A box with a square base and open top must have a volume of 42592 cm³. 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 , the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of .] Simplify your formula as much as possible. A(z) = 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 ² -] A'(x) = 0 when I = We next have to make sure that this value of a 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.)

A box with a square base and open top must have a volume of 42592 cm³. 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 , the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of .] Simplify your formula as much as possible. A(z) = 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 ² -] A'(x) = 0 when I = We next have to make sure that this value of a 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.)

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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Similar questions

A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in inches. a. Using the fact that the volume of the can is 25 cubic inches, express h in terms of x. b. Express the total surface area S of the can in terms of x.

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