Two cones have their heights in the ratio 1 : 3 and the radii of their bases in the ratio 3 : 1.

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 $=\frac{\text { volume of the first cone }}{\text { volume of the second cone }}$

$=\frac{\frac{1}{3} \pi \times(3 \mathrm{r})^{2} \times \mathrm{h}}{\frac{1}{3} \pi \times \mathrm{r}^{2} \times 3 \mathrm{~h}}$

$=\frac{9 r^{2} h}{3 r^{2} h}=3: 1$

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