A circle of radius $3\sqrt3$ inches has a $120^\circ$ sector cut out. The remaining $240^\circ$ sector, together with its interior, is rolled up to form the lateral surface of a right circular cone by taping its two straight edges together. What is the volume of the cone in cubic inches?

1. The remaining $240^\circ$ arc length is $\frac{2}{3}\cdot2\pi(3\sqrt3)=4\pi\sqrt3$, so the cone's base radius is $2\sqrt3$. Its slant height is $3\sqrt3$, hence its height is $\sqrt{27-12}=\sqrt{15}$.
2. The volume is $\frac13\pi(2\sqrt3)^2\sqrt{15}=\boxed{4\pi\sqrt{15}}$ cubic inches.