Question

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $=\sqrt{3}$ is :

• A. $\frac{4}{3}\sqrt{3}\pi$
• B. $2\pi$
• C. $\frac{8}{3}\sqrt{3}\pi$
• D. $4\pi$

Answer 1

Answer: (D)

$4\pi$

${r^2=R}^2-\frac{h^2}{4}$
$V_c=\pi r^{2}h$
$=\pi {(R}^2-\frac{h^2}{4})h=\pi R^{2}h-\frac{{\pi h}^3}{4}$
$\frac{dV}{dh}=\pi R^2-\frac{{3\pi h}^2}{4}$
${\left(\frac{dV}{dh}\right)}_{R=\sqrt{3}}=3\pi -\frac{{3\pi h}^2}{4}$
For maximum or minimum volume,$\frac{dV}{dh}=0$
$\Rightarrow h=2$

Also, $\frac{dV}{dh}$ changes sign from positive to negative in the neighbourhood of $h=2$. Hence, $h=2$ is a maximum point.
$\Rightarrow r=\sqrt{2}$
$\Rightarrow V=4\pi$ cubic units