Question

A solid hemisphere is mounted on a solid cylinder, both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume, then the ratio of the height of the cylinder to the common radius is

• A. $1:1$
• B. $1:2$
• C. $2:1$
• D. $\sqrt{2}:1$

Answer 1

The correct option is A $1:1$

$R=$ Radius of hemisphere and cylinder

$\Rightarrow$ $S=2\pi R^2+2\pi Rh+\pi R^2$

$\Rightarrow$ $\frac{S-2\pi R^2-\pi R^2}{2\pi R}=h$

$\therefore$ $h=\frac{S-3\pi R^2}{2\pi R}$ ----- ( 1 )

$V=\frac{2}{3}\pi R^3+\pi R^{2}h$

$\Rightarrow$ $V=\frac{2}{3}\pi R^3+\pi R^2\times \left(\frac{S-3\pi R^2}{2\pi R}\right)$ [ From ( 1 ) ]

$\Rightarrow$ $\frac{dV}{dR}=2\pi R^2+\frac{S}{2}-\frac{9\pi }{2}{R}^2$

For maximum and minimum $\frac{dV}{dR}=0$

$\Rightarrow$ $0=2\pi R^2+\frac{S}{2}-\frac{9\pi }{2}{R}^2$

$\Rightarrow$ $0=-5\pi R^2+S$

$\Rightarrow$ $5\pi R^2=S$

$\Rightarrow$ $5\pi R^2=3\pi R^2+2\pi Rh$

$\Rightarrow$ $2R=2h$

$\therefore$ $h:R=1:1$