A cistern has a leak which would empty 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. The cistern can hold

Options

A. 6840 litres
B. 7860 litres
C. 8640 litres
D. 1000 litres

Answer

To solve the problem step by step, we will analyze the information given and apply the concepts related to pipes and cisterns.

### Step-by-Step Solution:

1. **Understand the Problem**:
- A cistern has a leak that can empty it in 8 hours.
- A tap is turned on that fills the cistern at a rate of 6 liters per minute.
- With the tap on, the cistern is emptied in 12 hours.

2. **Calculate the Rate of the Leak**:
- If the leak can empty the cistern in 8 hours, the rate of the leak (let's call it A) is:
\[ \text{Rate of Leak (A)} = \frac{1 \text{ cistern}}{8 \text{ hours}} = \frac{1}{8} \text{ cistern per hour} \]

3. **Calculate the Rate with the Tap On**:
- When the tap is on, the cistern is emptied in 12 hours. Therefore, the combined rate of the leak and the tap (let's call it B) is:
\[ \text{Rate with Tap (B)} = \frac{1 \text{ cistern}}{12 \text{ hours}} = \frac{1}{12} \text{ cistern per hour} \]

4. **Set Up the Equation**:
- The rate of the tap filling the cistern (let's denote it as T) can be calculated from the flow rate:
- The tap fills at 6 liters per minute, which is:
\[ T = 6 \text{ liters/min} \times 60 \text{ min/hour} = 360 \text{ liters/hour} \]
- Now, we can express the relationship between the rates:
\[ \text{Rate of Leak (A)} - \text{Rate of Tap (T)} = \text{Rate with Tap (B)} \]
- Substituting the values we have:
\[ \frac{1}{8} - T = \frac{1}{12} \]

5. **Solve for T**:
- Rearranging the equation gives:
\[ T = \frac{1}{8} - \frac{1}{12} \]
- To solve this, find a common denominator (which is 24):
\[ T = \frac{3}{24} - \frac{2}{24} = \frac{1}{24} \text{ cistern per hour} \]

6. **Calculate the Capacity of the Cistern**:
- Since T represents the rate at which the tap fills the cistern, we can find the total capacity of the cistern by considering how long it would take to fill it completely.
- The time taken to fill the cistern at this rate is:
\[ \text{Time} = \frac{1 \text{ cistern}}{T} = \frac{1}{\frac{1}{24}} = 24 \text{ hours} \]

7. **Convert Time to Capacity**:
- Now, we calculate the total capacity of the cistern in liters:
\[ \text{Capacity} = \text{Rate of Tap} \times \text{Time} = 360 \text{ liters/hour} \times 24 \text{ hours} = 8640 \text{ liters} \]

### Final Answer:
The capacity of the cistern is **8640 liters**.