Two water tanks: capacity, cost per litre and days between refills
A primary school needs a new water tank. The supplier offers two shapes: a cylindrical tank with internal diameter 1.8 m and internal height 2.4 m, and a rectangular tank with internal dimensions 2.0 m by 1.5 m by 2.0 m. The cylindrical tank costs P4 800 and the rectangular one costs P4 500, both including delivery. The school uses about 450 litres of water on a school day. The caretaker is told to order a refill when the tank is down to 15% of its capacity. The head teacher says, 'Buy the cylinder - it is bigger, so it must be better value.' A parent points out that 1 m³ = 1 000 litres and asks for the working to be shown before any money is spent.
1. Calculate the capacity of each tank in litres. State which holds more and by what percentage (to 1 d.p.).
Read the worked answer
Cylinder: r = 1.8 / 2 = 0.9 m, h = 2.4 m
V = πr²h = π x 0.81 x 2.4 = 1.944π ≈ 6.108 m³ (using π ≈ 3.142)
Capacity ≈ 6 108 litres (a calculator's π gives 6 107 litres).
Rectangular tank: V = 2.0 x 1.5 x 2.0 = 6.0 m³ = 6 000 litres.
Difference: about 108 litres more in the cylinder.
Percentage: 108 / 6 000 x 100 ≈ 1.8%.
So the cylinder is bigger, but only by about one part in fifty-six - a difference that may well be swallowed by the rounding in the dimensions.
2. Compare the tanks by cost per litre and say whether the head teacher's reasoning holds.
Read the worked answer
- Cylinder: 4 800 / 6 108 ≈ P0.786 per litre
- Rectangular: 4 500 / 6 000 = P0.750 per litre
The rectangular tank is cheaper per litre by about 3.6 cents, roughly 4.6%.
The head teacher's reasoning fails because it looks at only one of the two quantities being compared. 'Bigger' is not 'better value' when the prices differ. A sharper way to see it: the extra 108 litres cost P300, which is about P2.78 for each additional litre - nearly four times the average cost per litre of the rectangular tank. The school is paying a high price for capacity it may not need. On this evidence the rectangular tank at P4 500 is the better buy.
3. Using the 15% refill rule and 450 litres per school day, find for how many whole school days each tank lasts before a refill is needed. Does the extra capacity help in practice?
Read the worked answer
Usable water is 85% of capacity.
Cylinder: 0.85 x 6 108 ≈ 5 192 litres. Days: 5 192 / 450 = 11.54, so 11 whole school days.
Rectangular: 0.85 x 6 000 = 5 100 litres. Days: 5 100 / 450 = 11.33, so 11 whole school days.
Both tanks need refilling after 11 school days. The 108-litre difference is small compared with one day's use (450 litres), so it cannot buy an extra day. The extra capacity therefore helps the school in no practical way, and the cheaper rectangular tank is the sensible order - saving P300 for the same refill cycle.
A further point worth raising with the caretaker: the 15% rule strands roughly 900 to 915 litres in the bottom of either tank, which is about two days of water. Lowering the refill level to 5% would recover that water without buying a larger tank.