The maximum volume (in cu. units) of the cylinder which can be inscribed in a sphere of radius 12 units is

• A

  $384 \sqrt{3} \pi$
• Correct answer
  B

  $768 \sqrt{3} \pi$
• C

  $\frac{768 \pi}{\sqrt{3}}$
• D

  $\frac{1152 \pi}{\sqrt{3}}$

Solution

$12^2=r^2+\left(\frac{h}{2}\right)^2$
$\begin{aligned} & \Rightarrow V=\pi r^2 h \\ & \Rightarrow V=\pi\left(144-\frac{h^2}{4}\right) h \\ & \Rightarrow V=144 \pi h-\frac{\pi}{4} h^3 \\ & \Rightarrow \quad \frac{d V}{d h}=144 \pi-\frac{3 \pi}{4} h^2\end{aligned}$ $\begin{aligned} & \Rightarrow \quad \frac{d V}{d h}=0 \Rightarrow 144 \pi=\frac{3 \pi}{4} h^2 \\ & \Rightarrow \quad h^2=48 \times 4 \Rightarrow h=8 \sqrt{3} \\ & \therefore \quad 12^2=r^2+48 \Rightarrow r^2=96\end{aligned}$ Volume $=\pi r^2 h=\pi \times 96 \times 8 \sqrt{3}=768 \sqrt{3} \pi \mathrm{cm}^3$.

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