Question
Inlet pipe can fill a swimming pool in nine hours and an outlet pipe can empty a pool in 12 hours through an air. Both pipes are left open assume the error was discovered after both pipes have been running for three hours, and the outlet pipe was then closed. How much more time would it then be required to fill up the pool?
Answers
GPT-4o mini
To determine how much more time is required to fill the pool after the outlet pipe has been closed, we can follow these steps:
1. **Calculate the rates of filling and emptying:**
- The inlet pipe fills the pool in 9 hours. Therefore, the rate of the inlet pipe is:
\[
\text{Rate of inlet pipe} = \frac{1}{9} \text{ pools per hour}
\]
- The outlet pipe empties the pool in 12 hours. Therefore, the rate of the outlet pipe is:
\[
\text{Rate of outlet pipe} = \frac{1}{12} \text{ pools per hour}
\]
2. **Determine the net rate when both pipes are open:**
When both pipes are open, the net rate of filling the pool is:
\[
\text{Net rate} = \text{Rate of inlet pipe} - \text{Rate of outlet pipe} = \frac{1}{9} - \frac{1}{12}
\]
To subtract these fractions, find a common denominator. The least common multiple of 9 and 12 is 36.
\[
\frac{1}{9} = \frac{4}{36} \quad \text{and} \quad \frac{1}{12} = \frac{3}{36}
\]
Thus, the net rate becomes:
\[
\text{Net rate} = \frac{4}{36} - \frac{3}{36} = \frac{1}{36} \text{ pools per hour}
\]
3. **Calculate how much of the pool is filled in 3 hours with both pipes open:**
The total amount filled in 3 hours is:
\[
\text{Amount filled} = \text{Net rate} \times \text{Time} = \frac{1}{36} \times 3 = \frac{3}{36} = \frac{1}{12} \text{ of the pool}
\]
4. **Determine how much of the pool remains to be filled:**
After 3 hours, the amount of the pool that still needs to be filled is:
\[
1 - \frac{1}{12} = \frac{11}{12}
\]
5. **Calculate how long it would take to fill the remaining part of the pool with only the inlet pipe:**
Now, with only the inlet pipe open, which has a filling rate of \(\frac{1}{9}\) pools per hour, we can find the time required to fill \(\frac{11}{12}\) of the pool:
Let \(t\) be the time in hours required:
\[
\text{Rate} \times t = \frac{11}{12}
\]
Thus:
\[
\frac{1}{9} \times t = \frac{11}{12}
\]
To solve for \(t\), we multiply both sides by 9:
\[
t = 9 \times \frac{11}{12} = \frac{99}{12} = 8.25 \text{ hours}
\]
Therefore, it would take an additional **8.25 hours** to fill the pool after the outlet pipe is closed.
1. **Calculate the rates of filling and emptying:**
- The inlet pipe fills the pool in 9 hours. Therefore, the rate of the inlet pipe is:
\[
\text{Rate of inlet pipe} = \frac{1}{9} \text{ pools per hour}
\]
- The outlet pipe empties the pool in 12 hours. Therefore, the rate of the outlet pipe is:
\[
\text{Rate of outlet pipe} = \frac{1}{12} \text{ pools per hour}
\]
2. **Determine the net rate when both pipes are open:**
When both pipes are open, the net rate of filling the pool is:
\[
\text{Net rate} = \text{Rate of inlet pipe} - \text{Rate of outlet pipe} = \frac{1}{9} - \frac{1}{12}
\]
To subtract these fractions, find a common denominator. The least common multiple of 9 and 12 is 36.
\[
\frac{1}{9} = \frac{4}{36} \quad \text{and} \quad \frac{1}{12} = \frac{3}{36}
\]
Thus, the net rate becomes:
\[
\text{Net rate} = \frac{4}{36} - \frac{3}{36} = \frac{1}{36} \text{ pools per hour}
\]
3. **Calculate how much of the pool is filled in 3 hours with both pipes open:**
The total amount filled in 3 hours is:
\[
\text{Amount filled} = \text{Net rate} \times \text{Time} = \frac{1}{36} \times 3 = \frac{3}{36} = \frac{1}{12} \text{ of the pool}
\]
4. **Determine how much of the pool remains to be filled:**
After 3 hours, the amount of the pool that still needs to be filled is:
\[
1 - \frac{1}{12} = \frac{11}{12}
\]
5. **Calculate how long it would take to fill the remaining part of the pool with only the inlet pipe:**
Now, with only the inlet pipe open, which has a filling rate of \(\frac{1}{9}\) pools per hour, we can find the time required to fill \(\frac{11}{12}\) of the pool:
Let \(t\) be the time in hours required:
\[
\text{Rate} \times t = \frac{11}{12}
\]
Thus:
\[
\frac{1}{9} \times t = \frac{11}{12}
\]
To solve for \(t\), we multiply both sides by 9:
\[
t = 9 \times \frac{11}{12} = \frac{99}{12} = 8.25 \text{ hours}
\]
Therefore, it would take an additional **8.25 hours** to fill the pool after the outlet pipe is closed.