= Optimal cylinder inscribed in a cone
{title2=$x_{V}=h/3,\quad x_A=h/2$}
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 $V'=\pi\tan^2\alpha(h-x)(h-3x)$, 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.
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