A box with a square base and open top must have a volume of 32000 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 I, 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(x) = Next, find the derivative, A'(x). A'(x) =
A box with a square base and open top must have a volume of 32000 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 I, 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(x) = Next, find the derivative, A'(x). A'(x) =
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Question 5**
A box with a square base and open top must have a volume of \(32000 \, \text{cm}^3\). 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.*
\[ A(x) = \text{________} \]
Next, find the derivative, \( A'(x) \).
\[ A'(x) = \text{________} \]
Now, calculate when the derivative equals zero, that is, when \( A'(x) = 0 \). *[Hint: multiply both sides by \( x^2 \).*
\[ A'(x) = 0 \text{ when } x = \text{________} \]
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 \( A''(x) \).
\[ A''(x) = \text{________} \]
Evaluate \( A''(x) \) at the \( x \)-value you gave above.
\[ \text{________} \]
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%2Fc29c810b-c4cc-47ca-9579-623151f0bc7a%2F31a6d330-2d5a-4d68-a3ee-902bd1813130%2F6rfbp99_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 5**
A box with a square base and open top must have a volume of \(32000 \, \text{cm}^3\). 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.*
\[ A(x) = \text{________} \]
Next, find the derivative, \( A'(x) \).
\[ A'(x) = \text{________} \]
Now, calculate when the derivative equals zero, that is, when \( A'(x) = 0 \). *[Hint: multiply both sides by \( x^2 \).*
\[ A'(x) = 0 \text{ when } x = \text{________} \]
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 \( A''(x) \).
\[ A''(x) = \text{________} \]
Evaluate \( A''(x) \) at the \( x \)-value you gave above.
\[ \text{________} \]
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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