To solve the problem step by step, we will analyze the filling and emptying rates of the pipes involved.
### Step 1: Determine the rates of the inlet and outlet pipes.
- **Inlet A** fills the tank in 4 hours. Therefore, its rate is:
\[
\text{Rate of A} = \frac{1 \text{ tank}}{4 \text{ hours}} = \frac{1}{4} \text{ tank/hour}
\]
- **Inlet B** fills the tank in 6 hours. Therefore, its rate is:
\[
\text{Rate of B} = \frac{1 \text{ tank}}{6 \text{ hours}} = \frac{1}{6} \text{ tank/hour}
\]
- **Outlet C** empties the tank in 8 hours. Therefore, its rate is:
\[
\text{Rate of C} = \frac{1 \text{ tank}}{8 \text{ hours}} = \frac{1}{8} \text{ tank/hour}
\]
### Step 2: Calculate the combined rate of both inlets when opened together.
When both inlets A and B are opened together, their combined rate is:
\[
\text{Combined rate of A and B} = \frac{1}{4} + \frac{1}{6}
\]
To add these fractions, we find a common denominator, which is 12:
\[
\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}
\]
Thus,
\[
\text{Combined rate of A and B} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \text{ tank/hour}
\]
### Step 3: Calculate the amount filled after 1 hour with both inlets open.
In 1 hour, both inlets fill:
\[
\text{Amount filled in 1 hour} = \frac{5}{12} \text{ tank}
\]
### Step 4: Determine the remaining part of the tank.
Since the tank is full (1 tank) and we filled \(\frac{5}{12}\) of it, the remaining part is:
\[
\text{Remaining part} = 1 - \frac{5}{12} = \frac{12}{12} - \frac{5}{12} = \frac{7}{12} \text{ tank}
\]
### Step 5: Calculate the combined rate when all three pipes are opened.
Now we will open all three pipes (A, B, and C). The combined rate is:
\[
\text{Combined rate of A, B, and C} = \frac{1}{4} + \frac{1}{6} - \frac{1}{8}
\]
Finding a common denominator for 4, 6, and 8, which is 24:
\[
\frac{1}{4} = \frac{6}{24}, \quad \frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24}
\]
Thus,
\[
\text{Combined rate} = \frac{6}{24} + \frac{4}{24} - \frac{3}{24} = \frac{7}{24} \text{ tank/hour}
\]
### Step 6: Calculate the time to fill the remaining part of the tank.
To find out how long it will take to fill the remaining \(\frac{7}{12}\) of the tank at the combined rate of \(\frac{7}{24}\) tank/hour, we set up the equation:
\[
\text{Time} = \frac{\text{Remaining part}}{\text{Combined rate}} = \frac{\frac{7}{12}}{\frac{7}{24}} = \frac{7}{12} \times \frac{24}{7} = \frac{24}{12} = 2 \text{ hours}
\]
### Step 7: Calculate the total time taken.
The total time taken is the time both inlets were open plus the time taken to fill the remaining part:
\[
\text{Total time} = 1 \text{ hour} + 2 \text{ hours} = 3 \text{ hours}
\]
### Final Answer:
Thus, the remaining part of the tank fills in **3 hours**, which corresponds to option (d).
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