Calculus

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. let the sides of the base be x cm each
    let the height be y

    V= x^2 y
    4000/x^2 = y

    surface area = x^2 + 4xy
    = x^2 + 4x(4000/x^2
    = x^2 + 16000/x
    d(surface area)/dx = 2x - 16000/x^2 = 0 for a min surface area

    2x = 16000/x^2
    x^3 = 16000
    x = appr 25.2 cm
    y = 4000/25.2^2 = appr 6.3 cm

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  2. V= x^2 y
    4000/x^2 = y

    surface area = x^2 + 4xy
    = x^2 + 4x(4000/x^2
    = x^2 + 16000/x
    d(surface area)/dx = 2x - 16000/x^2 = 0 for a min surface area

    2x = 16000/x^2
    x^3 = 8000
    x = 20cm
    y = 4000/400 = 10cm

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