Final answer:
It would take 24 hours to fill the pool with the hose filling and the drainpipe open, calculated by the net rate of the combined effects of filling and draining.
Step-by-step explanation:
The student asked how long it will take to fill a swimming pool with a hose, which can fill it in 8 hours, and a drainpipe open, which can empty it in 12 hours. To solve this, we work with the rates at which the pool is filled and emptied.
Firstly, we consider the rate of the hose, which can fill the pool in 8 hours, so it fills 1/8 of the pool per hour. Secondly, we consider the rate of the drainpipe, which can empty the pool in 12 hours, which means it empties 1/12 of the pool per hour.
When both the hose and drainpipe are working together, the net rate of filling the pool is the fill rate minus the empty rate, which is (1/8 - 1/12) pools per hour. To find a common denominator, we get, 3/24 - 2/24 = 1/24. Therefore, the net rate at which the pool is filled is 1/24 of the pool per hour.
Finally, to fill the pool, we need to divide the whole pool (1 pool) by the net rate (1/24 pools per hour), which gives us 24 hours. Thus, it would take 24 hours to fill the pool with both the hose filling and the drainpipe open.