The base radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5:3. The ratio of their volumes is :

Options

A. `27:20`
B. `20:27`
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 Given Ratios:**
- The base radii of the two cylinders are in the ratio \( R_1 : R_2 = 2 : 3 \).
- The heights of the two cylinders are in the ratio \( H_1 : H_2 = 5 : 3 \).

2. **Express the Radii and Heights:**
- Let \( R_1 = 2x \) and \( R_2 = 3x \) for some value \( x \).
- Let \( H_1 = 5y \) and \( H_2 = 3y \) for some value \( y \).

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

4. **Calculate the Volumes:**
- Volume of Cylinder 1:
\[ V_1 = \pi (R_1^2) H_1 = \pi (2x)^2 (5y) = \pi (4x^2) (5y) = 20\pi x^2 y \]
- Volume of Cylinder 2:
\[ V_2 = \pi (R_2^2) H_2 = \pi (3x)^2 (3y) = \pi (9x^2) (3y) = 27\pi x^2 y \]

5. **Find the Ratio of the Volumes:**
- The ratio of the volumes \( V_1 : V_2 \) is:
\[ \frac{V_1}{V_2} = \frac{20\pi x^2 y}{27\pi x^2 y} \]
- The \( \pi x^2 y \) terms cancel out, leading to:
\[ \frac{V_1}{V_2} = \frac{20}{27} \]

6. **Conclusion:**
- Therefore, the ratio of the volumes of the two cylinders is \( 20 : 27 \).