SOLUTION: Information: A rectangular box which is open at the top, has a square base with sides of x cm and a height of h cm. Question: An open rectangular box

Algebra ->  Volume -> SOLUTION: Information: A rectangular box which is open at the top, has a square base with sides of x cm and a height of h cm. Question: An open rectangular box       Log On


   

Question 889186: Information: A rectangular box which is open at the top, has a square base with sides of x cm and a height of h cm. Question: An open rectangular box is to be constructed so that it can hold 256cm cube when completely full. What must its dimensions be so that as little material as possible is used in its construction?
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Volume, h%2Ax%5E2=256.

Total one-sided area, x%5E2%2B4xh, represents the amount of material making the box. A is the amount of material.

A=x%5E2%2B4xh

Solve the volume equation for h.
h=256%2Fx%5E2 and substitute into A equation.
A=x%5E2%2B4x%28256%2Fx%5E2%29
highlight%28A=x%5E2%2B4%2A256%2Fx%29.

The derivative way to continue for the minimum is dA%2Fdx=0. Not sure if you are studying Calculus so here is a try with just arithmetic.

A=%28x%2Ax%5E2%2B4%2A256%29%2Fx-----------No. I am uncomfortable with this method.

DIFFENTIATING A:
dA%2Fdx=2x%2B%28-1%294%2A256%2Ax%5E%28-2%29
dA%2Fdx=2x-4%2A256%2Fx%5E2
%282x%5E3-4%2A256%29%2Fx%5E2
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dA%2Fdx=0=2x%5E3-4%2A256, we only need to focus on the numerator to be zero.
2x%5E3=4%2A256
x%5E3=2%2A256
highlight%28x=8%29----------assuming that this extreme will be for the minimum area.
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h=256%2F64
highlight%28h=4%29-----------the height dimension for this expected minimum material area.