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

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

$R$

$\frac{R}{2}$

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

answer is C.

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Detailed Solution

$R^{2} = r^{2} + \frac{h^{2}}{4} \Rightarrow v = π (R^{2} h - \frac{h^{3}}{4})$

$\Rightarrow$ For maxima or minima $\frac{d v}{d h} = 0 \Rightarrow h = \frac{2 R}{\sqrt{3}}$
At $h = \frac{2 R}{\sqrt{3}}, \frac{d^{2} v}{d h^{2}} < 0, ∴ r = \sqrt{\frac{2}{3}} R$

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