Given a right circular cone, you put an upside-down cone inside it so that its vertex is at the center of the base of the larger cone and its base is parallel to the base of the larger cone. If you choose the upside-down cone to have the largest possible volume, what fraction of the volume of the larger cone does it occupy? (Let H and R be the height and base radius of the larger cone, and let h and r be the height and base radius of the smaller cone. Hint: Use similar triangles to get an equation relating h and r.)

I'm guessing I should start with this? I'm not sure how to express the function using only one variable, though.

$$ V = \pi r^ 2 \frac{h}{3} \\ r \in (0, R) \\ h \in (0, H) $$