x y and z
 

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The area of the 4 sides totals 4xy and the area of the base is x² so the total surface area is 4xy+x²=48, therefore y=(48-x²)/4x. The volume V=x²y=x²(48-x²)/4x=x(48-x²)/4 or 12x-x³/4. dV/dx=12-3x²/4=0 when V is max or min.

So 3x²/4=12, x²=16, x=4ft (reject x=-4). Therefore y=(48-16)/16=2ft. Volume is 32 cu ft.

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