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calculus-max & min

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an open box with a square base is to be constructed from 48 square inches of material. What dimensions will produce a box with maximum volume?

  • calculus-max & min -

    The way I read your question:
    You have 48 square inches available in total to make the box.

    so let the base be x by x inches and the height y inches
    total area available = x^2 + 4xy = 48
    y = (48 - x^2)/(4x)

    volume = x^2 y
    = x^2 (48 - x^2)/(4x)
    = 12x - x^3/4

    d(volume)/dx = 12 - (3/4)x^2
    = 0 for a max of volume
    (3/4)x^2 = 12
    x^2 = 16
    x = √16 = 4

    if x = 4, then y = (48 - 16)/16 = 2

    the box should have a base of 4 inches by 4 inches and a height of 2 inches

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