A cistern has a leak which would empty it in 6 hours. A tap is turned on which fills the cistern @ 10 liters per hour and then it is emptied in IS hours. What is the capacity of the cistern?

Options

A. 100 litres
B. 166.66 litre
C. 60.66 litre
D. none of these

Answer

To solve the problem step by step, we will analyze the situation with the leak and the tap filling the cistern.

### Step 1: Determine the capacity of the cistern
Let the capacity of the cistern be \( C \) liters.

### Step 2: Calculate the rate at which the leak empties the cistern
The leak can empty the cistern in 6 hours. Therefore, the rate of the leak is:
\[ \text{Rate of leak} = \frac{C}{6} \text{ liters per hour} \]

### Step 3: Calculate the rate at which the tap fills the cistern
The tap fills the cistern at a rate of 10 liters per hour.

### Step 4: Determine the time taken to empty the cistern when both the leak and the tap are operating
It is given that the cistern is emptied in 15 hours when the tap is on. This means that the net effect of the leak and the tap is to empty the cistern in 15 hours.

### Step 5: Calculate the effective rate when both the leak and the tap are working
The effective rate when both the leak and the tap are working can be expressed as:
\[ \text{Effective rate} = \text{Rate of tap} - \text{Rate of leak} \]
Substituting the known values:
\[ \text{Effective rate} = 10 - \frac{C}{6} \]

### Step 6: Relate the effective rate to the time taken to empty the cistern
Since the cistern is emptied in 15 hours, we can express the effective rate as:
\[ \text{Effective rate} = \frac{C}{15} \]

### Step 7: Set up the equation
Now we can set the two expressions for the effective rate equal to each other:
\[ 10 - \frac{C}{6} = \frac{C}{15} \]

### Step 8: Solve the equation for \( C \)
To eliminate the fractions, we can multiply the entire equation by 30 (the least common multiple of 6 and 15):
\[ 30 \cdot 10 - 30 \cdot \frac{C}{6} = 30 \cdot \frac{C}{15} \]
This simplifies to:
\[ 300 - 5C = 2C \]
Now, combine like terms:
\[ 300 = 7C \]
Thus,
\[ C = \frac{300}{7} \approx 42.86 \text{ liters} \]

### Step 9: Conclusion
The capacity of the cistern is approximately 42.86 liters.