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. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $50\%$ of the volume of the frozen ice cream. What is the ratio of the cone's radius to its height? (Note: a cone with radius $r$ and height $h$ has volume $\pi r^{2}h/3$ and a sphere with radius $r$ has volume $4\pi r^3/3$.)