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 Definition:
The objective is to calculate the internal curved surface area of a building composed of a cylinder and a cone. The cylinder has a base radius of 12 m and a height of \( 15 \) m (derived from \( 3 \times 5 \)). The cone shares the same base radius of 12 m and has a slant height of 14 m.
The total internal curved surface area is the sum of the lateral surface areas of the cylinder and the cone.

Step 2: Cylinder Lateral Surface Area Calculation:
The formula for the lateral surface area of a cylinder is:
\[A_{\text{cylinder}} = 2 \pi r h\]Using \( r = 12 \) m and \( h = 15 \) m:
\[A_{\text{cylinder}} = 2 \pi \times 12 \times 15 = 360 \pi \, \text{m}^2\]

Step 3: Cone Lateral Surface Area Calculation:
The formula for the lateral surface area of a cone is:
\[A_{\text{cone}} = \pi r l\]Using \( r = 12 \) 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:
Summing the lateral surface areas:
\[A_{\text{total}} = A_{\text{cylinder}} + A_{\text{cone}} = 360 \pi + 168 \pi = 528 \pi \, \text{m}^2\]

Step 5: Conclusion:
The internal curved surface area of the building is \( 528 \pi \, \text{m}^2 \).
Approximate value:
\[528 \pi \approx 528 \times 3.1416 = 1658.6 \, \text{m}^2\]The approximate internal curved surface area is \( 1658.6 \, \text{m}^2 \).