Question

A cylinder of the greatest volume is inscribed in a sphere. How many times is the volume of the sphere greater than that of the cylinder?

Answer 1

Let R be the radius of sphere and r and h be the radius and height of the cylinder respectively

Let $\alpha$ be the angle between radius and axis of cylinder. Then

$r=R\sin \alpha$ & $h=2R\cos \alpha$

$V=\pi r^{2}h=\pi {\left(R\sin \alpha \right)}^{2}2R\cos \alpha =2\pi R^3{\sin }^2\alpha \cos \alpha$

$\frac{dV}{d\alpha }=0\Rightarrow 2\pi R^3[{\sin }^3\alpha -2\sin \alpha {\cos }^2\alpha ]=0$

$2\pi R^3\ne 0\therefore {\sin }^3\alpha -2\sin \alpha {\cos }^2\alpha =0\Rightarrow {\sin }^3\alpha =2\sin \alpha {\cos }^2\alpha$

$\Rightarrow {\tan }^2\alpha =2\Rightarrow \tan \alpha =\sqrt{2}\Rightarrow \alpha =54.74°$

$\therefore r=R\sin 54.74°=0.816R$

$h=2R\cos 54.74°=1.155R$

Volume of cylinder$=\pi \times {\left(0.816R\right)}^2\times 1.155R=2.42R^3$

Volume of sphere$=\frac{4}{3}\pi R^3=\frac{4}{3}\times \frac{22}{7}\times R^3=4.20R^3$

Hence, value of sphere is $(4.20R^3-2.42R^3=1.78R^3)$ greater than that of cylinder.