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

The hemisphere and the cone have equal bases

$$=$$>  Radius of the hemisphere = Radius of the cone

Let Radius of the hemisphere = Radius of the cone = r

Height of hemisphere is equal to the radius of the of the hemisphere

Given, Heights of hemisphere and cone are equal

$$=$$>  Height of the cone (h) = r

$$\therefore\ $$Ratio of their curved surfaces = $$2\pi\ r^2:\pi\ r\left(\sqrt{r^2+h^2}\right)$$

$$=2\pi\ r^2:\pi\ r\left(\sqrt{r^2+r^2}\right)$$

$$=2\pi\ r^2:\sqrt{2}\pi\ r^2$$

$$=2:\sqrt{2}$$

$$=\sqrt{2}.\sqrt{2}:\sqrt{2}$$

$$=\sqrt{2}:1$$

Hence, the correct answer is Option C