Question

The radius of a right circular cylinder of maximum volume which can be inscribed in a sphere of radius R is

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

Answer 1

Answer: (C)

$\sqrt{\frac{2}{3}}R$

$r^2+\frac{h^2}{4}=R^2$

$Volume=\pi r^{2}h$

$v=\pi (R^2-\frac{h^2}{4})h$

$\frac{dv}{dh}=\pi (R^2-\frac{h^2}{4}-\frac{2h^2}{4})$

$R^2=\frac{3h^2}{4}$

$h=\frac{2R}{\sqrt{3}}$

$r=\sqrt{\frac{2}{3}}R$