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7 August, 13:33

A box with a square base and open top must have a volume of 4,000 cm3. Find the dimensions of the box that minimize the amount of material used.

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  1. 7 August, 13:43
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    Let the side of the square base be x, and the height of the box be h.

    The material of the base is x^2, and the material of the four sides is 4xh.

    4000 = hx^2

    h = 4000/x^2

    The total material is

    M = x^2 + 4x (4000/x^2) = x^2 + 16000/x

    Take the first derivative of M and set equal to 0.

    M' = 2x - 16000/x^2 = 0

    Multiply by x^2:

    2x^3 = 16000

    x^3 = 8000

    x = 20; h = 10; M = 1200
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