# The ratio of volumes of two cones is 4:5 and the ratio of the radii of their bases is 2:3.

Question:

The ratio of volumes of two cones is 4:5 and the ratio of the radii of their bases is 2:3. Find the ratio of their vertical heights.

Solution:

Let the ratio of the radius be x

Radius of $1^{5 \mathrm{st}}$ cone $=2 \mathrm{x}$

Radius of $2^{\text {nd }}$ cone $=3 x$

Let the ratio of the volume be y

Volume of $1^{\text {st }}$ cone $=4 \mathrm{y}$

Volume of $2^{\text {nd }}$ cone $=5 y$

$y_{1} / y_{2}=4 y / 5 y$

$=4 / 5$

$\Rightarrow \frac{\frac{1}{3} \pi r_{a}^{2} h_{a}}{\frac{1}{3} \pi r_{b}^{2} h_{b}}=\frac{4}{5}$

$\Rightarrow \frac{\mathrm{h}_{\mathrm{a}} *(2 \mathrm{x})^{2}}{\mathrm{hb} *(3 \mathrm{x})^{2}}=\frac{4}{5}$

$\Rightarrow \frac{\mathrm{h}_{\mathrm{a}} * 4 \mathrm{x}^{2}}{\mathrm{~h}_{\mathrm{b}} * 9 \mathrm{x}^{2}}=\frac{4}{5}$

$\Rightarrow \frac{\mathrm{h}_{\mathrm{a}}}{\mathrm{h}_{\mathrm{b}}}=\frac{36}{20}=\frac{9}{5}$

Therefore the heights are in the ratio of 9:5.