A pipe can fill an empty tank in 8 h. After it is kept open for 2 h, another pipe which can fill the tank in 6 h is opened. In how many more hours will the tank be filled?

Options

A. `18/7h`
B. `19/7h`
C. `19/9h`
D. `18/8h`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Determine the filling rates of both pipes.
- **Pipe P1** can fill the tank in 8 hours. Therefore, in 1 hour, it fills:
\[ \text{Rate of P1} = \frac{1}{8} \text{ of the tank per hour} \]

- **Pipe P2** can fill the tank in 6 hours. Therefore, in 1 hour, it fills:
\[ \text{Rate of P2} = \frac{1}{6} \text{ of the tank per hour} \]

### Step 2: Calculate the volume filled by Pipe P1 in 2 hours.
- In 2 hours, Pipe P1 will fill:
\[ \text{Volume filled by P1 in 2 hours} = 2 \times \frac{1}{8} = \frac{2}{8} = \frac{1}{4} \text{ of the tank} \]

### Step 3: Determine the remaining volume to be filled.
- The total volume of the tank is considered as 1 (full tank). Therefore, the remaining volume after 2 hours is:
\[ \text{Remaining volume} = 1 - \frac{1}{4} = \frac{3}{4} \text{ of the tank} \]

### Step 4: Calculate the combined rate of both pipes when both are open.
- When both pipes are open, the combined filling rate is:
\[ \text{Combined rate} = \text{Rate of P1} + \text{Rate of P2} = \frac{1}{8} + \frac{1}{6} \]
- To add these fractions, find a common denominator (which is 24):
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{6} = \frac{4}{24} \]
- Therefore,
\[ \text{Combined rate} = \frac{3}{24} + \frac{4}{24} = \frac{7}{24} \text{ of the tank per hour} \]

### Step 5: Calculate the time required to fill the remaining volume.
- Let \( t \) be the time in hours required to fill the remaining \(\frac{3}{4}\) of the tank. Using the formula:
\[ \text{Volume filled} = \text{Rate} \times \text{Time} \]
- We can set up the equation:
\[ \frac{7}{24} \times t = \frac{3}{4} \]
- To solve for \( t \), multiply both sides by \( \frac{24}{7} \):
\[ t = \frac{3}{4} \times \frac{24}{7} = \frac{72}{28} = \frac{18}{7} \text{ hours} \]

### Step 6: Convert the time into a mixed number.
- To convert \( \frac{18}{7} \) into a mixed number:
\[ 18 \div 7 = 2 \quad \text{(remainder 4)} \]
- Thus, \( \frac{18}{7} = 2 \frac{4}{7} \) hours.

### Final Answer:
The tank will be filled in \( 2 \frac{4}{7} \) hours after the second pipe is opened.

---