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A cylinder is inscribed in a right circular cone of height 5.5 and radius (at the base) equal to 7. What are the dimensions of such a cylinder which has maximum volume?

  • Calculus -

    Center a cylinder of radius r on the axis of the cone of height H and base radius R. If the cylinder has height h, using similar triangles,

    H/R = (H-h)/r
    h = H - Hr/R

    volume of cylinder is

    v = πr2h = πr2(H - Hr/R)
    = πHr2 - πH/R r3

    dv/dr = 2πHr - 3πH/R r2
    = πHr(2 - 3r/R)

    max at r = 2R/3, h = H/3

    volume = πr2h = π(2R/3)2(H/3) = 4πR2H/27

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