A cistern has a leak which would empty it in 8 hours. A tap is turned on, which admits 6 litres a minute into the cistern and it is now emptied in 12 hours. How many litres can the cistern hold?

Options

A. 8000 litres
B. 8400 litres
C. 8640 litres
D. 8650 litres

Answer

To solve the problem step by step, we will first analyze the information given and then calculate the capacity of the cistern.

### Step 1: Understand the Problem
We know that:
- A leak can empty the cistern in 8 hours.
- A tap adds 6 liters per minute to the cistern.
- With the tap on, the cistern is emptied in 12 hours.

### Step 2: Calculate the Rate of the Leak
The leak empties the cistern in 8 hours. Therefore, the rate of the leak can be calculated as:
- If the cistern's total capacity is \( C \) liters, the leak's rate is \( \frac{C}{8} \) liters per hour.

### Step 3: Calculate the Rate of the Tap
The tap adds 6 liters per minute. To convert this to an hourly rate:
- \( 6 \text{ liters/minute} \times 60 \text{ minutes/hour} = 360 \text{ liters/hour} \).

### Step 4: Calculate the Combined Rate with the Tap On
When the tap is on, the cistern is emptied in 12 hours. Thus, the effective rate at which the cistern is emptied (considering both the leak and the tap) is:
- \( \frac{C}{12} \) liters per hour.

### Step 5: Set Up the Equation
Now we can set up the equation based on the rates:
- Rate of leak + Rate of tap = Rate of emptying with tap on.
- \( \frac{C}{8} - 360 = \frac{C}{12} \).

### Step 6: Solve the Equation
To solve for \( C \), we first eliminate the fractions by finding a common denominator, which is 24:
- Multiply the entire equation by 24:
\[ 24 \left(\frac{C}{8}\right) - 24 \times 360 = 24 \left(\frac{C}{12}\right) \]
This simplifies to:
\[ 3C - 8640 = 2C \]
- Rearranging gives:
\[ 3C - 2C = 8640 \]
\[ C = 8640 \text{ liters} \]

### Conclusion
The capacity of the cistern is **8640 liters**.