It takes pump A 2 hours less time that pump B to empty a swimming pool. Pump A is started at 8:00 a.m. and pump B is started at 10:00 a.m. At 12:00 p.m. 60% of the pool is empty when pump B broke down. How much time after 12:00 p.m. would it take pump A to empty the pool?
Question
It takes pump A 2 hours less time that pump B to empty a swimming pool. Pump A is started at 8:00 a.m. and pump B is started at 10:00 a.m. At 12:00 p.m. 60% of the pool is empty when pump B broke down. How much time after 12:00 p.m. would it take pump A to empty the pool?
Solution
To solve this problem, we need to first understand the rates at which the pumps can empty the pool.
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Let's denote the time it takes for pump B to empty the pool as 't'. Therefore, the time it takes for pump A to empty the pool is 't - 2' hours.
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The rate at which pump A can empty the pool is 1/(t - 2) pools per hour, and the rate for pump B is 1/t pools per hour.
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From 8:00 a.m. to 12:00 p.m., pump A has been working for 4 hours and pump B has been working for 2 hours. Therefore, the amount of the pool that has been emptied is 4/(t - 2) + 2/t = 0.6 (since 60% of the pool is empty).
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We can solve this equation for 't'. However, since it's a bit complex, we can make an educated guess. If we try t = 5, we get 4/(5 - 2) + 2/5 = 4/3 + 2/5 = 1.33 + 0.4 = 1.73, which is too high. If we try t = 6, we get 4/(6 - 2) + 2/6 = 1 + 1/3 = 1.33, which is too low. If we try t = 7, we get 4/(7 - 2) + 2/7 = 0.8 + 0.29 = 1.09, which is still too low. If we try t = 8, we get 4/(8 - 2) + 2/8 = 0.67 + 0.25 = 0.92, which is too low. If we try t = 9, we get 4/(9 - 2) + 2/9 = 0.57 + 0.22 = 0.79, which is too low. If we try t = 10, we get 4/(10 - 2) + 2/10 = 0.5 + 0.2 = 0.7, which is too low. Therefore, we can conclude that t is between 5 and 6.
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Now, we know that 60% of the pool has been emptied by 12:00 p.m. and pump B is broken. Therefore, pump A needs to empty the remaining 40% of the pool.
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The rate at which pump A can empty the pool is 1/(t - 2) pools per hour. Since we know that 't' is between 5 and 6, the rate is between 1/3 and 1/2 pools per hour.
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Therefore, to empty the remaining 40% of the pool, it would take pump A between 0.4/(1/3) = 1.2 hours and 0.4/(1/2) = 0.8 hours. So, it would take pump A between 48 minutes and 72 minutes to empty the pool after 12:00 p.m.
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