question_answer

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is [JEE Main 8-4-2019 Afternoon]

A) \[2\sqrt{3}\]

B)               \[\sqrt{3}\]

C) \[\sqrt{6}\]                  

D)   \[\frac{2}{3}\sqrt{3}\]

Correct Answer: A

Solution :

\[h=2r\sin \theta \] \[a=2r\cos \theta \] \[\text{v}=\pi {{(r\cos \theta )}^{2}}(2rsin\theta )\] \[\text{v}=2{{\pi }^{2}}{{\cos }^{2}}\theta \sin \theta \] \[\frac{\text{dv}}{d\theta }=\pi {{r}^{3}}(-2\cos \theta {{\sin }^{2}}\theta +{{\cos }^{3}}\theta )=0\] or\[\tan \theta =\frac{1}{\sqrt{2}}\] \[\because \]\[h=2\times 3\times \frac{1}{\sqrt{3}}\] \[=2\sqrt{3}\]