Question

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

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

Answer 1

The correct option is D $\sqrt{2}:1$

Curved surface Area of a Hemisphere $=2\pi r^2$

Curved surface Area of a cone $=\pi rs$

Let the radius of the hemisphere base radius of cone their height are also equal of the hemisphere $=r$

Therefore height of the cone, $h=r$

Ratio curved surface Area $=2\pi r^2:\pi rs$

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

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

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

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

$=2:\sqrt{2}$

$=\sqrt{2}:1$

$\therefore$ The ratio of curved surface area $=\sqrt{2}:1$