A hemisphere and a cone have equal bases. If their heights are also equal, then the ratio of their curved surface will be :

Answer: Option A

Let,
$$\begin{array}{rl}& OP=OQ=OR=r\\ & \therefore OR=h=r\end{array}$$
∴ Curved surface area of the hemisphere = $$2\pi r^2$$
Curved surface area of a cone = $$\pi rl$$
Where,
$$\begin{array}{rl}& l=\sqrt{h^2+r^2}\\ & =\sqrt{r^2+r^2}\\ & =r\sqrt{2}\end{array}$$
∴ Required ratio :
$$\begin{array}{rl}& =\frac{2\pi r^2}{\pi rl}\\ & =\frac{2\pi r^2}{\pi r\times r\sqrt{2}}\\ & =\frac{2}{\sqrt{2}}\\ & =\frac{2\times \sqrt{2}}{\sqrt{2}\times \sqrt{2}}\\ & =\frac{2\sqrt{2}}{2}\\ & =\frac{\sqrt{2}}{1}Or\sqrt{2}:1\end{array}$$