An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. The sphere has radius $3$ inches and the cone has height $8$ inches. When the ice cream melts, the melted ice cream occupies $50\%$ of the volume of the frozen ice cream. What fraction of the cone does the melted ice cream fill?

1. The sphere has volume $\frac43\pi(3)^3=36\pi$, so the melted ice cream has volume $50\%\cdot36\pi=18\pi$.
2. The cone has radius $3$ and height $8$, so its volume is $\frac13\pi(3)^2(8)=24\pi$.
3. The melted ice cream fills $\dfrac{18\pi}{24\pi}=\boxed{\dfrac34}$ of the cone.