Introduction / Context:
This problem involves a tank being filled by a pump while simultaneously losing water through a leak. It tests understanding of net rates: the pump contributes a positive filling rate, while the leak contributes a negative emptying rate. From the difference between the ideal filling time and the actual filling time, we deduce the leak's rate and then compute how long it alone would take to empty the tank.
Given Data / Assumptions:
The pump alone can fill an empty tank in 2 hours.
With the leak present, the tank is filled in 2.5 hours.
The leak alone empties the tank at a constant rate.
The tank capacity remains fixed, and there are no other inlets or outlets.
Concept / Approach:The method is to convert times into rates in terms of tanks per hour. The pump has a filling rate, and the leak has an emptying rate. When operating together, the net rate is the difference between these rates. The actual filling time with the leak is known, so we equate this net rate to the reciprocal of 2.5 hours. From that equation, we solve for the leak's rate, then invert that rate to find the time the leak would take to empty a full tank by itself.
Step-by-Step Solution:Let the tank capacity be 1 unit.Rate of the pump alone = 1/2 tank per hour.Let the leak's emptying rate be L tanks per hour.When both pump and leak are active, net rate = 1/2 - L tank per hour.With the leak, the tank is filled in 2.5 hours, that is 5/2 hours.So net rate = 1 / (5/2) = 2/5 tank per hour.Set up the equation: 1/2 - L = 2/5.Rearrange: L = 1/2 - 2/5.Convert to a common denominator of 10: 1/2 = 5/10 and 2/5 = 4/10.So L = 5/10 - 4/10 = 1/10 tank per hour.Therefore, the leak alone would empty a full tank in 1 / (1/10) = 10 hours.Verification / Alternative check:Check the net rate using L = 1/10.Net rate = 1/2 - 1/10.1/2 = 5/10, so net rate = 5/10 - 1/10 = 4/10 = 2/5 tank per hour.Time to fill one tank at 2/5 tank per hour = 1 / (2/5) = 5/2 hours = 2.5 hours.This matches the given information, so the calculation is confirmed.Why Other Options Are Wrong:8 hours and 9 hours imply a leak that is stronger than 1/10 tank per hour, which would prevent the pump from filling the tank in 2.5 hours under the given conditions. 10.5 hours and 12 hours suggest a weaker leak, which would slow down the pump less than the observed difference between 2 hours and 2.5 hours. Only 10 hours fits the data exactly.
Common Pitfalls:Some students mistakenly add the leak's rate instead of subtracting it, leading to a net rate greater than the pump's rate, which cannot be correct if filling is taking longer. Others mishandle the mixed number 2.5 hours and treat it as 2 hours and 50 minutes. It is safer to convert 2.5 hours to a fraction 5/2 before forming equations. Keeping track of signs and units at each step helps avoid these errors.
Final Answer:The leak alone would drain the full tank in 10 hours.
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