The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is:

1. $\mathrm{ }\sqrt{3}$
2. $\frac{2}{3}\sqrt{3}$
3. $\sqrt{6}$
4. $2 \sqrt{3}$

Solution

From the diagram, we get radius of cylinder $=\sqrt{R^2-{\left(\frac{h}{2}\right)}^2}$
$\Rightarrow$ Volume of cylinder $V=\pi \left(R^2-\frac{h^2}{4}\right)h$
For maximum volume: $\frac{dV}{dh}=0$
$\Rightarrow \pi \left(R^2-\frac{{3h}^2}{4}\right)=0$
$\Rightarrow h^2=\frac{4R^2}{3}=\frac{4\times 9}{3}=12$
$\Rightarrow h=2\sqrt{3}$

Asked in: JEE Main 2019 (08 Apr Shift 2)

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