The ratio of the radii of two cylinder s is `2:3` and the ratio of their heights is 5:3. The ratio of their volumes will be

Options

A. `4:9`
B. `9:4`
C. `20:27`
D. `27:20`

Answer

To find the ratio of the volumes of two cylinders given the ratios of their radii and heights, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Ratios**:
- The ratio of the radii of the two cylinders is given as \( r_1 : r_2 = 2 : 3 \).
- The ratio of their heights is given as \( h_1 : h_2 = 5 : 3 \).

2. **Write the Volume Formula**:
- The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
- Let \( V_1 \) be the volume of the first cylinder and \( V_2 \) be the volume of the second cylinder.

3. **Express the Volumes**:
- For the first cylinder:
\[ V_1 = \pi r_1^2 h_1 \]
- For the second cylinder:
\[ V_2 = \pi r_2^2 h_2 \]

4. **Set Up the Volume Ratio**:
- The ratio of the volumes \( \frac{V_1}{V_2} \) can be expressed as:
\[ \frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} \]
- The \( \pi \) cancels out:
\[ \frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2} \]

5. **Substitute the Ratios**:
- Substitute \( r_1 = 2k \) and \( r_2 = 3k \) (for some \( k \)):
\[ \frac{r_1}{r_2} = \frac{2}{3} \implies \frac{r_1^2}{r_2^2} = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]
- Substitute \( h_1 = 5m \) and \( h_2 = 3m \) (for some \( m \)):
\[ \frac{h_1}{h_2} = \frac{5}{3} \]

6. **Combine the Ratios**:
- Now substitute these values into the volume ratio:
\[ \frac{V_1}{V_2} = \frac{4}{9} \times \frac{5}{3} \]

7. **Calculate the Final Ratio**:
- Multiply the fractions:
\[ \frac{V_1}{V_2} = \frac{4 \times 5}{9 \times 3} = \frac{20}{27} \]

8. **Conclusion**:
- Therefore, the ratio of the volumes of the two cylinders is:
\[ V_1 : V_2 = 20 : 27 \]