Question

The interior of a building is in the form of a cylinder of base radius 12 m and height 3×5 m surmounted by a cone of equal base and slant height 14 m. Find the internal curved surface area of the building.

Answer 1

Step 1: Problem Identification
The building comprises a cylinder and a cone. The objective is to determine the internal curved surface area.
This area is the sum of the curved surface areas of the cylindrical and conical sections.

Step 2: Relevant Formulas
1. Cylinder's Curved Surface Area (CSA):
\[\text{CSA}_{\text{cylinder}} = 2\pi rh\]
where $r$ is the radius and $h$ is the height.

2. Cone's Curved Surface Area (CSA):
\[\text{CSA}_{\text{cone}} = \pi r l\]
where $r$ is the radius and $l$ is the slant height.

Step 3: Provided Data
- Radius (cylinder & cone): $r = 12$ m
- Cylinder height: $h = 3 \times 5 = 15$ m
- Cone slant height: $l = 14$ m

Step 4: Cylinder CSA Calculation
Applying the formula:
\[\text{CSA}_{\text{cylinder}} = 2\pi \times 12 \times 15 = 360\pi \text{ m}^2\]

Step 5: Cone CSA Calculation
Applying the formula:
\[\text{CSA}_{\text{cone}} = \pi \times 12 \times 14 = 168\pi \text{ m}^2\]

Step 6: Total Internal Curved Surface Area Calculation
Summing the individual CSAs:
\[\text{Total CSA} = 360\pi + 168\pi = 528\pi \text{ m}^2\]

Using $\pi \approx 3.1416$:
\[\text{Total CSA} = 528 \times 3.1416 \approx 1658.3 \text{ m}^2\]

Conclusion
The internal curved surface area of the building is approximately 1658.3 m².