} and look for the minimum of this function. To find its minimum, we take partial derivatives of f(x,y) over x and y and equate them to zero. It gives us this system of equations y - {{{250/x^2}}} = 0, (2) (x-derivative) x - {{{250/y^2}}} = 0. (3) (y-derivative) From (2) and (3) x^2*y = 250 (4) x*y^2 = 250. (5) Dividing (4) by (5), we get {{{x/y}}} = 1, or x = y. Working similarly with the other pair of independent variables, we can get similarly x = z, which tells us that the minimum surface value is achieved for the cube x = y = z. Its dimension is {{{x^3}}} = 250, or x = {{{root(3,250)}}} = 6.3 (rounded to one decimal place). <U>ANSWER</U>. The minimum surface area is a cube with the edge size of {{{root(3,250)}}} = 6.3 cm. <U>CHECK</U>. {{{6.3^3}}} = 250.047 cm^3. </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/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/Couple-of-non-standard-Calculus-problems.lesson>Couple of non-standard Calculus problems</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-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.
