Question:
Two cones have their heights in the ratio 1 : 3 and the radii of their bases in the ratio 3 : 1. Show that their volumes are in the ratio 3 : 1.
Solution:
Let the heights of the first and second cones be h and 3h, respectively .
Also, let the radius of the first and second cones be 3r and r, respectively.
$\therefore$ Ratio of volumes of the cones $=\frac{\text { volume of the first cone }}{\text { volume of the second cone }}$
$=\frac{\frac{1}{3} \pi \times(3 \mathrm{r})^{2} \times \mathrm{h}}{\frac{1}{3} \pi \times \mathrm{r}^{2} \times 3 \mathrm{~h}}$
$=\frac{9 r^{2} h}{3 r^{2} h}=3: 1$