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Optimization of Box Dimensions

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Suppose that you are to make a rectangular box with a square base from two different materials. The material for the top and four sides of the box costs $1/ft^2; the material for the base costs $2/ft^2. FInd the dimensions of the box of greatest possible volume if you are allowed to spend $144 for the material to make it.

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Let the base be a square of side x and height of the box be y.

Then, volume, V = x^2y. For maximum volume, dV/dx = 0

x^2(dy/dx) + 2xy = 0 ...

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