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A box with a square base and open top must have a volume of 32000cm3 find the dimensions of the box that minimize the amount of material used?
Suppose the sides of the base are x cm, and the height is h cm.
Then the volume = x2*h = 32000 cm3
The amount of material used for the base and 4 sides (no top) is x2 + 4xh cm2
So the requirement is to minimise x2 + 4xh
subject to x2h = 32000 or h = 32000/x2
So minimise x2 + 4x*32000/x2
or x2 + 128000/x
Setting the derivative = 0,
2x - 128000/x2 = 0
or 2x3 = 128000
or x3 = 64000 or x = 40 cm.
And then h = 32000/x2 = 20 cm.
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