If the radii of the bases of a cylinder and a cone are in the ratio $\displaystyle 3$ : $\displaystyle 4$ and their heights are in the ratio $\displaystyle 2$ : $\displaystyle 3$, find the ratio of their volumes.

Let radius of cylinder \(\displaystyle =r_{1}\) & height of cylinder \(\displaystyle =h_{1}\) Let radius of cone \(\displaystyle =r_{2}\) & height of cone \(\displaystyle =h_{2}\) \[\frac{r_{1}}{r_{2}}=\frac{3}{4}, \quad \frac{h_{1}}{h_{2}}=\frac{2}{3} \] \(\displaystyle \frac{\text { volume of cylinder }}{\text { volume of cone }}=\frac{\pi r_{1}{ }^{2} h_{1}}{\dfrac{1}{3} \pi r_{2}{ }^{2} h_{2}}=3 \times\left(\frac{r_{1}}{r_{2}}\right)^{2} \times\left(\frac{h_{1}}{h_{2}}\right)\) \[\begin{aligned} & =3 \times\left(\frac{3}{4}\right)^{2} \times\left(\frac{2}{3}\right) \\ & =\frac{9}{8} \end{aligned} \] Hence, required ratio is $\displaystyle 9$:$\displaystyle 8$