Water Tank Leak and Refill Decision
A boarding school in Bamenda has one rectangular concrete water tank measuring 3.2 m long, 2.5 m wide and 1.8 m deep. A supply pipe fills the tank at 45 litres per minute. The tank has a crack that loses water at a steady 0.02 litres per second whenever water is in the tank. The school has 120 boarders, and each boarder uses 18 litres of water per day on average. The bursar can pay a plumber 18,000 XAF to seal the crack permanently. Alternatively, lost water can be replaced by buying from a vendor at 250 XAF per 100 litres. The school refills the tank only when it becomes empty. The bursar wants to know whether the tank can last four days between refills and whether repairing the crack now is financially sensible.
1. Calculate the tank's capacity in litres and the net time needed to fill it from empty while the crack is leaking. Give the time in hours and minutes, correct to the nearest minute.
Read the worked answer
Capacity = 3.2 × 2.5 × 1.8 = 14.4 m³. Since 1 m³ = 1,000 L, capacity = 14,400 L. Leak rate = 0.02 L/s × 60 = 1.2 L/min. Net filling rate = 45 − 1.2 = 43.8 L/min. Time = 14,400 ÷ 43.8 = 328.767... min. 0.767 min ≈ 46 s, so the time is about 5 h 29 min to the nearest minute. Without the leak it would be 14,400 ÷ 45 = 320 min = 5 h 20 min, so the leak adds about 9 minutes per fill.
2. Using the daily water use and the leak, determine whether a full tank can last the four days between refills. Show the daily loss and the four-day requirement.
Read the worked answer
Daily use by boarders = 120 × 18 = 2,160 L. Daily leak loss = 1.2 × 60 × 24 = 1,728 L. Total daily draw = 2,160 + 1,728 = 3,888 L. Four-day requirement = 4 × 3,888 = 15,552 L. Capacity is 14,400 L, so the tank cannot last four days. It lasts 14,400 ÷ 3,888 = 3.70 days, about 3 days 17 hours. The school would run out before the fourth day.
3. Advise the bursar whether to repair the crack now. Use the cost of lost water and the repair cost to find a payback period, and state any assumption.
Read the worked answer
Lost water costs 1,728 L/day. At 250 XAF per 100 L, cost per day = (1,728 ÷ 100) × 250 = 17.28 × 250 = 4,320 XAF/day. Repair cost = 18,000 XAF. Payback time = 18,000 ÷ 4,320 = 4.17 days, about 4 days 4 hours. If the school will keep using the tank for more than about 4.2 days, repairing is cheaper than replacing lost water. Since a school term is far longer than that, the repair is financially sensible. Assumption: the leak rate stays at 0.02 L/s and the vendor price stays at 250 XAF per 100 L. The repair also shortens filling time by about 9 minutes per fill, which is a practical benefit.