Pipe A is an inlet pipe that can fill an empty cistern in 69 hours. Pipe B can drain the filled cistern in 46 hours. When the cistern was filled the two pipes are opened one at a time for an hour each, starting with Pipe B. how long will it take for the cistern to be empty?

Options

A. 11 days 10 hours
B. 11 days 7 hours
C. 11 days 12 hours
D. 1 days 13 hours

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the capacities of the pipes
- **Pipe A** fills the cistern in 69 hours.
- **Pipe B** drains the cistern in 46 hours.

### Step 2: Calculate the total capacity of the cistern
To find the efficiency of each pipe, we can assume the capacity of the cistern is the least common multiple (LCM) of 69 and 46.

- The LCM of 69 and 46 is 138.

### Step 3: Calculate the efficiency of Pipe A
- Efficiency of Pipe A = Total capacity / Time taken by Pipe A
- Efficiency of Pipe A = 138 / 69 = 2 units/hour (filling)

### Step 4: Calculate the efficiency of Pipe B
- Efficiency of Pipe B = Total capacity / Time taken by Pipe B
- Efficiency of Pipe B = 138 / 46 = 3 units/hour (draining)
- Since Pipe B is draining, its efficiency will be negative: -3 units/hour.

### Step 5: Determine the pattern of operation
The pipes are opened one at a time for one hour each, starting with Pipe B:
1. First hour: Pipe B (drains) = -3 units
2. Second hour: Pipe A (fills) = +2 units

### Step 6: Calculate the net efficiency for two hours
- Net efficiency for two hours = Efficiency of Pipe B + Efficiency of Pipe A
- Net efficiency for two hours = -3 + 2 = -1 unit

### Step 7: Calculate the total units drained in two hours
- In two hours, the total units drained = 2 hours * -1 unit/hour = -2 units.

### Step 8: Determine how many cycles are needed to empty the cistern
Since the cistern has a total capacity of 138 units, we need to find out how many cycles (2 hours each) it will take to empty it:
- Total units to be drained = 138 units.
- Units drained in one cycle (2 hours) = -2 units.
- Number of cycles needed = 138 / 2 = 69 cycles.

### Step 9: Calculate total time taken
- Each cycle takes 2 hours.
- Total time taken = 69 cycles * 2 hours/cycle = 138 hours.

### Step 10: Add the last hour of operation
After 138 hours, the cistern is still not empty, and we need to account for the last hour of operation:
- In the 139th hour, Pipe B will drain another 3 units.
- Therefore, the total time taken to empty the cistern = 138 + 1 = 139 hours.

### Final Answer
The total time taken for the cistern to be empty is **139 hours**.

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