The Water Tank Reversal
A town's primary water tank holds 50,000 liters when full and drains at a constant 400 liters per hour through normal demand. At time t = 0 the tank is full. A maintenance crew plans to refill it using a pump rated at 900 liters per hour, but the pump cannot start until the tank has dropped to 30,000 liters. A safety rule requires the tank never fall below 20,000 liters. The crew chief writes the model: volume V(t) = 50000 - 400t before pumping, and V(t) = 50000 - 400t + 900(t - t_start) after pumping begins, where t_start is the moment the pump starts. A council member argues the tank will never be at risk because the pump adds more than the drain removes.
1. Determine t_start and the earliest time the tank would hit the 20,000-liter floor if the pump never started. Then evaluate whether the council member's claim is correct.
Read the worked answer
Finding t_start: Set 50000 - 400t = 30000, so 400t = 20000, giving t_start = 50 hours.
Floor time without the pump: Set 50000 - 400t = 20000, so 400t = 30000, giving t = 75 hours.
Evaluating the claim: The council member is only partly right. After t = 50 the net rate is -400 + 900 = +500 liters per hour, so the tank rises. But between t = 0 and t = 50 the tank falls unchecked. At t = 50 the volume is 30,000, which is still above 20,000, so the floor is never breached. The claim holds for this scenario, but only because t_start (50 h) is less than the floor time (75 h). The margin is 25 hours. If demand rose to 600 L/h, the pump would start at t = 20000/600 ≈ 33.3 h, still safe; but if the trigger were 15,000 L instead of 30,000, the pump would start at t = 87.5 h - after the floor is already breached at t = 75 h. The reasoning must check the ordering of events, not just the sign of the net rate.
2. Suppose demand spikes to 700 liters per hour during the first 50 hours but the pump still cannot start until volume reaches 30,000. Does the safety rule hold? Show the calculation.
Read the worked answer
With a drain of 700 L/h, the volume model is V(t) = 50000 - 700t.
Time to reach the pump trigger (30,000 L): 50000 - 700t = 30000, so 700t = 20000, giving t ≈ 28.57 hours.
Time to reach the safety floor (20,000 L): 50000 - 700t = 20000, so 700t = 30000, giving t ≈ 42.86 hours.
Since 28.57 < 42.86, the pump starts before the floor is reached, so the safety rule holds. The margin is about 14.3 hours.
Why this matters: The council member's intuition that 'adding more than we remove keeps us safe' is a statement about the post-pump regime only. Safety depends on the pre-pump regime, where the drain acts alone. The correct test is whether t_start < t_floor, which requires comparing 20000/drain against 30000/drain - always satisfied here because the trigger volume (30,000) exceeds the floor (20,000). The real risk would come from a trigger set below the floor, which no drain rate can rescue.