Cone Frustum Height Problem

A right circular cone has a radius of $3$ cm and height of $10$ cm. The cone is cut by a plane parallel to the base forming a smaller cone and a frustum. If the smaller cone and the frustum have equal lateral surface areas, what is the height (in cm) of the smaller cone? The answer is in the form $a\sqrt{b}$ for where $a$ and $b$ are positive integers, and $b$ is square-free. Input the answer as the sum $a+b$.