The cap of a joker is conical in which `840 cm^(2)` cloth is used. Find its slant height if the circumference of its base is 56 cm.

Answer

To find the slant height of the conical joker cap, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the given information**:
- The curved surface area (CSA) of the cone is given as \( 840 \, \text{cm}^2 \).
- The circumference of the base of the cone is given as \( 56 \, \text{cm} \).

2. **Use the formula for the curved surface area of a cone**:
\[ \text{Curved Surface Area (CSA)} = \pi r l \]
where \( r \) is the radius of the base and \( l \) is the slant height.

3. **Find the radius \( r \)**:
- The circumference \( C \) of the base is given by the formula:
\[ C = 2\pi r \]
- We can rearrange this to find \( r \):
\[ r = \frac{C}{2\pi} = \frac{56}{2\pi} = \frac{28}{\pi} \]

4. **Substitute \( r \) into the CSA formula**:
- Now substitute \( r \) into the CSA formula:
\[ 840 = \pi \left(\frac{28}{\pi}\right) l \]
- Simplifying this gives:
\[ 840 = 28l \]

5. **Solve for \( l \)**:
- Rearranging the equation to find \( l \):
\[ l = \frac{840}{28} = 30 \, \text{cm} \]

6. **Conclusion**:
- The slant height \( l \) of the conical joker cap is \( 30 \, \text{cm} \).