Estimating a Rectangular Water Tank Refill
A base camp stores water in a rectangular tank measuring 2.4 m long, 1.5 m wide, and 1.2 m deep. The tank is currently one-third full. A pump fills it at 18 litres per minute, but the pump loses efficiency: after every 10 minutes of running it delivers 10% less per minute than in the previous 10-minute block (so block 1 = 18 L/min, block 2 = 16.2 L/min, and so on). The camp needs the tank at least 90% full before a 06:00 inspection. It is now 04:30. A logistician says 'the tank is small; the pump will easily make it.'
1. Compute the tank capacity, the current volume, and the volume needed to reach 90% full.
Read the worked answer
Capacity = 2.4 × 1.5 × 1.2 = 4.32 m³ = 4320 L. Current = 1/3 × 4320 = 1440 L. 90% target = 0.9 × 4320 = 3888 L. Volume needed = 3888 − 1440 = 2448 L. Need 2448 L.
2. The pump runs for 90 minutes (04:30 to 06:00). Using the block-efficiency rule (10-minute blocks), determine whether the target is met.
Read the worked answer
90 minutes = 9 blocks of 10 min. Block n delivers 18 × 0.9^(n−1) L/min × 10 min = 180 × 0.9^(n−1) L. Sum = 180 × (1 + 0.9 + 0.9² + … + 0.9⁸). Geometric sum = (1 − 0.9⁹)/(1 − 0.9) = (1 − 0.3874)/0.1 = 6.126. Total ≈ 180 × 6.126 ≈ 1102.6 L. That is far short of 2448 L. Target is NOT met. The logistician's 'easily make it' claim is wrong - the pump delivers roughly 1103 L, about 45% of what is needed.
3. Given the shortfall, what is the minimum extra time (to the nearest minute) the pump would need beyond 06:00, assuming the same block rule continues?
Read the worked answer
After 9 blocks, delivered ≈ 1102.6 L; remaining need ≈ 2448 − 1102.6 = 1345.4 L. Block 10 rate = 180 × 0.9⁹ ≈ 180 × 0.3874 ≈ 69.7 L per 10 min. Block 11 ≈ 62.8 L, block 12 ≈ 56.5 L, block 13 ≈ 50.8 L, block 14 ≈ 45.8 L, block 15 ≈ 41.2 L, block 16 ≈ 37.1 L, block 17 ≈ 33.4 L, block 18 ≈ 30.0 L, block 19 ≈ 27.0 L, block 20 ≈ 24.3 L. Cumulative from block 10: 69.7, 132.5, 189.0, 239.8, 285.6, 326.8, 363.9, 397.3, 424.3, 448.6, 472.9… This converges slowly. Sum of remaining blocks ≈ 180 × 0.9⁹ × 1/(1−0.9) = 180 × 0.3874 × 10 ≈ 697 L total possible from block 10 onward - which is less than 1345.4 L. The pump can never reach 90% under this rule; the shortfall cannot be closed by waiting. The correct judgement is that the efficiency decay makes the target unattainable and an alternative water source or pump is required.