Calculus

A rectangular box with a square base and cover is to have a volume of 2500 cubic feet. If the cost per square foot for the bottom is $2, for the top is $3, and for the sides is $1, what should the dimensions be in order to minimize the cost?

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  1. If the base has side x, and the height is h, then

    x^2h = 2500
    h = 2500/x^2

    So, the cost to make the box is

    c = 2(x^2) + 3(x^2) + 1*4*xh
    = 5x^2 + 10000/x
    dc/dx = 10x-10000/x^2
    dc/dx=0 when x=10

    so, the box is 10x10x25 ft

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