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 :

Answer: Option D

let their radii be x and 2x, and their heights be h and H respectively
Then,
$$\begin{array}{rl}& \frac{\frac{1}{3}\times \pi \times x^2\times h}{\frac{1}{3}\times \pi \times {\left(2x\right)}^2\times H}=\frac{2}{3}\\ & \iff \frac{h}{H}=\frac{2}{3}\times 4\\ & \iff \frac{h}{H}=\frac{8}{3}Or8:3\end{array}$$