An inlet pipe on a swimming pool can be used to fill the pool in 22
hours. The drain pipe can be used to empty the pool in 24
hours. If the pool is 14
filled and then the inlet pipe and drain pipe are opened, how many more hours would it take to fill the pool? Round your answer to two decimal places, if needed.

To solve this problem, we can use the concept of rates of work. Let's define the problem in terms of rates:

1. **Inlet Pipe**: It fills the pool in 22 hours. Therefore, the rate of the inlet pipe is:
   \[
   \text{Rate of Inlet Pipe} = \frac{1 \text{ pool}}{22 \text{ hours}} = \frac{1}{22} \text{ pools per hour}
   \]

2. **Drain Pipe**: It empties the pool in 24 hours. Therefore, the rate of the drain pipe is:
   \[
   \text{Rate of Drain Pipe} = \frac{1 \text{ pool}}{24 \text{ hours}} = \frac{1}{24} \text{ pools per hour}
   \]

3. **Combined Rate**: When both the inlet and drain pipes are open, their rates are combined. Since one is filling and the other is draining, the combined rate is the difference between the inlet rate and the drain rate:
   \[
   \text{Combined Rate} = \frac{1}{22} - \frac{1}{24}
   \]

To find the combined rate, let's calculate it:

\[
\text{Combined Rate} = \frac{1}{22} - \frac{1}{24} = \frac{24 - 22}{22 \times 24} = \frac{2}{528} = \frac{1}{264} \text{ pools per hour}
\]

This means that the pool fills at a rate of \(\frac{1}{264}\) pools per hour when both pipes are open.

4. **Amount of Pool Already Filled**: The pool is already \(\frac{1}{4}\) filled. Therefore, the amount of pool left to be filled is:
   \[
   1 - \frac{1}{4} = \frac{3}{4} \text{ of the pool}
   \]

5. **Time to Fill the Remaining Pool**: To find out how many more hours it will take to fill the remaining \(\frac{3}{4}\) of the pool at the combined rate, we use the formula:

   \[
   \text{Time} = \frac{\text{Amount to Fill}}{\text{Rate}}
   \]

   Plugging in the values:

   \[
   \text{Time} = \frac{\frac{3}{4} \text{ pools}}{\frac{1}{264} \text{ pools per hour}} = \frac{3}{4} \times 264 = 198 \text{ hours}
   \]

Thus, it would take **198 hours** to fill the remaining part of the pool when both the inlet and drain pipes are open.

Do you want details or have any questions?

Here are 5 related questions for further exploration:

1. If the inlet pipe's rate was doubled, how would that affect the time needed to fill the pool?
2. What would happen if both the inlet and drain pipes had the same rate?
3. How would the filling time change if the pool started at half full?
4. If a third pipe could either fill or drain the pool, how could it be used to optimize the filling time?
5. How would the situation change if the drain pipe worked twice as fast as it does now?

**Tip:** In problems involving rates, combining rates often means adding or subtracting the rates depending on whether the tasks are working together or against each other.

Learn how to calculate the time required to fill a swimming pool given the rates of an inlet pipe and a drain pipe. This problem involves using rates of work to determine the combined rate of filling and draining the pool. Suitable for advanced high school students.

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