The Water Tank Puzzle
A farm has a cylindrical water tank with a radius of 2 m and a height of 3 m. It is currently 40 percent full. A pump fills the tank at 500 litres per minute. A leak at the base drains it at 120 litres per minute. The farmer needs at least 20,000 litres in the tank before nightfall, which is 3 hours away. Use π ≈ 3.14 and 1 cubic metre = 1000 litres.
1. Calculate the tank's capacity, the current volume, and the time needed to reach 20,000 litres.
Read the worked answer
Capacity: Volume = πr²h = 3.14 × 2² × 3 = 3.14 × 12 = 37.68 m³, which is 37,680 litres.
Current volume: 40% of 37,680 = 15,072 litres.
Net fill rate: 500 − 120 = 380 litres per minute.
Water needed: 20,000 − 15,072 = 4,928 litres.
Time: 4,928 ÷ 380 ≈ 12.97 minutes, about 13 minutes.
This is well within the 3 hours (180 minutes), so the farmer will reach the target comfortably, assuming the pump and leak rates stay constant.
2. The farmer worries the leak might worsen. Find the maximum leak rate that would still allow the target to be reached in 3 hours, and judge how much risk that leaves.
Read the worked answer
In 180 minutes the farmer must add 4,928 litres net, so the required net rate is 4,928 ÷ 180 ≈ 27.4 litres per minute.
With the pump at 500 L/min, the maximum tolerable leak is 500 − 27.4 ≈ 472.6 litres per minute.
Judgement: The current leak is 120 L/min, well below the 472.6 L/min limit. The farmer has a large safety margin: the leak could nearly quadruple and the target would still be met. The real risk is not the leak rate but a pump failure or a sudden large breach, which the margin does not cover. A careful answer notes the assumption that the pump runs continuously and that the tank does not overflow (it will not, since 20,000 litres is below the 37,680-litre capacity).