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

The curved surface area of a hemisphere is given by 2πr², where r is the radius of the hemisphere.

The curved surface area of a cone is given by πrl, where r is the radius and l is the slant height of the cone.

Given that the hemisphere and the cone have equal bases, their radii are equal. Also, their heights are equal. In a cone, the slant height, l, is the hypotenuse of a right triangle whose other two sides are the radius, r, and the height, h. Since the height of the cone is equal to the radius (because the height of the cone is equal to the height of the hemisphere which is a radius), the slant height l is √2r (by Pythagoras theorem).

Substituting l = √2r in the formula for the curved surface area of the cone, we get πr*√2r = √2πr².

Therefore, the ratio of the curved surface area of the hemisphere to the cone is 2πr² : √2πr² = 2 : √2 = √2 : 1 after rationalizing the denominator.

So, the ratio of their curved surface areas is √2 : 1.