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 = (Hh)/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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