The base radii of two cylinders, are-in the ratio `2 : 5` and their heights are in the ratio `5 : 7`. The ratio of their volumes is:

Options

A. `27:20`
B. `4:35`
C. `9:4`
D. `4:9`

Answer

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

### Step-by-Step Solution:

1. **Identify the Ratios**:
- The base radii of the two cylinders are in the ratio \(2:5\).
- The heights of the two cylinders are in the ratio \(5:7\).

2. **Assign Variables**:
- Let the radius of the first cylinder be \(r_1 = 2k\) and the radius of the second cylinder be \(r_2 = 5k\), where \(k\) is a common multiplier.
- Let the height of the first cylinder be \(h_1 = 5m\) and the height of the second cylinder be \(h_2 = 7m\), where \(m\) is another common multiplier.

3. **Volume Formula**:
- The volume \(V\) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]

4. **Calculate the Volumes**:
- Volume of the first cylinder \(V_1\):
\[ V_1 = \pi (r_1^2) h_1 = \pi (2k)^2 (5m) = \pi (4k^2)(5m) = 20\pi k^2 m \]
- Volume of the second cylinder \(V_2\):
\[ V_2 = \pi (r_2^2) h_2 = \pi (5k)^2 (7m) = \pi (25k^2)(7m) = 175\pi k^2 m \]

5. **Find the Ratio of the Volumes**:
- The ratio of the volumes \(V_1 : V_2\) is given by:
\[ \frac{V_1}{V_2} = \frac{20\pi k^2 m}{175\pi k^2 m} \]
- Simplifying this gives:
\[ \frac{20}{175} = \frac{4}{35} \]

6. **Final Ratio**:
- Therefore, the ratio of the volumes of the two cylinders is:
\[ \text{Ratio of volumes} = 4 : 35 \]