Two fill pipes A and B can fill cistern in 56 and 63 min respectively. Both fill pipes are opened together. But 7 minutes before the cistern will fill pipe B is closed. In how much time will the cistern take to fill?

Options

A. `29(7)/(17)`min
B. `37(11)/(17)`min
C. `35(5)/(17)`min
D. `32(16)/(17)`min

Answer

To solve the problem, we need to determine how long it takes to fill the cistern when both pipes A and B are opened together, with pipe B closing 7 minutes before the cistern is completely filled.

### Step-by-Step Solution:

1. **Determine the Filling Rates of Pipes A and B:**
- Pipe A can fill the cistern in 56 minutes.
- Pipe B can fill the cistern in 63 minutes.
- Therefore, the filling rates (efficiencies) of the pipes are:
- Rate of Pipe A = \( \frac{1}{56} \) of the cistern per minute.
- Rate of Pipe B = \( \frac{1}{63} \) of the cistern per minute.

2. **Calculate the Least Common Multiple (LCM):**
- To find a common capacity for calculations, we can use the LCM of the times taken by both pipes.
- LCM of 56 and 63 = 504 (this is the capacity of the cistern).

3. **Calculate the Efficiency in Terms of the Capacity:**
- Efficiency of Pipe A:
- In 1 minute, Pipe A fills \( \frac{504}{56} = 9 \) liters.
- Efficiency of Pipe B:
- In 1 minute, Pipe B fills \( \frac{504}{63} = 8 \) liters.

4. **Set Up the Equation:**
- Let the total time taken to fill the cistern be \( X \) minutes.
- Pipe A runs for the entire time \( X \) minutes.
- Pipe B runs for \( X - 7 \) minutes (since it closes 7 minutes before the cistern is full).
- The total amount of cistern filled can be expressed as:
\[ 9X + 8(X - 7) = 504 \]

5. **Simplify the Equation:**
- Expand the equation:
\[ 9X + 8X - 56 = 504 \]
- Combine like terms:
\[ 17X - 56 = 504 \]

6. **Solve for X:**
- Add 56 to both sides:
\[ 17X = 560 \]
- Divide by 17:
\[ X = \frac{560}{17} \approx 32.94 \text{ minutes} \]

7. **Convert to Minutes and Seconds:**
- \( 32.94 \) minutes can be expressed as \( 32 \) minutes and \( 0.94 \times 60 \approx 56.4 \) seconds.
- Therefore, the time taken to fill the cistern is approximately \( 32 \) minutes and \( 56 \) seconds.

### Final Answer:
The cistern will take approximately **32 minutes and 56 seconds** to fill.