Rainwater Tank Capacity
A rural school installs a rainwater tank for its garden. The tank is a cylinder of radius 1.2 m and height 2.4 m, with a hemispherical top that also stores water. The school must keep 15% of the tank's total capacity as freeboard, so that water does not overflow in heavy rain. The garden irrigation system uses 850 litres per day. The school wants to know whether one tank can supply 14 days of irrigation without refilling, and whether two identical tanks would supply more than 30 days. Rainfall during the test period is ignored, and no other water is drawn from the tank. The caretaker also asks how many litres are usable after the freeboard is allowed. A local supplier quotes capacity in cubic metres, but the school's irrigation records are in litres. Use 1 m³ = 1000 L and π = 3.142. The board will decide whether to buy one tank or two based on the answers.
1. Calculate the total capacity of the tank in litres using π = 3.142.
Read the worked answer
The tank is a cylinder plus a hemisphere.
Cylinder volume $= \pi r^2 h = 3.142(1.2^2)(2.4)$.
$1.2^2 = 1.44$, and $1.44 \times 2.4 = 3.456$.
So cylinder volume $= 3.142 \times 3.456 = 10.858752$ m³.
Hemisphere volume $= \frac23 \pi r^3 = \frac23(3.142)(1.2^3)$.
$1.2^3 = 1.728$, so hemisphere volume $= \frac23(3.142)(1.728) = 3.619584$ m³.
Total volume $= 10.858752 + 3.619584 = 14.478336$ m³.
In litres: $14.478336 \times 1000 = 14478.336$ L, about 14,478 L.
2. Find the usable capacity after 15% freeboard and decide whether one tank supplies 14 days at 850 L/day.
Read the worked answer
Freeboard is 15%, so usable capacity is 85% of total:
$0.85 \times 14478.336 = 12306.5856$ L, about 12,307 L.
Water needed for 14 days is $14 \times 850 = 11900$ L.
Since $12306.6 > 11900$, one tank does supply 14 days. The surplus is $12306.6 - 11900 = 406.6$ L, which is about half a day's use. This is a very small margin: if daily use rose above $12306.6/14 \approx 879$ L/day, one tank would not last 14 days.
3. Evaluate the claim that two identical tanks would supply more than 30 days.
Read the worked answer
Usable capacity for two tanks is
$2 \times 12306.5856 = 24613.1712$ L.
Water needed for 30 days is
$30 \times 850 = 25500$ L.
Two tanks fall short by $25500 - 24613.17 = 886.83$ L.
The supply lasts $24613.17/850 = 28.96$ days, just under 29 days. The claim of more than 30 days is false under the 15% freeboard rule. Without freeboard, two tanks would hold about 28,957 L and last about 34.1 days, but that would ignore the required safety margin.