A conical water tower with a radius of 9 metres and a height of 12 metres is…

Concept — For a right circular cone the slant height is l = √(r2 + h2), the curved (lateral) face has area πrl, and the flat circular base has area πr2. A cone that is closed at the bottom is wrapped by both surfaces, so its total surface area is πrl + πr2 = πr(l + r). A cost then follows from total area × rate per unit area.

Application

1. Given: radius r = 9 m, height h = 12 m, rate = ₹350 per m2, π = 3.14.

2. Slant height: l = √(92 + 122) = √(81 + 144) = √225 = 15 m.

3. Curved surface area: πrl = 3.14 × 9 × 15 = 423.90 m2.

4. Base area: πr2 = 3.14 × 81 = 254.34 m2.

5. The tower is concreted on its curved face and on its base, so the total concreted area is 423.90 + 254.34 = 678.24 m2.

6. Cost = area × rate = 678.24 × 350 = ₹2,37,384.

Cross-check

• Factorised route: πr(l + r) = 3.14 × 9 × (15 + 9) = 28.26 × 24 = 678.24 m2 — the same area obtained a second way.

• 9, 12, 15 is a Pythagorean triple, so the slant height is exactly 15 m and no rounding enters the area.

• Splitting the multiplication: 678.24 × 300 = 2,03,472 and 678.24 × 50 = 33,912; adding gives ₹2,37,384.

So the concrete for the curved face together with the base costs ₹2,37,384.