please check the screenshot an answer A box with a square base and open top must have a volume of 42592 cm. We wish to find the dimensionsof the box that minimize the amount of material used.First, find a formula for the surface area of the box in terms of only æ, the length of one side of the squarebase.[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 æ?A' (x) = 0 when æ =We next have to make sure that this value of x gives a minimum value for the surface area. Let's use thesecond derivative test. Find A"(x).A"(2) =Evaluate A"(x) at the x-value you gave above.NOTE: Since your last answer is positive, this means that the graph of A(æ) is concave up around thatvalue, so the zero of A' (x) must indicate a local minimum for A(x). (Your boss is happy now.)

In this question, the concept of maxima and minima is applied. Maxima and Minima The extrema of a…

the box that minimize the amount of material used. st, find a formula for the surface area of the box in terms of se. int: use the volume formula to express the height of the box mplify your formula as much as possible. (z) = | xt, find the derivative, A'(x). "(z) =| '(x)

the box that minimize the amount of material used. st, find a formula for the surface area of the box in terms of se. int: use the volume formula to express the height of the box mplify your formula as much as possible. (z) = | xt, find the derivative, A'(x). "(z) =| '(x)

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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Question

please check the screenshot an answer

A box with a square base and open top must have a volume of 42592 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 æ, 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 æ?
A' (x) = 0 when æ =
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"(2) =
Evaluate A"(x) at the x-value you gave above.
NOTE: Since your last answer is positive, this means that the graph of A(æ) 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.)

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