The ratio of radii of two right circular cylinders (A and B) is 2:3. The ratio of volumes of the cylinders A and B is 9:7, then what is the ratio of the heights of the cylinders A and B?

81:28

The correct answer is Option (4) → 81:28

To find the ratio of the heights of the two cylinders, we use the formula for the volume of a right circular cylinder:

$V = \pi r^2 h$

1. Identify the given information

Let the radii of cylinders A and B be $r_A$ and $r_B$, and their heights be $h_A$ and $h_B$.

• Ratio of radii: $\frac{r_A}{r_B} = \frac{2}{3}$
• Ratio of volumes: $\frac{V_A}{V_B} = \frac{9}{7}$

2. Set up the volume ratio equation

The ratio of their volumes is:

$\frac{V_A}{V_B} = \frac{\pi r_A^2 h_A}{\pi r_B^2 h_B} = \frac{9}{7}$

Cancel $\pi$ from both the numerator and the denominator:

$\left( \frac{r_A}{r_B} \right)^2 \times \frac{h_A}{h_B} = \frac{9}{7}$

3. Substitute the radius ratio and solve for height

Substitute $\frac{2}{3}$ for $\frac{r_A}{r_B}$:

$\left( \frac{2}{3} \right)^2 \times \frac{h_A}{h_B} = \frac{9}{7}$

$\frac{4}{9} \times \frac{h_A}{h_B} = \frac{9}{7}$

To isolate $\frac{h_A}{h_B}$, multiply both sides by $\frac{9}{4}$:

$\frac{h_A}{h_B} = \frac{9}{7} \times \frac{9}{4}$

$\frac{h_A}{h_B} = \frac{81}{28}$

Conclusion

The ratio of the heights of cylinders A and B is 81:28.