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Calculus

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An open top bos with a square base is to be constructed from 192 sq ft of material. What should be the dimensions of the box if the volume is to be a maximum? What is the maximum volume?

  • Calculus - ,

    base --- x by x ft
    height -- y ft

    area = 4xy + x^2
    192 = 4xy + x^2
    y = (192 - x^2)/(4x)

    V = x^2 y
    = x^2 (192-x^2)/(4x)
    = 48x - (1/4)x^3
    dV/dx = 48 - (3/4)x^2 = 0 for a max of V
    (3/4)x^2 = 48
    x^2 = 64
    x = 8
    then y = (192-64)/(32) = 4

    max Volume = (8^2)(4) = 256

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