From a thin piece of cardboard 72 in by 72 in, square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume?
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If the side of the squares cut out from the corners is x inches, then the dimensions of the box are length=width=72-2x and height=x, and volume V(x)=(72-2x)²x=4(36-x)²x=4x(1296-72x+x²)=5184x-288x²+4x³.

The graph shows V(x) where x is in feet. The maximum and minimum can be clearly seen.

dV/dx=5184-576x+12x²=0 at extrema. This quadratic reduces to:

432-48x+x²=0=(x-36)(x-12), making x=36 or 12. We can reject x=36 because V(36)=0 (minimum).

V(12)=27648 cubic inches=16 cubic feet.

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