Painting and Filling a Domed Colony Water Tank
A residential colony installs a water tank made of a cylinder of radius 1.4 m and height 3 m, with a hemisphere of the same radius fixed on top. The flat circular base of the cylinder rests on a concrete platform and will not be painted; the curved surfaces of the cylinder and the hemisphere, and nothing else, are to be given two coats of paint. One litre of paint covers 8 m², and paint is sold only in one-litre tins costing ₹280 each. The colony secretary has budgeted ₹1,200 for painting and claims that 'the tank holds about 50,000 litres, enough for 240 flats at 90 litres a day for a full day, with a large surplus.' A resident objects that the estimate is inflated because the top hemisphere is empty space and holds no water. (Use π = 22/7.)
1. Calculate the area to be painted for two coats, the number of tins required and the cost, and judge whether the ₹1,200 budget is adequate.
Read the worked answer
Painted surfaces: the base circle is excluded (it rests on the platform) and there is no flat top circle, because the hemisphere covers the cylinder completely.
Curved surface of cylinder = 2πrh = 2 × 22/7 × 1.4 × 3 = 26.4 m²
Curved surface of hemisphere = 2πr² = 2 × 22/7 × 1.96 = 12.32 m²
One coat = 38.72 m²; two coats = 77.44 m²
Paint needed = 77.44/8 = 9.68 L. Tins are whole litres, so 10 tins → 10 × 280 = ₹2,800.
The budget of ₹1,200 is short by ₹1,600. Typical traps: adding the base circle (πr² = 6.16 m²) gives 44.88 m² per coat and 12 tins (₹3,360); forgetting the second coat gives 5 tins (₹1,400). Both are wrong.
2. Find the capacity of the tank in litres and test both the secretary's 50,000-litre figure and the resident's objection.
Read the worked answer
Volume = πr²h + (2/3)πr³
= (22/7)(1.96)(3) + (2/3)(22/7)(2.744)
= 18.48 + 5.75 = 24.23 m³ ≈ 24,230 litres (1 m³ = 1000 L).
Demand = 240 × 90 = 21,600 litres, so the tank does meet one day's demand, with a surplus of about 2,630 litres.
Judgements: the secretary's 'about 50,000 litres' is roughly double the true capacity and is wrong, even though his conclusion about meeting demand happens to hold. The resident is also wrong: the hemisphere is part of the closed shell and holds water - it is a domed roof contributing about 5,750 litres, not empty space. Both parties are wrong for different reasons, which is exactly why the calculation must be done rather than argued.
3. The secretary says that if funds are short, a single coat will cost about ₹1,200. Check this claim.
Read the worked answer
One coat = 38.72 m², so paint = 38.72/8 = 4.84 L.
If only whole tins can be bought: 5 tins = ₹1,400.
If the painter charges for the exact quantity used: 4.84 × 280 = ₹1,355.20.
In both cases the figure exceeds ₹1,200 - by ₹200 or ₹155.20 respectively - so the claim is false.
The cheapest honest option is one coat using 5 tins, which supply 40 m² of coverage against the 38.72 m² needed, leaving 1.28 m² spare; the colony should budget at least ₹1,355 and cannot reach the target without reducing the painted area or finding cheaper paint.