A box with a square base and open top must have a volume of 55296 cm. We wish to find the dimensions of the box that minimize the amount of material used.
A box with a square base and open top must have a volume of 55296 cm. We wish to find the dimensions of the box that minimize the amount of material used.
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter9: Surfaces And Solids
Section9.CR: Review Exercises
Problem 20CR: If the radius length of one sphere is three times as long as the radius length of another sphere,...
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![A box with a square base and open top must have a volume of 55296 cm'. We wish to find the
dimensions of the box that minimize the amount of mnaterial used.
First, find a formula for the surface area of the box in terms of only a, 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 z?]
A'(r) = 0 when a =
We next have to make sure that this value of e gives a minimum value for the surface area. Let's use
the second derivative test. Find A"(x).
A"(z) =
Evaluate A"() at the r-value you gave above.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9f44317-7e6b-4763-9be3-6ee721697e17%2F1d75dfd3-b6aa-4c3a-945d-216baa6c0ccf%2Fmuxnmel_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A box with a square base and open top must have a volume of 55296 cm'. We wish to find the
dimensions of the box that minimize the amount of mnaterial used.
First, find a formula for the surface area of the box in terms of only a, 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 z?]
A'(r) = 0 when a =
We next have to make sure that this value of e gives a minimum value for the surface area. Let's use
the second derivative test. Find A"(x).
A"(z) =
Evaluate A"() at the r-value you gave above.
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