The leaking school tank: net filling rate and overnight overflow
A primary school in Machinga has a rectangular water tank for its hand-washing station. Inside, the tank measures 1.5 m long, 1 m wide and 0.6 m deep. Water runs in from a tap at a steady 12 litres per minute. A crack near the bottom lets water out at a steady 2 litres per minute whenever there is water above it, and nobody has repaired it. The tank is empty on Monday at 06:00, when the caretaker opens the tap. On Tuesday the tank starts the day one-third full, so the caretaker opens the tap at 06:00 again. The school has 180 pupils, and each pupil needs 5 litres of water for washing hands during the day. A villager suggests that the caretaker should simply leave the tap open all night, from 20:00 to 06:00, so that the tank is certainly full by morning.
1. Find the capacity of the tank in litres and the time needed to fill it from empty on Monday. State the finishing time.
Read the worked answer
Volume = 1.5 m × 1 m × 0.6 m = 0.9 m³. Since 1 m³ = 1 000 litres, the tank holds 900 litres.
Net rate = water in − water out = 12 − 2 = 10 litres per minute.
Time = 900 ÷ 10 = 90 minutes = 1 hour 30 minutes, so the tank is full at 07:30.
Check against need: 180 pupils × 5 litres = 900 litres, so one full tank is exactly one day's water. Using 900 ÷ 12 = 75 minutes would be wrong because it ignores the crack.
2. On Tuesday the tank starts one-third full. How long does it take to fill from 06:00, and what mistake would a pupil make by working out 900 ÷ 12?
Read the worked answer
One third of 900 = 300 litres already in the tank, so 900 − 300 = 600 litres must still go in.
Time = 600 ÷ 10 = 60 minutes, so the tank is full at 07:00.
Shortcut check: two-thirds of the 90 minutes is 60 minutes - the same answer, because the tank starts one-third full.
The 900 ÷ 12 = 75 minutes error makes two mistakes at once: it ignores the 2 litres per minute lost through the crack, and it ignores the 300 litres already in the tank. That answer would have the caretaker closing the tap 15 minutes early and leaving the tank short. The tap must be closed at 07:00; if it stays open the tank overflows and the school still pays for the wasted water.
3. Test the villager's suggestion (tap open 20:00 to 06:00). How much water is drawn, how much is stored, and how much is wasted? Give a better plan.
Read the worked answer
20:00 to 06:00 is 10 hours = 600 minutes.
Water drawn: 12 × 600 = 7 200 litres.
But the tank holds only 900 litres, and the net fill rate is 10 litres per minute, so it is full after 90 minutes (at 21:30). For the remaining 510 minutes the tap keeps running and the water simply overflows.
Accounting for the 7 200 litres:
- stored: 900 litres
- lost through the crack: 2 × 600 = 1 200 litres
- overflow: 7 200 − 900 − 1 200 = 5 100 litres (i.e. 10 litres per minute × 510 minutes).
That is 5 100 ÷ 7 200 ≈ 71% wasted.
Better plan: open the tap for 90 minutes at 06:00 - full tank, no overflow. Even better, repair the crack: the net rate becomes 12 litres per minute and filling takes 900 ÷ 12 = 75 minutes, and the school stops losing the 1 200 litres a day that the crack drains.