[Math] show that the height of the cylinder of maximum volume that can be inscribed within a cone of height $h$ is $\frac{h}3$.

Question

show that the height of the cylinder of maximum volume that can be inscribed within a cone of height $h$ is $\frac{h}3$.

I have tried solving this sum but am unable to substitute the radius of the cylinder in terms of $h$.

Answer 1

Hint:

Look at the figure. With: $$ BH=h \quad HC=R \quad FG=x \quad HG=r $$

from the similarity of the triangles $BHC$ and $FGC$ we have:

$$ h:x=R:(R-r) $$ so that $r=\frac{R}{h}(h-x)$

Now you can express the volume of the cylinder as $V=\pi r^2x=\pi \frac{R^2}{h^2}(h-x)^2x $.

Maximize this function.