jmc

The two bases are circles, and the area of a circle is $\pi r^2$. If the area of the upper base (which is also the base of the small cone) is $25\pi$ sq cm, then its radius is $5$ cm, and the radius of the lower base is $15$ cm. The upper base, therefore, has a radius that is $\frac{1}{3}$ the size of the radius of the smaller base. Because the slope of the sides of a cone is uniform, the frustum must have been cut off $\frac{2}{3}$ of the way up the cone, so $x$ is $\frac{1}{3}$ of the total height of the cone, $H$. We can now solve for $x$, because we know that the height of the frustum, $24$ cm is $\frac{2}{3}$ of the total height. $$\begin{array}{rl}\frac{2}{3}H& =24\\ H& =36\\ x& =H\times \frac{1}{3}\\ x& =36\times \frac{1}{3}\\ x& =12\end{array}$$ Therefore, the height of the small cone is $\boxed{12}$ centimeters.