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: Understanding the problem:
The building consists of two parts: a cylindrical base and a conical top. We are tasked with finding the internal curved surface area of the building.
The formula for the internal curved surface area of the building is the sum of the curved surface areas of the cylinder and the cone.

Step 2: Formulae for curved surface areas:
1. The curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA}_{\text{cylinder}} = 2\pi rh \] where $r$ is the radius and $h$ is the height of the cylinder.

2. The curved surface area (CSA) of a cone is given by:
\[ \text{CSA}_{\text{cone}} = \pi r l \] where $r$ is the radius and $l$ is the slant height of the cone.

Step 3: Given values:
- Radius of the base of the cylinder and the cone, $r = 12$ m
- Height of the cylinder, $h = 3 \times 5 = 15$ m
- Slant height of the cone, $l = 14$ m

Step 4: Calculate the CSA of the cylinder:
Using the formula for the CSA of the cylinder:
\[ \text{CSA}_{\text{cylinder}} = 2\pi \times 12 \times 15 = 360\pi \text{ m}^2 \]

Step 5: Calculate the CSA of the cone:
Using the formula for the CSA of the cone:
\[ \text{CSA}_{\text{cone}} = \pi \times 12 \times 14 = 168\pi \text{ m}^2 \]

Step 6: Find the total internal curved surface area:
The total internal curved surface area of the building is the sum of the curved surface areas of the cylinder and the cone:
\[ \text{Total CSA} = 360\pi + 168\pi = 528\pi \text{ m}^2 \]

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.