Question

The ratio of the bases radii of two right cones is $2:3$ and the ratio of their height is $3:2$, then the ratio of their volumes is:

• A. $1:3$
• B. $3:2$
• C. $\sqrt{2}:\sqrt{3}$
• D. $2:3$

Answer 1

The correct option is D $2:3$

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.
Given, ratio of radii $=2:3$

and ratio of heights $=3:2$.
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.

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

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

$=\frac{{r_1}^2}{{r_2}^2}:\frac{h_2}{h_1}$

$={\left(\frac{2}{3}\right)}^2:\frac{2}{3}$

$=\frac{4}{9}:\frac{2}{3}$ $=2:3$.

Hence, their volumes are in the ratio $2:3$.

Therefore, option $D$ is correct.