The diameters of two cones are equal. If their slant heights are in the ratio 5:4, find the ratio of their curved surface areas.

Answer

To solve the problem of finding the ratio of the curved surface areas of two cones with equal diameters and slant heights in the ratio of 5:4, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The diameters of the two cones are equal.
- The ratio of their slant heights (l1:l2) is 5:4.

2. **Relate the Diameters to Radii**:
- Since the diameters are equal, we can denote the diameter of both cones as \( d \).
- Therefore, the radius \( r \) of both cones can be expressed as:
\[ r_1 = \frac{d}{2} \quad \text{and} \quad r_2 = \frac{d}{2} \]
- This implies \( r_1 = r_2 \).

3. **Use the Formula for Curved Surface Area of a Cone**:
- The formula for the curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi r l \]
- For the first cone, the curved surface area (CSA1) is:
\[ \text{CSA}_1 = \pi r_1 l_1 \]
- For the second cone, the curved surface area (CSA2) is:
\[ \text{CSA}_2 = \pi r_2 l_2 \]

4. **Set Up the Ratio of the Curved Surface Areas**:
- The ratio of the curved surface areas of the two cones is:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{\pi r_1 l_1}{\pi r_2 l_2} \]
- Since \( r_1 = r_2 \), the \( \pi \) and \( r \) terms cancel out:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{l_1}{l_2} \]

5. **Substitute the Ratio of Slant Heights**:
- We know that the ratio of slant heights \( l_1:l_2 = 5:4 \). Thus:
\[ \frac{l_1}{l_2} = \frac{5}{4} \]

6. **Conclusion**:
- Therefore, the ratio of the curved surface areas of the two cones is:
\[ \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{5}{4} \]
- We can write this as:
\[ \text{Required Ratio} = 5:4 \]

### Final Answer:
The required ratio of the curved surface areas of the two cones is \( 5:4 \).