Filling a garden tank with two taps and a leak
A community garden has a 1,200-litre water tank. Tap A alone can fill the empty tank in 6 hours. Tap B alone can fill it in 9 hours. There is also a leak at the base; when the tank is full, the leak alone would empty it in 18 hours. The gardeners open both taps at the same time with the leak still present. They need the tank full before a Saturday morning planting event, which starts 5 hours after they begin. They also want to know whether repairing the leak would make a useful difference to the filling time. The water supply pressure is constant, and the tank starts empty. The gardeners pay for water at €1.80 per cubic metre; 1 cubic metre is 1,000 litres.
1. Calculate the net filling rate in litres per hour and the time to fill the tank with both taps open and the leak present. Will the tank be full before the event starts?
Read the worked answer
Tap A rate = 1200 ÷ 6 = 200 litres per hour. Tap B rate = 1200 ÷ 9 = 133.33 litres per hour. Leak rate = 1200 ÷ 18 = 66.67 litres per hour removed.
Net rate = 200 + 133.33 - 66.67 = 266.66 litres per hour.
Time to fill = 1200 ÷ 266.66 = 4.5 hours.
4.5 hours is less than the 5 hours available, so the tank will be full before the planting event starts, with about 0.5 hour (30 minutes) to spare.
2. Find the filling time if the leak is repaired. How much earlier is the tank full, and what is the water cost for one full tank?
Read the worked answer
With the leak repaired, the combined rate is 200 + 133.33 = 333.33 litres per hour.
Time to fill = 1200 ÷ 333.33 = 3.6 hours.
Earlier by 4.5 - 3.6 = 0.9 hours, which is 0.9 × 60 = 54 minutes.
A full tank is 1200 litres = 1.2 cubic metres. Water cost = 1.2 × €1.80 = €2.16.
Repairing the leak saves 54 minutes and also stops water being wasted while the tank is full, so it is worth doing even though the event deadline can already be met.