Through an inlet, a tank takes 8 h to get filled up. Due to a leak in the bottom, it takes 2 h more to get it filled completely. If the tank is full, how much time will the leak take to empty it?

Options

A. 16h
B. 20h
C. 32h
D. 40h

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the problem
The tank can be filled by an inlet in 8 hours. Due to a leak, it takes 2 hours longer (10 hours) to fill the tank. We need to find out how long it takes for the leak to empty the tank.

### Step 2: Define the tank's capacity
Let's assume the capacity of the tank is 40 units (this is a convenient number since it is a common multiple of the filling times).

### Step 3: Calculate the filling rates
1. **Inlet filling rate**:
- The inlet fills the tank in 8 hours.
- Therefore, the filling rate of the inlet = Total capacity / Time taken = 40 units / 8 hours = 5 units per hour.

2. **Combined filling rate (inlet + leak)**:
- The tank takes 10 hours to fill when both the inlet and leak are working.
- Therefore, the combined filling rate = Total capacity / Time taken = 40 units / 10 hours = 4 units per hour.

### Step 4: Set up the equation for the leak
Let the rate of the leak be \( L \) units per hour. The equation for the combined filling rate is:
\[ \text{Inlet rate} - \text{Leak rate} = \text{Combined rate} \]
\[ 5 - L = 4 \]

### Step 5: Solve for the leak rate
Rearranging the equation gives:
\[ L = 5 - 4 = 1 \text{ unit per hour} \]

### Step 6: Calculate the time taken by the leak to empty the tank
Since the leak empties the tank at a rate of 1 unit per hour, the time taken to empty the entire tank (40 units) is:
\[ \text{Time} = \frac{\text{Total capacity}}{\text{Leak rate}} = \frac{40 \text{ units}}{1 \text{ unit/hour}} = 40 \text{ hours} \]

### Final Answer
The leak will take **40 hours** to empty the tank.

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