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A box with a square base and open top must have a volume of 186624 cm. We wish to find the dimensions of the box that minimize the amount of material used.

A box with a square base and open top must have a volume of 186624 cm. We wish to find the dimensions of the box that minimize the amount of material used.

Mathematics For Machine Technology
Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter59: Areas Of Rectangles, Parallelograms, And Trapezoids
Section: Chapter Questions
Problem 79A
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A box with a square base and open top must have a volume of 186624 cm°. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of æ.] Simplify your formula as much as possible. A(x) = Next, find the derivative, A'(x). A'(x) = Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x.] A'(x) = 0 when a = We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"(x) Evaluate A"(x) at the x-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that value, so the zero of A'(x) must indicate a local minimum for A(x). (Your boss is happy now.) Question Help: Message instructor Add Work Submit Question
Transcribed Image Text:A box with a square base and open top must have a volume of 186624 cm°. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of æ.] Simplify your formula as much as possible. A(x) = Next, find the derivative, A'(x). A'(x) = Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x.] A'(x) = 0 when a = We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"(x) Evaluate A"(x) at the x-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that value, so the zero of A'(x) must indicate a local minimum for A(x). (Your boss is happy now.) Question Help: Message instructor Add Work Submit Question
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