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A box with a square base and open top must have a volume of 340736 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 r, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of r.] Simplify your formula as much as possible. A(z) = %3D Next, find the derivative, A'(x). A'(z) = %3D Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by a.] %3D A'(z) = 0 when z = %3! We next have to make sure that this value of I gives a minimum value for the surface area. Let's use the second derivative test. Find A"(1). A"(z) = Evaluate A"(z) at the z-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(z) is concave up around that value, so the zero of A'(z) must indicate a local minimum for A(z). (Your boss is happy now.) Add Work Check Answer

A box with a square base and open top must have a volume of 340736 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 r, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of r.] Simplify your formula as much as possible. A(z) = %3D Next, find the derivative, A'(x). A'(z) = %3D Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by a.] %3D A'(z) = 0 when z = %3! We next have to make sure that this value of I gives a minimum value for the surface area. Let's use the second derivative test. Find A"(1). A"(z) = Evaluate A"(z) at the z-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(z) is concave up around that value, so the zero of A'(z) must indicate a local minimum for A(z). (Your boss is happy now.) Add Work Check Answer

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 340736 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 x.] 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 a.] A'(x) = 0 when x = 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(a) 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.) Add Work Check Answer
Transcribed Image Text:A box with a square base and open top must have a volume of 340736 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 x.] 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 a.] A'(x) = 0 when x = 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(a) 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.) Add Work Check Answer
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