Grade 12 Calculus

A rectangular piece of paper with perimeter
100 cm is to be rolled to form a cylindrical
tube. Find the dimensions of the paper that
will produce a tube with maximum volume.

I have made it up to getting an equation for V(w).

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  1. if the sheet is x by y, and is rolled along the y axis,

    2x+2y = 100
    v = pi r^2 y
    where 2pi r = x, or r = x/(2pi), so
    v = pi (x/(2pi))^2 (100-2x)/2
    = x^2(50-x)/(4pi)

    dv/dx = x(100-3x)/(4pi)
    dv/dx=0 when x = 100/3

    at that point, max v is pi*(50/3)^3

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