Question

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

Answer 1

It is given that the heights of two cones are in the ratio $1:3$ that is:

$\frac{h_1}{h_2}=\frac{1}{3}.......\left(1\right)$

It is also given that the ratio of the radius of the cones is $3:1$ that is:

$\frac{r_1}{r_2}=\frac{3}{1}.......\left(2\right)$

We know that the volume of a cone with radius $r$ and height $h$ is $V=\frac{1}{3}\pi r^{2}h$, therefore, using equations $1$ and $2$, we have:

$\frac{V_1}{V_2}=\frac{\frac{1}{3}\pi r_1^2{h}_1}{\frac{1}{3}\pi r_2^2{h}_2}$

$\Rightarrow \frac{V_1}{V_2}=\frac{r_1^2}{r_2^2}\times \frac{h_1}{h_2}$

$\Rightarrow \frac{V_1}{V_2}={\left(\frac{r_1}{r_2}\right)}^2\times \frac{h_1}{h_2}$

$\Rightarrow \frac{V_1}{V_2}=3^2\times \frac{1}{3}$

$\Rightarrow \frac{V_1}{V_2}=\frac{3}{1}$

$\Rightarrow V_1:V_2=3:1$

Hence, the ratio of the volume of two cones is $3:1$.