A circus tent is cylindrical to a height of 8 m surmounted by a conical part. If total height of the tent is 13 m and the diameter of its base is 24 m, calculate :
total surface area of the tent,

Answer

To calculate the total surface area of the circus tent, we need to find the surface areas of both the cylindrical and conical parts of the tent and then add them together.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Height of the cylindrical part (h_cylinder) = 8 m
- Total height of the tent = 13 m
- Diameter of the base = 24 m

2. **Calculate the Radius:**
- Radius (r) = Diameter / 2 = 24 m / 2 = 12 m

3. **Calculate the Height of the Conical Part:**
- Height of the conical part (h_cone) = Total height - Height of the cylindrical part
- h_cone = 13 m - 8 m = 5 m

4. **Calculate the Slant Height of the Cone:**
- The slant height (l) can be calculated using the Pythagorean theorem:
- \( l = \sqrt{(r^2 + h_{cone}^2)} \)
- \( l = \sqrt{(12^2 + 5^2)} = \sqrt{(144 + 25)} = \sqrt{169} = 13 m \)

5. **Calculate the Surface Area of the Cylinder:**
- The formula for the curved surface area of a cylinder is:
- \( \text{Surface Area}_{cylinder} = 2 \pi r h_{cylinder} \)
- \( \text{Surface Area}_{cylinder} = 2 \pi (12) (8) = 192 \pi \)

6. **Calculate the Surface Area of the Cone:**
- The formula for the curved surface area of a cone is:
- \( \text{Surface Area}_{cone} = \pi r l \)
- \( \text{Surface Area}_{cone} = \pi (12) (13) = 156 \pi \)

7. **Calculate the Total Surface Area of the Tent:**
- The total surface area of the tent is the sum of the surface areas of the cylinder and the cone:
- \( \text{Total Surface Area} = \text{Surface Area}_{cylinder} + \text{Surface Area}_{cone} \)
- \( \text{Total Surface Area} = 192 \pi + 156 \pi = 348 \pi \)

### Final Answer:
The total surface area of the tent is \( 348 \pi \) square meters.

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