Final answer:
In this problem involving rates, it's found that when a pump that fills a pool in 4 hours and another that empties the full pool in 6 hours are both operating on a half-full pool, it will take 6 hours for the pool to become full.
Step-by-step explanation:
The subject of this question is Mathematics, specifically an application of rates and time. This problem can be solved using the concept of rates, which states that 'rate * time= work'. The rate of the pump that can fill the pool is 1 pool/4 hours, and the rate of the pump that empties it is -1 pool/6 hours (it is negative because it is emptying, not filling). When both pumps are operating, their combined rate is 1/4 - 1/6 = 1/12 pools/hour. Since the pool is half full, we only need to fill 0.5 pool. Using the formula 'rate * time = work', we solve the equation (1/12) * t = 0.5 for t, and find that t = 6 hours. Therefore, it will take 6 hours for the pool to become full.
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