Question

If the ratio of the radius of a cone and a cylinder of equal volume is 3:5, then find the ratio of their heights.

• A. $\frac{25}{3}$
• B. $\frac{28}{3}$
• C. $\frac{23}{3}$
• D. $7$

Answer 1

The correct option is A

$\frac{25}{3}$

Let $r_1$ and $h_1$ be the radius and height of the cone and $r_2$ and $h_2$ be the radius and height of the cylinder.
Then, volume of cone = $\frac{1}{3}\pi (r_1{)}^2{h}_1$
and, volume of cylinder = $\pi {r_2}^2{h}_2$
It is given that the volume of cone is equal to the volume of cylinder.
$\therefore$ $\frac{1}{3}\pi (r_1{)}^2{h}_1=\pi (r_2{)}^2{h}_2$

${r_1}^2{h}_1=3{r_2}^2{h}_2$

$\frac{h_1}{h_2}=3\times \frac{{r_2}^2}{{r_1}^2}$

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

$\frac{h_1}{h_2}=3\times {\left(\frac{5}{3}\right)}^2$

$\frac{h_1}{h_2}=3\times \frac{25}{9}$

$h_1:h_2=25:3$

Thus, the ratio of their heights is 25 : 3.