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

• A

  $\mathrm{ }\sqrt{3}$
• B

  $\frac{2}{3}\sqrt{3}$
• C

  $\sqrt{6}$
• Correct answer
  D

  $2 \sqrt{3}$

Solution

From the diagram, we get radius of cylinder $=\sqrt{R^2-{\left(\frac{h}{2}\right)}^2}$
$\Rightarrow$ Volume of cylinder $V=\pi \left(R^2-\frac{h^2}{4}\right)h$
For maximum volume: $\frac{dV}{dh}=0$
$\Rightarrow \pi \left(R^2-\frac{{3h}^2}{4}\right)=0$
$\Rightarrow h^2=\frac{4R^2}{3}=\frac{4\times 9}{3}=12$
$\Rightarrow h=2\sqrt{3}$

Hence, the correct answer is option: 'D'.