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:
We are asked to find the internal curved surface area of a building. The building has two parts: a cylinder and a cone. The cylinder has a base radius of 12 m and a height of \( 3 \times 5 = 15 \) m. The cone has the same base radius of 12 m and a slant height of 14 m.
The internal curved surface area consists of the curved surface area of both the cylinder and the cone.

Step 2: Formula for the curved surface area of the cylinder:
The formula for the curved surface area of a cylinder is:
\[ A_{\text{cylinder}} = 2 \pi r h \] where \( r \) is the radius of the base and \( h \) is the height of the cylinder.
Substitute \( r = 12 \) m and \( h = 15 \) m:
\[ A_{\text{cylinder}} = 2 \pi \times 12 \times 15 = 360 \pi \, \text{m}^2 \]

Step 3: Formula for the curved surface area of the cone:
The formula for the curved surface area of a cone is:
\[ A_{\text{cone}} = \pi r l \] where \( r \) is the radius of the base and \( l \) is the slant height of the cone.
Substitute \( r = 12 \) m and \( l = 14 \) m:
\[ A_{\text{cone}} = \pi \times 12 \times 14 = 168 \pi \, \text{m}^2 \]

Step 4: Total internal curved surface area:
The total internal curved surface area is the sum of the curved surface areas of the cylinder and the cone:
\[ A_{\text{total}} = A_{\text{cylinder}} + A_{\text{cone}} = 360 \pi + 168 \pi = 528 \pi \, \text{m}^2 \]

Step 5: Final answer:
The internal curved surface area of the building is \( 528 \pi \, \text{m}^2 \), which is approximately:
\[ 528 \pi \approx 528 \times 3.1416 = 1658.6 \, \text{m}^2 \] Thus, the internal curved surface area of the building is approximately \( 1658.6 \, \text{m}^2 \).