Question

Two cones have their heights in the ratio $1:3$ and the radii of their bases are in the ratio $3:1$, then the ratio of their volumes is:

• A. $1:3$
• B. $27:1$
• C. $3:1$
• D. $1:27$

Answer 1

The correct option is C $3:1$

We know, Volume of a cone $=\frac{1}{3}\pi r^{2}h$, where $r$ is the radius of the base of the cone and $h$ is the height.
Let $r_1$ and $h_1$ be the radius and height of the first cone and $r_2$ and $h_2$ be the radius and height of the second cone.
Given, ratio of heights $h_1:h_2=1:3$

and ratio of radii$r_1:r_2=3:1$.

Now, ratio of their volumes is calculated as:
$V_1:V_2$
$=\frac{1}{3}\pi {r_1}^2{h}_1:\frac{1}{3}\pi {r_2}^2{h}_2$

$={r_1}^2{h}_1:{r_2}^2{h}_2$

$=\frac{{r_1}^2}{{r_2}^2}:\frac{h_2}{h_1}$
$={\left(\frac{3}{1}\right)}^2:\frac{3}{1}$
$=\frac{9}{1}:\frac{3}{1}$

$=3:1$.
Hence, their volumes are in the ratio $3:1$.

Therefore, option $C$ is correct.