An inlet pipe on a swimming pool can be used to fill the pool in 28 hours. The drain pipe can be used to empty the pool in 42 hours. If the pool is one-third filled and then the drain pipe is accidentally opened, how long will it take to fill the pool?
To solve this problem, we need to calculate the rate at which each pipe can fill or empty the pool. Let's begin by finding the rates of each pipe.
The rate at which the inlet pipe fills the pool is 1 pool per 28 hours. Therefore, we can express this rate as 1/28 pool per hour.
Similarly, the rate at which the drain pipe empties the pool is 1 pool per 42 hours. So, the drain pipe's rate can be expressed as 1/42 pool per hour.
Now, let's consider the scenario where the pool is already one-third filled. This means that 2/3 of the pool still needs to be filled.
Since the drain pipe is accidentally opened, it will now act as a drain, emptying the pool while the inlet pipe is trying to fill it.
Now, we need to calculate the net rate of filling the pool by subtracting the drain pipe's rate from the inlet pipe's rate. This can be done as follows:
Net rate = Inlet pipe rate - Drain pipe rate
Net rate = (1/28) - (1/42)
= 3/84 - 2/84
= 1/84 pool per hour
From the net rate, we can conclude that 1/84 of the pool can be filled every hour.
Since we need to fill 2/3 of the pool, we can calculate the time it will take to fill the remaining portion by setting up a proportion:
(1/84) pool per hour = (2/3) pool / x hours
Cross multiplying, we get:
(1/84) * x = (2/3)
To solve for x, multiply both sides by 84:
x = (2/3) * 84
x = 56
Therefore, it will take 56 hours to fill the pool after the drain pipe is accidentally opened.