A right circular cylinder is inscribed in a right circular cone of height 12 and base radius 6, with the cylinder's base on the cone's base. What is the maximum possible volume of the cylinder?

Core Concept

By similar triangles, a cylinder of radius $r$ has height $h=12-2r$, so $V=\pi r^2(12-2r)$. $V'=6\pi r(4-r)$, so $V$ is maximized at $r=4$, giving $V=\pi(16)(4)=64\pi$.