Question

If two cones have their heights in the ratio $1:3$ and radii $3:1$, then the ratio of their volumes is:

• A. $1:3$
• B. $3:1$
• C. $2:3$
• D. $3:2$

Answer 1

The correct option is B $3:1$

The ratio of heights of cone is $h_1:h_2=1:3$.
The ratio of radius of cone is $r_1:r_2=3:1$.
We know that the volume of the cone,
$V=\frac{1}{3}\pi r^{2}h$
Therefore,
$\frac{V_1}{V_2}=\frac{\frac{1}{3}\pi r_1^2{h}_1}{\frac{1}{3}\pi r_2^2{h}_2}$

$⟹$ $\frac{V_1}{V_2}=\frac{r_1^2{h}_1}{r_2^2{h}_2}$

$⟹$ $\frac{V_1}{V_2}=\frac{9\times 1}{1\times 3}$

$⟹$ $\frac{V_1}{V_2}=\frac{3}{1}=3:1$.
Hence, the ratio of the volume of this cone is $3:1$ and option $B$ is correct.