A swimming pool can be filled by a vent in 10 hours. When the pool is filled, it can be emptied bya drain pipe in 20 hours. The pool manager mistakenly leaves the drain pipe open while trying tofill the pool. How long will it take to fill the pool?

Net rate of filling = 1/10 - 1/20 pool/hour = 1/20 pool/hour

Fill time = (1 pool)/(1/20 pool/h)
= 20 hours

To find out how long it will take to fill the pool, we need to calculate the net rate at which the pool is being filled.

Let's denote the rate at which the vent fills the pool as R1 (in pools per hour) and the rate at which the drain pipe empties the pool as R2 (in pools per hour).

Given that the vent fills the pool in 10 hours, we can say that R1 = 1/10 pools per hour.

Similarly, the drain pipe empties the pool in 20 hours, so R2 = 1/20 pools per hour.

Now, to calculate the net rate of filling the pool, we subtract the rate of emptying from the rate of filling:

Net Rate = R1 - R2 = 1/10 - 1/20 = 2/20 - 1/20 = 1/20 pools per hour.

Therefore, the pool is being filled at a rate of 1/20 pools per hour.

To find out how long it will take to fill the pool, we divide the total volume of the pool (which we'll assume to be 1 pool) by the net rate of filling:

Time to fill pool = 1 pool / (1/20 pools per hour).

When we divide 1 by 1/20, we get:

Time to fill pool = 20 hours.

Therefore, it will take 20 hours to fill the pool when the drain pipe is mistakenly left open.