If the radii of two cylinders are in the ratio 2:3 and their heights are in the ratio 5:3, then find the ratio of their volumes.

Options

A. `10:17`
B. `20:27`
C. `17: 27`
D. `20: 37`

Explanation

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. **Define the Variables**:
Let the radius of the first cylinder be \( R_1 \) and the radius of the second cylinder be \( R_2 \).
Let the height of the first cylinder be \( H_1 \) and the height of the second cylinder be \( H_2 \).

2. **Set Up the Ratios**:
According to the problem:
- The ratio of the radii is given as \( \frac{R_1}{R_2} = \frac{2}{3} \).
- The ratio of the heights is given as \( \frac{H_1}{H_2} = \frac{5}{3} \).

3. **Volume Formula**:
The volume \( V \) of a cylinder is calculated using the formula:
\[ V = \pi R^2 H \]
Therefore, the volumes of the two cylinders can be expressed as:
- Volume of the first cylinder: \( V_1 = \pi R_1^2 H_1 \)
- Volume of 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 the ratios of the radii and heights into the equation:
\[ \frac{V_1}{V_2} = \frac{R_1^2}{R_2^2} \cdot \frac{H_1}{H_2} \]
Using the given ratios:
\[ \frac{R_1^2}{R_2^2} = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]
and
\[ \frac{H_1}{H_2} = \frac{5}{3} \]

6. **Combine the Ratios**:
Now, combine the ratios:
\[ \frac{V_1}{V_2} = \frac{4}{9} \cdot \frac{5}{3} = \frac{4 \cdot 5}{9 \cdot 3} = \frac{20}{27} \]

7. **Final Result**:
Therefore, the ratio of the volumes of the two cylinders is:
\[ \frac{V_1}{V_2} = \frac{20}{27} \]