A box with a square base and open top must have a volume of 171500 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 x.] A'(x) = 0 when z = 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.
A box with a square base and open top must have a volume of 171500 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 x.] A'(x) = 0 when z = 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.
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 171500 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 x.]
A'(x) = 0 when z =
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 r-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.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1477abb9-d688-4e57-8562-687cc6a266e8%2F666fc4b7-fc98-4479-b6bb-41bcf80cdace%2Fezmgxq4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A box with a square base and open top must have a volume of 171500 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 x.]
A'(x) = 0 when z =
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 r-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.)
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