Question

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius of $3$ is :

• A. $\frac{2}{3}\sqrt{3}$
• B. $2\sqrt{3}$
• C. $\sqrt{6}$
• D. $\sqrt{3}$

Answer 1

The correct option is B $2\sqrt{3}$


Let $\angle AOB=\theta$ and radius of sphere is $r$.
$\therefore$ Radius of cylinder $=r\sin \theta$
and Height of cylinder $=2r\cos \theta$

Volume of cylinder, $V=\pi (r\sin \theta {)}^2\times 2r\cos \theta$
$\Rightarrow V=\pi 2r^3{\sin }^2\theta \cos \theta$
$\frac{dV}{d\theta }=\pi 2r^3(2\sin \theta {\cos }^2\theta -{\sin }^3\theta )$

For maxima or minima, $\frac{dV}{d\theta }=0$
$\Rightarrow \sin \theta (2{\cos }^2\theta -{\sin }^2\theta )=0$
$\Rightarrow \tan \theta =\sqrt{2}$
$\Rightarrow \cos \theta =\frac{1}{\sqrt{3}}$
$\therefore$ Height of cylinder $=2r\cos \theta =2\times 3\times \frac{1}{\sqrt{3}}=2\sqrt{3}$