Calculus

An open top box with a square base is to be made so that it holds 3 cubic feet. Assuming the material on the base costs $3 per square foot and the material on the sides costs $2 per square foot, determine the size of the base that minimizes the total cost.

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  1. let be base be x ft by x ft
    let the height be y ft

    (x^2)(y) = 3
    y = 3/x^2

    Cost = 3(x^2) + 2(4xy)
    = 3x^2 + 8x(3/x^2)
    = 3x^2 + 24/x

    d(cost)/dx = 6x - 24/x^2
    = 0 for a min of cost

    6x = 24/x^2
    x^3 = 4
    x = cuberoot(4)
    = appr 1.6 ft

    the base should be appr 1.6 ft by 1.6 ft

    (the height is appr 1.19 ft)

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