problem: Two cones have the same volume. If one has a base with radius 3 times as large as the other's and a height of 24 inches, how many inches tall is the other?
  
  Note: The volume of a cone is $\frac{1}{3} \pi r^2 h,$ where $r$ is the radius and $h$ is the height.
solution: The volume is proportional to the square of the base radius and to the height, so if these have the same volume, their heights are inversely proportional to the square of the radii. This means that with a radius 1/3 as big as the first, the second cone has a height of $24\left(\frac1{1/3}\right)^2=24\cdot9=\boxed{216}$ inches.