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)
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 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.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F401c5c17-00ca-4e37-8d4f-d84636e780cf%2Fd323276c-8322-4473-8940-1157f88198fe%2Frmqdiv_processed.png&w=3840&q=75)
Transcribed Image Text: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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