Tank Capacity, Paint Cost and a Plumber's Suggestion
A housing society in Gaya installs a cylindrical overhead tank on its roof. The tank has radius 1.4 m and height 3 m, and its overflow pipe stops filling when the tank is 90% full. The society's 40 families together use about 1,540 litres of water per day. The secretary claims that the tank will last 12 days on a single filling. A painter quotes ₹35 per square metre to paint the outside curved surface and the flat top of the tank, and the secretary has kept a budget of ₹1,200 for this work. Later, a plumber suggests replacing the tank with another of the same height but radius 1.0 m, arguing that this will cut the stored water to almost exactly half. The secretary wants all three claims checked with figures before the next committee meeting, using π = 22/7.
1. Find the capacity of the tank, the quantity of water actually stored at 90% filling, and the number of days it will last.
Read the worked answer
Volume of a cylinder = πr²h = (22/7) × (1.4)² × 3.
(1.4)² = 1.96, and (22/7) × 1.96 = 22 × 0.28 = 6.16. So volume = 6.16 × 3 = 18.48 m³.
Since 1 m³ = 1,000 litres, capacity = 18,480 litres. At 90% filling: 0.9 × 18,480 = 16,632 litres.
Days of use = 16,632 ÷ 1,540 = 10.8 days.
So the tank gives 10 full days and part of the 11th, not 12 days. The secretary's claim is false by more than a day; at 1,540 litres per day, 12 days would need 18,480 litres, which is the whole tank, ignoring the 90% cut-off.
2. Compute the painting cost for the curved surface plus the flat top and test the ₹1,200 budget.
Read the worked answer
Curved surface area = 2πrh = 2 × (22/7) × 1.4 × 3 = 2 × 4.4 × 3 = 26.4 m².
Flat top area = πr² = (22/7) × 1.96 = 6.16 m².
Total area to be painted = 26.4 + 6.16 = 32.56 m².
Cost = 32.56 × 35 = ₹1,139.60.
Comparison with budget: 1,200 − 1,139.60 = ₹60.40 left over. The budget is sufficient, but only just - about 5% margin, so any rate revision or extra surface (such as the base ring) would break it.
Common error to avoid: using 4πr² or adding the bottom circular face; the bottom sits on the roof and is not painted.
3. Evaluate the plumber's claim that reducing the radius to 1.0 m keeps the height and cuts stored water to almost exactly half.
Read the worked answer
New volume = πr²h = (22/7) × (1.0)² × 3 = 66/7 ≈ 9.4286 m³ ≈ 9,428.6 litres.
Old capacity = 18,480 litres; exactly half would be 9,240 litres.
Difference = 9,428.6 − 9,240 = 188.6 litres, i.e. the new tank holds about 2% more than half.
Ratio = 9,428.6/18,480 ≈ 0.510, so the new tank holds about 51% of the old one.
Judgement: 'almost exactly half' is a fair everyday description, since the error is only about 2%, and the result follows directly from the area ratio (1.0/1.4)² = 1/1.96 ≈ 0.51 because the height is unchanged. But if the society needs precision for planning, the correct statement is that capacity falls to about 51%, not 50%.