If the ratio of volumes of two cones is 2 : 3 and the ratio of the radii of their bases is 1 : 2, then the ratio of their height will be :

let their radii be x and 2x, and their heights be h and H respectively
Then,
$$\eqalign{ & \frac{{\frac{1}{3} \times \pi \times {x^2} \times h}}{{\frac{1}{3} \times \pi \times {{\left( {2x} \right)}^2} \times H}} = \frac{2}{3} \cr & \Leftrightarrow \frac{h}{H} = \frac{2}{3} \times 4 \cr & \Leftrightarrow \frac{h}{H} = \frac{8}{3}\,Or\,8:3 \cr} $$