A cistern has three pipes A,B and C. Pipes Aand B can fill it in 3 and 4h,respectively,while pipe C can empty the completely filled cistern in 1 h. If the pipes are opened in order at 3:00 pm, 4:00 pm and 5:00 pm, respectively, at what time will the cistern be empty?

Options

A. 6:15pm
B. 7:12pm
C. 8:12pm
D. 8:35pm

Answer

To solve the problem step-by-step, we will first determine the filling and emptying rates of the pipes, then calculate how much water is in the cistern at different times, and finally find out when the cistern will be empty.

### Step 1: Determine the capacity of the cistern
To make calculations easier, we will assume the capacity of the cistern is the least common multiple (LCM) of the times taken by the pipes to fill or empty it.

- Pipe A fills the cistern in 3 hours.
- Pipe B fills the cistern in 4 hours.
- Pipe C empties the cistern in 1 hour.

Calculating the LCM of 3, 4, and 1:
- The LCM of 3 and 4 is 12.
- Therefore, the capacity of the cistern is 12 units.

### Step 2: Calculate the filling rates of pipes A and B
- Pipe A fills the cistern at a rate of \( \frac{12 \text{ units}}{3 \text{ hours}} = 4 \text{ units/hour} \).
- Pipe B fills the cistern at a rate of \( \frac{12 \text{ units}}{4 \text{ hours}} = 3 \text{ units/hour} \).
- Pipe C empties the cistern at a rate of \( \frac{12 \text{ units}}{1 \text{ hour}} = 12 \text{ units/hour} \).

### Step 3: Calculate the water filled by pipes A and B before pipe C opens
- Pipe A is opened at 3:00 PM and runs until 5:00 PM (2 hours).
- In 2 hours, Pipe A fills:
\[ 2 \text{ hours} \times 4 \text{ units/hour} = 8 \text{ units} \]

- Pipe B is opened at 4:00 PM and runs until 5:00 PM (1 hour).
- In 1 hour, Pipe B fills:
\[ 1 \text{ hour} \times 3 \text{ units/hour} = 3 \text{ units} \]

- Total water filled by 5:00 PM:
\[ 8 \text{ units} + 3 \text{ units} = 11 \text{ units} \]

### Step 4: Calculate the net rate of filling after 5:00 PM
- At 5:00 PM, Pipe C is opened, and all three pipes are working together.
- The combined rate of filling and emptying is:
\[ (4 \text{ units/hour} + 3 \text{ units/hour} - 12 \text{ units/hour}) = -5 \text{ units/hour} \]
This means that the cistern is emptying at a rate of 5 units per hour.

### Step 5: Calculate the time taken to empty the cistern
- The amount of water left in the cistern at 5:00 PM is 11 units.
- Time taken to empty 11 units at a rate of 5 units/hour:
\[ \text{Time} = \frac{\text{Capacity}}{\text{Rate}} = \frac{11 \text{ units}}{5 \text{ units/hour}} = 2.2 \text{ hours} = 2 \text{ hours} + 12 \text{ minutes} \]

### Step 6: Determine the final time when the cistern will be empty
- Starting from 5:00 PM, adding 2 hours and 12 minutes:
\[ 5:00 PM + 2 \text{ hours} + 12 \text{ minutes} = 7:12 PM \]

Thus, the cistern will be completely empty at **7:12 PM**.