# If the volumes of two cones are in the ratio of 1:4

Question:

If the volumes of two cones are in the ratio of 1:4 and their diameters are in the ratio of 4:5, then find the ratio of their heights.

Solution:

Let $r$ and $R$ be the base radii, $h$ and $H$ be the heights, $v$ and $V$ be the volumes of the two given cones.

We have,

$\frac{2 r}{2 R}=\frac{4}{5}$ or $\frac{r}{R}=\frac{4}{5} \quad \cdots$ (i)

and

$\frac{v}{V}=\frac{1}{4}$

$\Rightarrow \frac{\left(\frac{1}{3} \pi r^{2} h\right)}{\left(\frac{1}{3} \pi R^{2} H\right)}=\frac{1}{4}$

$\Rightarrow \frac{r^{2} h}{R^{2} H}=\frac{1}{4}$

$\Rightarrow\left(\frac{r}{R}\right)^{2} \times \frac{h}{H}=\frac{1}{4}$

$\Rightarrow\left(\frac{4}{5}\right)^{2} \times \frac{h}{H}=\frac{1}{4} \quad[$ Using $(\mathrm{i})]$

$\Rightarrow\left(\frac{16}{25}\right) \times \frac{h}{H}=\frac{1}{4}$

$\Rightarrow \frac{h}{H}=\frac{1 \times 25}{4 \times 16}$

$\Rightarrow \frac{h}{H}=\frac{25}{64}$

$\therefore h: H=25: 64$

So, the ratio of their heights is 25:64.