The radii of two cylinders are in the ratio of 3:5 and their heights are in the ratio 4:3 . The ratio of their volumes is :

Options

A. `12 : 25`
B. `13 : 25 `
C. `4:5`
D. `5:4`

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 \( R_1 : R_2 = 3 : 5 \).
- The ratio of the heights of the two cylinders is given as \( H_1 : H_2 = 4 : 3 \).

2. **Use the Volume Formula**:
- The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi R^2 H \]
- Therefore, the volumes of the two cylinders can be expressed as:
\[ V_1 = \pi R_1^2 H_1 \quad \text{and} \quad V_2 = \pi R_2^2 H_2 \]

3. **Set Up the Ratio of Volumes**:
- The ratio of the volumes \( V_1 : V_2 \) 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, so we have:
\[ \frac{V_1}{V_2} = \frac{R_1^2 H_1}{R_2^2 H_2} \]

4. **Substitute the Ratios**:
- Substitute the values of \( R_1 \) and \( R_2 \):
\[ \frac{R_1}{R_2} = \frac{3}{5} \]
- Therefore, \( \frac{R_1^2}{R_2^2} = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \).

5. **Substitute the Heights**:
- Substitute the values of \( H_1 \) and \( H_2 \):
\[ \frac{H_1}{H_2} = \frac{4}{3} \]

6. **Combine the Ratios**:
- Now, combine the ratios:
\[ \frac{V_1}{V_2} = \frac{R_1^2}{R_2^2} \cdot \frac{H_1}{H_2} = \frac{9}{25} \cdot \frac{4}{3} \]

7. **Calculate the Final Ratio**:
- Perform the multiplication:
\[ \frac{9 \cdot 4}{25 \cdot 3} = \frac{36}{75} \]
- Simplifying \( \frac{36}{75} \):
\[ \frac{36 \div 3}{75 \div 3} = \frac{12}{25} \]

8. **Final Result**:
- The ratio of the volumes of the two cylinders is:
\[ V_1 : V_2 = 12 : 25 \]

### Conclusion:
The ratio of the volumes of the two cylinders is \( 12 : 25 \).

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