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.

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

To do:

We have to find the ratio of their vertical heights.

Solution:

Ratio of the volumes of two cones $= 4:5$

Ratio of the radii of the cones $= 2:3$

Let the radius of the first cone $(r_1)$ be $2x$ and the radius of the second cone $(r_2)$ be $3x$

Let $h_1$ and $h_2$ be the heights of the cones respectively.

Therefore,

$\frac{1}{3} \pi r_{1}{ }^{2} h_{1}: \frac{1}{3} \pi r_{2}{ }^{2} h_{2}=4: 5$

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

$\frac{\pi r_{1}^{2} h_{1}}{\pi r_{2}^{2} h_{2}}=\frac{4}{5}$

$\frac{(2 x)^{2}}{(3 x)^{2}} \times \frac{h_{1}}{h_{2}}=\frac{4}{5}$

$\frac{4 x^{2}}{9 x^{2}} \times \frac{h_{1}}{h_{2}}=\frac{4}{5}$

$\Rightarrow \frac{h_{1}}{h_{2}}=\frac{4}{5} \times \frac{9 x^{2}}{4 x^{2}}$

$\frac{h_{1}}{h_{2}}=\frac{9}{5}$

The ratio of the vertical heights of the two cones is $9: 5$.