Question

A cylinder is inscribed in a sphere of radius r. What is the radius of the cylinder of maximum volume?

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

Answer 1

Answer: (B)

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

taking an angle $\theta$ with the vertical lines

we found that ,

$r=R\sin \theta$

$h=R\cos \theta$

now, Volume = $\pi R^3{\sin }^2\theta \cos \theta$

for $\theta$ to be maximum $\frac{dV}{d\theta }=0$

$\Rightarrow V^{\prime }=\pi R^3(2\sin \theta {\cos }^2\theta -{\sin }^3\theta )$

$\Rightarrow 0=(2{\cos }^2\theta -{\sin }^2\theta )$ or $\sin \theta =0$

second condition is assbsured this can't be happen

so, we will go with

$\Rightarrow 0=(2-2{\sin }^2\theta -{\sin }^2\theta )$

$\Rightarrow \sin \theta =\frac{\sqrt{2}}{\sqrt{3}}$

hence,

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