The radii of two cylinders are in the ratio 3:4 and their heights are in the ratio 6: 5. The ratio of their curved surface areas is

Options

A. ` 5 : 4`
B. `3 : 4`
C. `9 : 10`
D. `4 : 5`

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Ratios**:
- The ratio of the radii of the two cylinders is given as \( r_1 : r_2 = 3 : 4 \).
- The ratio of the heights of the two cylinders is given as \( h_1 : h_2 = 6 : 5 \).

2. **Write the Formula for Curved Surface Area**:
- The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2 \pi r h \]
- For the first cylinder, the curved surface area \( A_1 \) is:
\[ A_1 = 2 \pi r_1 h_1 \]
- For the second cylinder, the curved surface area \( A_2 \) is:
\[ A_2 = 2 \pi r_2 h_2 \]

3. **Find the Ratio of Curved Surface Areas**:
- We need to find the ratio \( \frac{A_1}{A_2} \):
\[ \frac{A_1}{A_2} = \frac{2 \pi r_1 h_1}{2 \pi r_2 h_2} \]
- The \( 2 \pi \) cancels out:
\[ \frac{A_1}{A_2} = \frac{r_1 h_1}{r_2 h_2} \]

4. **Substitute the Ratios**:
- Substitute the ratios of the radii and heights:
\[ \frac{A_1}{A_2} = \frac{r_1}{r_2} \cdot \frac{h_1}{h_2} = \frac{3}{4} \cdot \frac{6}{5} \]

5. **Calculate the Ratio**:
- Now, calculate the product:
\[ \frac{A_1}{A_2} = \frac{3 \cdot 6}{4 \cdot 5} = \frac{18}{20} = \frac{9}{10} \]

6. **Final Answer**:
- Therefore, the ratio of the curved surface areas of the two cylinders is:
\[ \frac{A_1}{A_2} = 9 : 10 \]