A leak in the bottom of a tank can empty the full tank in 6 hours. An inlet pipe fills water at the rate of 4 litres a minute. When the tank is full, the inlet is opened and due to the leak the tank is empty in 8 hours. Find the capacity of the tank.

Options

A. 5760 litres
B. 96 litres
C. 10 litres
D. 24 litres

Answer

To solve the problem step by step, we will break down the information given and use it to find the capacity of the tank.

### Step 1: Determine the leak rate
The leak can empty the full tank in 6 hours. Therefore, the rate at which the leak empties the tank (L) can be calculated as:
\[ L = \frac{1 \text{ tank}}{6 \text{ hours}} = \frac{1}{6} \text{ tank per hour} \]

### Step 2: Determine the inlet rate
The inlet pipe fills water at the rate of 4 liters per minute. To convert this to an hourly rate:
\[ \text{Inlet rate (I)} = 4 \text{ liters/min} \times 60 \text{ min/hour} = 240 \text{ liters/hour} \]

### Step 3: Determine the combined effect of the leak and the inlet
When the tank is full, the inlet is opened and due to the leak, the tank is empty in 8 hours. This means that the combined effect of the leak and the inlet leads to the tank emptying in 8 hours:
\[ \text{Net rate} = \frac{1 \text{ tank}}{8 \text{ hours}} = \frac{1}{8} \text{ tank per hour} \]

### Step 4: Set up the equation
The net rate can be expressed as the difference between the leak rate and the inlet rate:
\[ L - I = \frac{1}{8} \]
Substituting the values we have:
\[ \frac{1}{6} - \frac{240}{C} = -\frac{1}{8} \]
where \(C\) is the capacity of the tank in liters.

### Step 5: Solve for the capacity of the tank
To solve for \(C\), we first convert the leak rate to liters. The leak rate in liters per hour can be calculated as:
\[ L = \frac{C}{6} \]
Now substituting this back into the equation:
\[ \frac{C}{6} - 240 = -\frac{C}{8} \]
To eliminate the fractions, multiply through by 24 (the least common multiple of 6 and 8):
\[ 4C - 5760 = -3C \]
Combine like terms:
\[ 4C + 3C = 5760 \]
\[ 7C = 5760 \]
Now, divide by 7:
\[ C = \frac{5760}{7} = 823.14 \text{ liters} \]

### Step 6: Final calculation for the capacity
Since we need the capacity in whole liters, we round to the nearest whole number:
\[ C \approx 823 \text{ liters} \]

### Conclusion
The capacity of the tank is approximately **823 liters**.