Posted by **Anonymous** on Friday, October 28, 2011 at 12:59pm.

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 -
**Steve**, Friday, October 28, 2011 at 3:30pm
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 = πr^{2}h = πr^{2}(H - Hr/R)

= πHr^{2} - πH/R r^{3}

dv/dr = 2πHr - 3πH/R r^{2}

= πHr(2 - 3r/R)

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

volume = πr^{2}h = π(2R/3)^{2}(H/3) = 4πR^{2}H/27

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