Question

The height of right circular cylinder of maximum volume inscribed in a sphere of diameter $2a$ is

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

Answer 1

The correct option is C $\frac{2a}{\sqrt{3}}$

Let the radius and height of the cylinder are $r$ and $h$, respectively.
In $△AOM$,
$r^2+\left(\frac{h^2}{4}\right)=a^2$
$\therefore h^2=4(a^2-r^2)$
Now, $V=\pi r^{2}h=\pi (a^{2}h-\frac{1}{4}{h}^3)$
For max or min,
$\frac{dV}{dh}=\pi (a^2-\frac{3}{4}{h}^2)=0$
$\Rightarrow h=\left(\frac{2}{\sqrt{3}}\right)a$
Now, $\frac{d^{2}V}{d{h}^2}=-\frac{6h}{4}<0$
So, V is maximum at $h=\frac{2a}{\sqrt{3}}$.