Choosing a Rooftop Water Tank: Capacity First, Price Second
A housing society in Vijayawada must install a rooftop tank that holds at least 1,000 litres of water (1 m³ = 1,000 litres). Two quotations are on the table. Tank A is a closed cylinder of radius 0.50 m and height 1.40 m. Tank B is a closed cuboid measuring 1.00 m × 0.80 m × 1.20 m. Both are fabricated from the same 2 mm steel sheet at ₹480 per square metre of surface area, and the fabricator charges a flat ₹1,200 as labour for either tank. The secretary wants the cheapest tank that still meets the 1,000-litre minimum. The plumber has already built a circular concrete base of 1.30 m diameter and insists that a cylindrical tank must sit entirely within it. A 0.40 m diameter manhole is cut into the top of each tank, but for simplicity ignore the steel removed. Use π = 22/7 and treat all given costs as fixed for this calculation only.
1. Which tank meets the 1,000-litre requirement? Show the capacity calculation for both.
Read the worked answer
Tank A (cylinder) V = πr²h = (22/7) × (0.50)² × 1.40 = (22/7) × 0.25 × 1.40 = (22/7) × 0.35 = 1.10 m³ = 1,100 L Since 1,100 L ≥ 1,000 L, Tank A qualifies.
Tank B (cuboid) V = l × b × h = 1.00 × 0.80 × 1.20 = 0.96 m³ = 960 L 960 L < 1,000 L, so Tank B fails by 40 L (about 4%).
Conclusion: only Tank A satisfies the brief. The manhole is explicitly excluded from the calculation, so no correction is applied for it.
2. Compute the total cost of the qualifying tank and compare it with the rejected option, so the secretary can see what the extra money actually buys.
Read the worked answer
Tank A: total surface of a closed cylinder = 2πr² + 2πrh
- Top + bottom: 2 × (22/7) × 0.25 = 1.571 m²
- Curved wall: 2 × (22/7) × 0.50 × 1.40 = 4.400 m²
- Total = 5.971 m² Cost = 5.971 × 480 = ₹2,866 + ₹1,200 labour ≈ ₹4,066
Tank B: total surface of a closed cuboid = 2(lb + bh + hl) = 2(0.80 + 0.96 + 1.20) = 5.92 m² Cost = 5.92 × 480 = ₹2,842 + ₹1,200 ≈ ₹4,042
The rejected tank is cheaper by only about ₹25, yet it is 140 L short of the requirement. The secretary should reject the 'cheaper' quotation: the saving is negligible and the tank would fail the specification. Notice also that Tank B's surface area is almost equal to Tank A's even though its volume is smaller, so surface area alone is a poor guide to capacity.
3. The plumber's base is 1.30 m across. The society later asks for Tank A with the radius increased to 0.60 m (height unchanged, still closed). Check that it fits on the base and find the new capacity and new total cost. Does the cost grow in proportion to the radius?
Read the worked answer
Fit: diameter = 2 × 0.60 = 1.20 m, which is less than the 1.30 m base, so it fits with 5 cm clearance all round.
Capacity: V = (22/7) × 0.36 × 1.40 = (22/7) × 0.504 = 1.584 m³ = 1,584 L (a 44% rise, because V ∝ r²).
Cost: surface = 2 × (22/7) × 0.36 + 2 × (22/7) × 0.60 × 1.40 = 2.263 + 5.280 = 7.543 m² Cost = 7.543 × 480 = ₹3,621 + ₹1,200 = ₹4,821 (about 18.6% more than ₹4,066).
Judgement: growth is not proportional. The radius rises 20%, but capacity rises 44% while cost rises only about 19%, because the flat ends scale as r² and the curved wall as r. Enlarging the radius is clearly better value per litre than increasing the height, provided the base and the roof structure can take the extra mass.