To solve the problem, we need to determine how long it will take for the cistern to be emptied using the given pipes. Let's break down the solution step by step.
### Step 1: Determine the rates of the pipes
- **Pipe A** can fill the cistern in 12 hours. Therefore, its rate of filling is:
\[
\text{Rate of A} = \frac{1}{12} \text{ cisterns per hour}
\]
- **Pipe B** can fill the cistern in 36 hours. Therefore, its rate of filling is:
\[
\text{Rate of B} = \frac{1}{36} \text{ cisterns per hour}
\]
- **Pipe C** can drain the cistern in 6 hours. Therefore, its rate of draining is:
\[
\text{Rate of C} = \frac{1}{6} \text{ cisterns per hour}
\]
### Step 2: Calculate the net effect of the pipes
When all pipes are considered, we need to calculate the effective rate when they are used in the sequence C, A, B.
1. **When Pipe C is open for 1 hour:**
\[
\text{Amount drained by C in 1 hour} = \frac{1}{6}
\]
2. **When Pipe A is open for 1 hour:**
\[
\text{Amount filled by A in 1 hour} = \frac{1}{12}
\]
3. **When Pipe B is open for 1 hour:**
\[
\text{Amount filled by B in 1 hour} = \frac{1}{36}
\]
### Step 3: Calculate the total amount after one complete cycle (C, A, B)
In one complete cycle (C for 1 hour, A for 1 hour, B for 1 hour):
\[
\text{Total amount} = \left(-\frac{1}{6}\right) + \left(\frac{1}{12}\right) + \left(\frac{1}{36}\right)
\]
### Step 4: Find a common denominator to simplify
The least common multiple (LCM) of 6, 12, and 36 is 36. Thus, we can rewrite the fractions:
\[
-\frac{1}{6} = -\frac{6}{36}, \quad \frac{1}{12} = \frac{3}{36}, \quad \frac{1}{36} = \frac{1}{36}
\]
Now, substituting these into the total amount:
\[
\text{Total amount} = -\frac{6}{36} + \frac{3}{36} + \frac{1}{36} = -\frac{6 - 3 - 1}{36} = -\frac{4}{36} = -\frac{1}{9}
\]
### Step 5: Determine the time to empty the cistern
The total amount drained in one complete cycle is \(-\frac{1}{9}\) of the cistern. To empty the full cistern (1 cistern), we set up the equation:
\[
\text{Number of cycles} \times \left(-\frac{1}{9}\right) = -1
\]
This gives:
\[
\text{Number of cycles} = 9
\]
### Step 6: Calculate the total time taken
Each cycle consists of 3 hours (1 hour for C, 1 hour for A, and 1 hour for B):
\[
\text{Total time} = 9 \text{ cycles} \times 3 \text{ hours/cycle} = 27 \text{ hours}
\]
### Final Answer
The total time taken to empty the filled cistern is **27 hours**.
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