Question

Show that when curve surface of a cylinder inscribed in a sphere of radius $r$ is maximum, then height of the cylinder is $\sqrt{2}r$.

Answer 1

Radius of sphere $=r$

Let $\angle OAB=\varphi$

So radius of cylinder, $r^{\prime }=r\cos \varphi$

$h/2=r\sin \varphi$

$h=2r\sin \varphi$

C & A$=2\pi r^{\prime }h$

C & A$=2\pi \left(2r\sin \varphi \right)\left(r\cos \varphi \right)$

C & A $=2\pi r^2\sin 2\varphi$

C & A is maximum where $\sin 2\varphi$ is max.

i.e. $\sin 2\varphi =\pi /2$

$2\varphi =\pi /2$

$\varphi =\pi /4$.

$\therefore h=2r\sin \varphi$

$h=\sqrt{2}r$