The ratio of heights of two cylinders is 3:2 and the ratio of their radii is 6:7 . What is the ratio of their curved surface areas ?

Options

A. `9:7`
B. `1:1 `
C. `7:9`
D. `7:4`

Answer

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

### Step-by-Step Solution:

1. **Identify the Ratios**:
- The ratio of the heights of the two cylinders (H1:H2) is given as 3:2.
- The ratio of the radii of the two cylinders (R1:R2) is given as 6:7.

2. **Express the Ratios Mathematically**:
- We can express the heights and radii as:
- H1/H2 = 3/2
- R1/R2 = 6/7

3. **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, for Cylinder 1 (C1):
\[ \text{CSA}_1 = 2\pi R_1 H_1 \]
- And for Cylinder 2 (C2):
\[ \text{CSA}_2 = 2\pi R_2 H_2 \]

4. **Find the Ratio of the Curved Surface Areas**:
- We need to find the ratio of CSA1 to CSA2:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{2\pi R_1 H_1}{2\pi R_2 H_2} \]
- The \(2\pi\) cancels out:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{R_1 H_1}{R_2 H_2} \]

5. **Substituting the Ratios**:
- Substitute the values of R1/R2 and H1/H2:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{R_1}{R_2} \cdot \frac{H_1}{H_2} = \frac{6}{7} \cdot \frac{3}{2} \]

6. **Calculating the Final Ratio**:
- Now, multiply the fractions:
\[ \frac{6 \cdot 3}{7 \cdot 2} = \frac{18}{14} \]
- Simplifying this gives:
\[ \frac{18}{14} = \frac{9}{7} \]

7. **Final Result**:
- Therefore, the ratio of the curved surface areas of the two cylinders is:
\[ \text{CSA}_1 : \text{CSA}_2 = 9 : 7 \]