Optimal cylinder inscribed in a cone ($x_V=h/3,x_A=h/2$)

For a coaxial cylinder standing on the base of a right circular cone of height $h>0$ and semi-vertical angle $0<\alpha <\pi /2$, similar triangles give radius $r=(h-x)\tan \alpha$ at cylinder height $x$. Its volume is $V=\pi x(h-x{)}^2{\tan }^2\alpha$. Differentiation gives $V^{\prime }=\pi {\tan }^2\alpha (h-x)(h-3x)$, so the unique positive-volume maximum has $x=h/3$ and occupies $4/9$ of the cone's volume. Its curved surface area is instead $A=2\pi x(h-x)\tan \alpha$, maximized at $x=h/2$. Changing the optimized quantity changes the optimal shape, even though the geometric constraint is identical.