Question

A solid is hemispherical at the bottom and conical above. If the curved surface area of the two parts are equal, then what is the ratio of the height and radius of the conical part?

• A. $\frac{1}{3}$
• B. $\frac{\sqrt{3}}{1}$
• C. $\frac{3}{1}$
• D. $\frac{1}{\sqrt{3}}$

Answer 1

The correct option is B

$\frac{\sqrt{3}}{1}$

Note that the Curved Surface Area of a cone of radius $rcm$ and slant height $lcm$ = $\pi rl$
And, CSA of a hemisphere of radius $rcm$ = $2\pi r^2$
Given that $\pi \times r\times l=2\pi r^2$
$⟹l=2r$.........(1)
But by Pythagoras theorem, we have $l=\sqrt{r^2+h^2}$
Thus from (1), we have $\sqrt{r^2+h^2}=2r$
Squaring on both sides, we have $r^2+h^2=4r^2$
$⟹3r^2=h^2$
$⟹\frac{h^2}{r^2}=\frac{3}{1}$
$⟹\frac{h}{r}=\frac{\sqrt{3}}{1}$