There are five water pipes `P_1 P_2, P_3, P_4,P_5` connected to a tank, which can fill or empty the tank individually in 45, 40,30,20,15 minutes, respectively. A few of them are inlet pipes and others are the outlet pipes. If the combination of an inlet pipe and an outlet pipe can either empty or fill the whole tank in 1 hour, which one of the following can be the feasible combination of pipes as indicated by {inlet pipe, outlet pipe} in that order only?

Options

A. `{P_3,P_4}` and `{P_4,P_5}`
B. `{P_1,P_2}` and `{P_4,P_5}`
C. `{P_4,P_3}` and `{P_5,P_4}`
D. `{P_4,P_3}` and `{P_4,P_5}`

Answer

To solve the problem, we need to determine the feasible combinations of inlet and outlet pipes that can either fill or empty the tank in 1 hour. We will calculate the efficiency of each pipe and then find the combinations that meet the criteria.

### Step-by-Step Solution:

1. **Identify the Time Taken by Each Pipe:**
- Pipe P1 fills the tank in 45 minutes.
- Pipe P2 fills the tank in 40 minutes.
- Pipe P3 fills the tank in 30 minutes.
- Pipe P4 fills the tank in 20 minutes.
- Pipe P5 fills the tank in 15 minutes.

2. **Calculate the Filling Rate of Each Pipe:**
- To find the rate of each pipe, we can use the formula:
\[ \text{Rate} = \frac{1 \text{ tank}}{\text{Time in minutes}} \]
- Therefore, the rates are:
- Rate of P1 = \( \frac{1}{45} \) tanks/minute
- Rate of P2 = \( \frac{1}{40} \) tanks/minute
- Rate of P3 = \( \frac{1}{30} \) tanks/minute
- Rate of P4 = \( \frac{1}{20} \) tanks/minute
- Rate of P5 = \( \frac{1}{15} \) tanks/minute

3. **Convert Rates to Hourly Rates:**
- Since we need the rates per hour, we multiply each rate by 60 (minutes):
- Rate of P1 = \( \frac{60}{45} = \frac{4}{3} \) tanks/hour
- Rate of P2 = \( \frac{60}{40} = \frac{3}{2} \) tanks/hour
- Rate of P3 = \( \frac{60}{30} = 2 \) tanks/hour
- Rate of P4 = \( \frac{60}{20} = 3 \) tanks/hour
- Rate of P5 = \( \frac{60}{15} = 4 \) tanks/hour

4. **Determine the Efficiency of Each Outlet Pipe:**
- The outlet pipes will have negative rates. We will assume the outlet pipes have the same rates as the inlet pipes:
- Rate of P1 (outlet) = \( -\frac{4}{3} \) tanks/hour
- Rate of P2 (outlet) = \( -\frac{3}{2} \) tanks/hour
- Rate of P3 (outlet) = \( -2 \) tanks/hour
- Rate of P4 (outlet) = \( -3 \) tanks/hour
- Rate of P5 (outlet) = \( -4 \) tanks/hour

5. **Check Combinations of Inlet and Outlet Pipes:**
- We need to find pairs of pipes (inlet, outlet) such that their combined rate equals 1 tank/hour (to fill) or -1 tank/hour (to empty).
- For each combination, we calculate:
\[ \text{Combined Rate} = \text{Inlet Rate} + \text{Outlet Rate} \]

6. **Testing Combinations:**
- Let's check the combinations:
- **Combination (P4, P3)**:
- Inlet (P4) = 3 tanks/hour
- Outlet (P3) = -2 tanks/hour
- Combined Rate = \( 3 - 2 = 1 \) tank/hour (fills the tank)
- **Combination (P5, P4)**:
- Inlet (P5) = 4 tanks/hour
- Outlet (P4) = -3 tanks/hour
- Combined Rate = \( 4 - 3 = 1 \) tank/hour (fills the tank)

7. **Conclusion:**
- The feasible combinations are:
- (P4, P3) → Fills the tank in 1 hour
- (P5, P4) → Fills the tank in 1 hour

### Final Answer:
The feasible combinations of pipes are:
- {P4, P3}
- {P5, P4}