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

• A

  $2 \sqrt{3} a$
• B

  $\sqrt{3} a$
• Correct answer
  C

  $\frac{2 a}{\sqrt{3}}$
• D

  $\frac{a}{\sqrt{3}}$

Solution

Let the radius and height of the cylinder are and $h$, respectively.
In $\triangle A O M$ $\therefore$ $$ \begin{array}{l} r^{2}+\left(\frac{h^{2}}{4}\right)=a^{2} \\ h^{2}=4\left(a^{2}-r^{2}\right) \end{array} $$ Now, $V=\pi r^{2} h=\pi\left(a^{2} h-\frac{1}{4} h^{3}\right)$ For max or min, $$ \begin{aligned} & \frac{d V}{d h} &=\pi\left(a^{2}-\frac{3}{4} h^{2}\right)=0 \\ \Rightarrow & h &=\left(\frac{2}{\sqrt{3}}\right) a \end{aligned} $$ Now, $\quad \frac{d^{2} V}{d h^{2}}=-\frac{6 h}{4} < 0$ So, $V$ is maximum at $h=\frac{2 a}{\sqrt{3}}$.

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