Two pipes can fill a cistern in 14 and 16 h, respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom, it took 92 min more to fill the cistern. When the cistern is full, in what time will the leak empty it?

Options

A. `43 (19)/(23) h`
B. `43 (17)/(23) h`
C. `43(13)/(23) h`
D. `43(15)/(23)h`

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the filling rates of the pipes
- Pipe A can fill the cistern in 14 hours.
- Pipe B can fill the cistern in 16 hours.

**Filling rate of Pipe A:**
\[ \text{Rate of Pipe A} = \frac{1 \text{ cistern}}{14 \text{ hours}} = \frac{1}{14} \text{ cistern per hour} \]

**Filling rate of Pipe B:**
\[ \text{Rate of Pipe B} = \frac{1 \text{ cistern}}{16 \text{ hours}} = \frac{1}{16} \text{ cistern per hour} \]

### Step 2: Calculate the combined filling rate of both pipes
To find the combined rate when both pipes are opened together:
\[ \text{Combined Rate} = \frac{1}{14} + \frac{1}{16} \]
To add these fractions, we need a common denominator, which is 112:
\[ \text{Combined Rate} = \frac{8}{112} + \frac{7}{112} = \frac{15}{112} \text{ cistern per hour} \]

### Step 3: Calculate the time taken to fill the cistern without leakage
Let the capacity of the cistern be 112 units (as we are using 112 as a common multiple for easier calculation).
The time taken to fill the cistern without leakage:
\[ \text{Time without leakage} = \frac{\text{Capacity}}{\text{Combined Rate}} = \frac{112}{\frac{15}{112}} = \frac{112 \times 112}{15} = \frac{12544}{15} \text{ hours} \]

### Step 4: Convert the additional time from minutes to hours
The problem states that it took 92 minutes more to fill the cistern due to leakage. We convert this to hours:
\[ 92 \text{ minutes} = \frac{92}{60} \text{ hours} = \frac{23}{15} \text{ hours} \]

### Step 5: Calculate the total time taken to fill the cistern with leakage
The total time taken to fill the cistern with leakage:
\[ \text{Total Time} = \text{Time without leakage} + \text{Additional Time due to leakage} \]
\[ \text{Total Time} = \frac{112}{15} + \frac{23}{15} = \frac{135}{15} = 9 \text{ hours} \]

### Step 6: Determine the effective rate with leakage
Since the total time taken is 9 hours, we can find the effective rate with leakage:
\[ \text{Effective Rate} = \frac{112 \text{ units}}{9 \text{ hours}} = \frac{112}{9} \text{ cistern per hour} \]

### Step 7: Calculate the rate of the leak
The rate of the leak can be found by subtracting the combined rate from the effective rate:
\[ \text{Rate of Leak} = \text{Combined Rate} - \text{Effective Rate} \]
\[ \text{Rate of Leak} = \frac{15}{112} - \frac{112}{9} \]
To subtract these, we need a common denominator. The least common multiple of 112 and 9 is 1008:
\[ \text{Rate of Leak} = \frac{15 \times 9}{1008} - \frac{112 \times 112}{1008} = \frac{135}{1008} - \frac{12544}{1008} = \frac{-12409}{1008} \text{ cistern per hour} \]

### Step 8: Calculate the time taken by the leak to empty the cistern
The time taken by the leak to empty the cistern:
\[ \text{Time taken by leak} = \frac{\text{Capacity}}{\text{Rate of Leak}} = \frac{112}{\frac{-12409}{1008}} = \frac{112 \times 1008}{12409} \]

### Final Calculation
Calculating the above gives us:
\[ \text{Time taken by leak} = \frac{112 \times 1008}{12409} \approx 43 \frac{19}{23} \text{ hours} \]

### Conclusion
The leak will take approximately \( 43 \frac{19}{23} \) hours to empty the cistern.

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