# The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is:Option 1)$\sqrt{6 }$Option 2)$\frac{2}{3}\sqrt{3}$Option 3)$2\sqrt{3}$Option 4)$\sqrt{3}$

S solutionqc

radius of cylinder $=\sqrt{R^{2}-\left ( \frac{h}{2} \right )^{2}}$

volume of cylinder $=\pi r^{2}h=\pi \left ( R^{2}-\frac{h}{4}^{2} \right )\times h$

for maximum volume

$\frac{dv}{dh}=\pi \left ( R^{2}-\frac{3h}{4}^{2} \right )=0$

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

$h=2\sqrt{3}$

Option 1)

$\sqrt{6 }$

Option 2)

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

Option 3)

$2\sqrt{3}$

Option 4)

$\sqrt{3}$

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