2004 AIME I Problem 11

A solid in the shape of a right circular cone is 4 inches tall and its base has a 3 -inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid \mathcal{C} and a frustum-shaped solid \mathcal{F}, in such a way that the ratio between the areas of the painted surfaces of \mathcal{C} and \mathcal{F} and the ratio between the volumes of \mathcal{C} and \mathcal{F} are both equal to k. Given that k=m / n, where m and n are relatively prime positive integers, find m+n.