### Instruction: 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 $75\%$ of the volume of the frozen ice cream. What is the ratio of the cone's height to its radius? (Note: A cone with radius $r$ and height $h$ has volume $\pi r^2 h/3$, and a sphere with radius $r$ has volume $4\pi r^3/3$.).### Response: To find the ratio of the cone's height to its radius, I need to express $h$ in terms of $r$.
I also need to use the fact that the melted ice cream fills the cone, so the volume of the cone must equal $75\%$ of the volume of the sphere.
I can write this as an equation: $\pi r^2 h/3 = 0.75 \cdot 4\pi r^3/3$.
Simplifying this equation, I get $h = 0.75 \cdot 4r = 3r$.
This means that the cone's height is three times its radius, so the ratio of the cone's height to its radius is $3/1$.