Question

A solid is hemispherical at the bottom and conical above it. If the surface areas of the two parts are equal, then the ratio of its radius and the height of its conical part is
(a) 1 : 3
(b) $1:\sqrt{3}$
(c) $\sqrt{3}:1$
(d) 1 : 1

Answer 1

(b) $1:\sqrt{3}$
Surface area of the hemisphere$=2\mathrm{\pi }{r}^2$
Surface area of the cone$=\mathrm{\pi }rl$
$=\mathrm{\pi }r\sqrt{r^2+h^2}$
Therefore,

$$\begin{gathered} 2\mathrm{\pi }{r}^2=\mathrm{\pi }r\sqrt{r^2+h^2} \\ \Rightarrow 2r=\sqrt{r^2+h^2} \\ \Rightarrow 4r^2=r^2+h^2\left[\text{Squaring both sides}\right] \\ \Rightarrow 3r^2=h^2 \\ \Rightarrow \frac{r^2}{h^2}=\frac{1}{3} \\ \Rightarrow \frac{r}{h}=\frac{1}{\sqrt{3}} \\ \Rightarrow \frac{r}{h}=1:\sqrt{3} \end{gathered}$$