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Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 96 in. Find the dimensions of a rectangular package that has a square cross section and the largest volume that may be sent via priority mail. Hint: The length plus the girth is 4x + h (see the figure below).

  • Math - ,

    well, we know

    4x+h = 96, so h=96-4x
    v = hx^2
    = x^2(96-4x)
    = 96x^2 - 4x^3
    = 4(24x^2 - x^3)

    dv/dx = 4(48x - 3x^2)
    = 12x(16 - x)

    dv/dx = 0 when x = 0 or 16
    so, h = 96-4*16 = 32

    So, if the package is 16x16x32
    v = 8192 in^3

    Now, technically, 4x+h <= 96, but it should be clear that max volume occurs when 4x+h is as large as possible.

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