Question

A cylinder with the greatest lateral area is inscribed in a sphere. Find the ratio of the radius of the sphere to that of the base of the cylinder.

Answer 1

Let radius of sphere be R, and radius and height of cylinder be r and h respectively.

$\alpha$ be the angle between radius and axis of cylinder.

Then $S=2\pi rh$

$r=R\sin \alpha$ & $h=2R\cos \alpha$

$\therefore S=2\pi R^2\sin \alpha \cos \alpha$

$\frac{dS}{d\alpha }=0$

$\Rightarrow 2\pi R^2[{\cos }^2\alpha -{\sin }^2\alpha ]=0$

$\Rightarrow {\cos }^2\alpha -{\sin }^2\alpha =0...........[\because 2\pi R^2\ne 0]$

$\Rightarrow {\tan }^2\alpha =1\Rightarrow \alpha =45°$

$\therefore r=R\sin 45°\Rightarrow r=\frac{R}{\sqrt{2}}\Rightarrow \frac{R}{r}=\sqrt{2}$