Consider a cylinder and a cone with a common base such that the volume of the
part of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder.
Coz the small cone is similar to the large one, so the ratio of the height of these two cones is $ 1: \sqrt[3]{2}$.
So the required ratio is $ \sqrt[3]{2}: \sqrt[3]{2}-1$.