A box with a square base and open top must have a volume of 296352 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): %3D Next, find the derivative, A'(x). = (x),V Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x] A'(x) = 0 when a 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"() at the x-value you gave above.
A box with a square base and open top must have a volume of 296352 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): %3D Next, find the derivative, A'(x). = (x),V Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x] A'(x) = 0 when a 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"() at the x-value you gave above.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![A box with a square base and open top must have a volume of 296352 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),V
Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x]
A'(x) = 0 when a =
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F51cb03b1-322d-48ee-bdd2-053ff413b0a8%2F55f722ab-1ff6-4f61-8025-8fb8dbc022ef%2F3p8x5z_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A box with a square base and open top must have a volume of 296352 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),V
Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x]
A'(x) = 0 when a =
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.
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