A cone and a hemisphere have equal curved surface areas and equal radii. Find the ratio of their volumes.

Let the common radius be r.

Curved surface area (CSA) of cone
$CSA = \pi r l$

Curved surface area (CSA) of hemisphere
$CSA = 2\pi r^2$

Given both curved surface areas are equal:

$\pi r l = 2\pi r^2$

$l = 2r$

For a cone:

$l^2 = r^2 + h^2$

$(2r)^2 = r^2 + h^2$

$4r^2 = r^2 + h^2$

$h^2 = 3r^2$

$h = r\sqrt{3}$

Volume of cone

$V_c = \frac{1}{3}\pi r^2 h$

$V_c = \frac{1}{3}\pi r^2 (r\sqrt{3}) = \frac{\sqrt{3}}{3}\pi r^3$

Volume of hemisphere

$V_h = \frac{2}{3}\pi r^3$

Ratio of volumes

$V_c : V_h = \frac{\sqrt{3}}{3}\pi r^3 : \frac{2}{3}\pi r^3$

Cancel common terms:

$= \sqrt{3} : 2$

Ratio of volumes (Cone : Hemisphere) = $\sqrt{3} : 2$