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a rectagular sheet of metal has perimeter 36 meters. maximize the volume if the sheet is rolled into a cylinder

  • calculus -

    if the sheet is x by y,

    x = 18-y

    rolled parallel to x,

    v = pi r^2 x
    where r = y/2pi
    v = pi (y/2pi)^2 (18-y)
    = y/4pi (18-y)

    dv/dy = (9-y)/2pi
    dv/dy = 0 when y=9

    as expected, it's a 9x9 square sheet.

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