Two pipes A and B can fill an empty cistem in 4.8 and 7.2 hours, respectively. Pipe C can drain the entire cistern in 9.6 hours when no other pipe is in operation. Initially when the cisten was empty. Pipe A Pipe C were turned on. After a few hours. Pipe A was turned off and Pipe B was turned on instantly. In all it took 16.8 hours to fill the cisten. For how many hours was Pipe B turned on?

Options

A. 11.55
B. 12.6
C. 10.5
D. 10.8

Answer

To solve the problem step by step, we need to determine how many hours Pipe B was turned on after Pipe A was turned off. Let's break it down:

### Step 1: Determine the rates of filling and draining
- **Pipe A** fills the cistern in 4.8 hours. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1}{4.8} \text{ cisterns per hour} \]

- **Pipe B** fills the cistern in 7.2 hours. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1}{7.2} \text{ cisterns per hour} \]

- **Pipe C** drains the cistern in 9.6 hours. Therefore, its rate is:
\[ \text{Rate of C} = -\frac{1}{9.6} \text{ cisterns per hour} \]

### Step 2: Find the Least Common Multiple (LCM)
To work with the rates, we find the LCM of the times:
- LCM of 4.8, 7.2, and 9.6 is 28.8 hours.

### Step 3: Convert rates to a common basis
Now we convert the rates to how much of the cistern each pipe fills or drains in 28.8 hours:
- For Pipe A:
\[ \text{Amount filled by A} = \frac{28.8}{4.8} = 6 \text{ parts} \]

- For Pipe B:
\[ \text{Amount filled by B} = \frac{28.8}{7.2} = 4 \text{ parts} \]

- For Pipe C:
\[ \text{Amount drained by C} = \frac{28.8}{9.6} = 3 \text{ parts} \]

### Step 4: Set up the equation
Let \( x \) be the number of hours Pipe B was on. Then, Pipe A was on for \( 16.8 - x \) hours, and Pipe C was on for the entire 16.8 hours. The total work done can be represented as:
\[ 6(16.8 - x) + 4x - 3(16.8) = 28.8 \]

### Step 5: Simplify the equation
Expanding the equation:
\[ 100.8 - 6x + 4x - 50.4 = 28.8 \]
Combine like terms:
\[ 50.4 - 2x = 28.8 \]

### Step 6: Solve for \( x \)
Rearranging gives:
\[ 50.4 - 28.8 = 2x \]
\[ 21.6 = 2x \]
\[ x = \frac{21.6}{2} = 10.8 \]

### Conclusion
Pipe B was turned on for **10.8 hours**.

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