The interior of a right, circular cone is 8 inches tall with a 2-inch radius at the opening. The interior of the cone is filled with ice cream, and the cone has a hemisphere of ice cream exactly covering the opening of the cone. What is the volume of ice cream? Express your answer in terms of $\pi$.
The volume of the cone is $\frac13\cdot4\pi\cdot8=\frac{32}{3}\pi$ cubic inches, and the volume of the hemisphere is $\frac23\cdot8\pi=\frac{16}{3}\pi$. The sum is \[
\left(\frac{16}{3}+\frac{32}{3}\right)\pi=\boxed{16\pi}.
\]