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, strarting 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 need to analyze the rates at which the pipes fill and drain the cistern, and then determine how long it takes to empty the cistern when both pipes are operated alternately.

### Step 1: Determine the filling rate of Pipe A
Pipe A can fill the cistern in 69 hours.

- **Filling rate of Pipe A** = 1 cistern / 69 hours = \( \frac{1}{69} \) cisterns per hour.

### Step 2: Determine the draining rate of Pipe B
Pipe B can drain the cistern in 46 hours.

- **Draining rate of Pipe B** = 1 cistern / 46 hours = \( \frac{1}{46} \) cisterns per hour.

### Step 3: Calculate the effective rate when both pipes are used
When both pipes are opened one after the other, we need to calculate the net effect after 2 hours (1 hour for each pipe).

- In the first hour, Pipe B is opened (draining):
- Amount drained = \( \frac{1}{46} \) cisterns.

- In the second hour, Pipe A is opened (filling):
- Amount filled = \( \frac{1}{69} \) cisterns.

### Step 4: Calculate the net effect after 2 hours
Now, we find the net amount of the cistern after 2 hours:

\[ \text{Net effect in 2 hours} = \text{Amount filled by A} - \text{Amount drained by B} \]
\[ = \frac{1}{69} - \frac{1}{46} \]

To perform this calculation, we need a common denominator. The least common multiple (LCM) of 69 and 46 is 3174.

Converting both fractions:
\[ \frac{1}{69} = \frac{46}{3174} \]
\[ \frac{1}{46} = \frac{69}{3174} \]

Now substituting back:
\[ \text{Net effect} = \frac{46}{3174} - \frac{69}{3174} = \frac{46 - 69}{3174} = \frac{-23}{3174} \]

This means that every 2 hours, the cistern is effectively losing \( \frac{23}{3174} \) of its capacity.

### Step 5: Calculate total time to empty the cistern
The total capacity of the cistern is 1. We need to find out how many such 2-hour cycles are needed to empty the cistern completely.

Let \( x \) be the number of 2-hour cycles needed:
\[ x \cdot \left( \frac{23}{3174} \right) = 1 \]
\[ x = \frac{3174}{23} \approx 138 \]

Since each cycle takes 2 hours, the total time taken to empty the cistern is:
\[ \text{Total time} = 138 \cdot 2 = 276 \text{ hours} \]

### Step 6: Convert hours into days and hours
To convert 276 hours into days and hours:
- Days = \( \frac{276}{24} = 11 \) days
- Remaining hours = \( 276 \mod 24 = 12 \) hours

### Final Answer
Therefore, it will take **11 days and 12 hours** to empty the cistern.

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