A box with a square base and open top must have a volume of 108000 cm'. We wish to find the dimensions of the box that minimize the amount of material used. ст 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 x²] A'(x) = 0 when r = We next have to make sure that this value of a 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.
A box with a square base and open top must have a volume of 108000 cm'. We wish to find the dimensions of the box that minimize the amount of material used. ст 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 x²] A'(x) = 0 when r = We next have to make sure that this value of a 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.
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 108000 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 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) =
%3D
Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x]
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 r-value you gave above.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18dba5cd-32eb-4f32-9d0b-47f44f310a0a%2F9bdd2fb1-8a89-45f1-b4e1-2c2a0a7d5cfb%2Fi9t4xzp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A box with a square base and open top must have a volume of 108000 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 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) =
%3D
Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by x]
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 r-value you gave above.
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