Two right circular solid cylinders have radii in the ratio 3 : 5 and heights in the ratio 2:3. Find the ratio between their curved surface areas.

Answer

To solve the problem of finding the ratio between the curved surface areas of two right circular solid cylinders with given ratios of radii and heights, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given 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 \( h_1 : h_2 = 2 : 3 \).

2. **Formula for Curved Surface Area**:
- The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2 \pi r h \]
- Therefore, the curved surface areas of the two cylinders can be expressed as:
\[ \text{CSA}_1 = 2 \pi r_1 h_1 \]
\[ \text{CSA}_2 = 2 \pi r_2 h_2 \]

3. **Set Up the Ratio of Curved Surface Areas**:
- The ratio of the curved surface areas of the two cylinders can be written as:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{2 \pi r_1 h_1}{2 \pi r_2 h_2} \]
- The \( 2 \pi \) terms will cancel out:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{r_1 h_1}{r_2 h_2} \]

4. **Substituting the Ratios**:
- Substitute the ratios of the radii and heights into the equation:
\[ \frac{r_1}{r_2} = \frac{3}{5} \quad \text{and} \quad \frac{h_1}{h_2} = \frac{2}{3} \]
- Therefore, we can express the ratio of the curved surface areas as:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{3}{5} \times \frac{2}{3} \]

5. **Calculating the Final Ratio**:
- Now, multiply the two fractions:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{3 \times 2}{5 \times 3} = \frac{6}{15} \]
- Simplifying \( \frac{6}{15} \) gives:
\[ \frac{6}{15} = \frac{2}{5} \]

6. **Final Answer**:
- The ratio of the curved surface areas of the two cylinders is:
\[ \text{CSA}_1 : \text{CSA}_2 = 2 : 5 \]