The diameters of two cones are equal. If their slant heights be in the ratio of 5:7, then find the ratio of their curved surface areas.

Options

A. `25:7`
B. `25:49`
C. `5:49`
D. `5:7`

Answer

To find the ratio of the curved surface areas of two cones with equal diameters and slant heights in the ratio of 5:7, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two cones with equal diameters. This means their radii are also equal.
- Let the radius of both cones be \( R \).

2. **Identifying the Slant Heights**:
- Let the slant height of cone 1 be \( L_1 \) and the slant height of cone 2 be \( L_2 \).
- According to the problem, the ratio of the slant heights is given as:
\[ \frac{L_1}{L_2} = \frac{5}{7} \]

3. **Curved Surface Area Formula**:
- The formula for the curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi r L \]
- For cone 1, the curved surface area (CSA1) can be expressed as:
\[ \text{CSA}_1 = \pi R L_1 \]
- For cone 2, the curved surface area (CSA2) can be expressed as:
\[ \text{CSA}_2 = \pi R L_2 \]

4. **Finding the Ratio of Curved Surface Areas**:
- To find the ratio of the curved surface areas of the two cones, we can write:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{\pi R L_1}{\pi R L_2} \]
- Since \( \pi \) and \( R \) are common in both the numerator and denominator, they cancel out:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{L_1}{L_2} \]

5. **Substituting the Ratio of Slant Heights**:
- Now, substituting the ratio of the slant heights:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{5}{7} \]

6. **Conclusion**:
- Therefore, the ratio of the curved surface areas of the two cones is:
\[ \text{CSA}_1 : \text{CSA}_2 = 5 : 7 \]

### Final Answer:
The ratio of the curved surface areas of the two cones is \( 5 : 7 \).