A pipe can fill a tank in 36 minutes & another pipe can-fill it in 48 minutes, but a third Pipe can empty it in 18 minutes. The first two pipes are kept open for 16 minutes in the beginning then the third Pipe is also opened. In what time is the cistern emptied?

Options

A. A)120 min
B. B)80 min
C. C)96 min
D. D)112 min

Answer

To solve the problem step by step, we will break down the calculations involved in determining how long it takes to empty the tank after the first two pipes have been opened for a while.

### Step-by-Step Solution:

1. **Identify the Rates of the Pipes:**
- Pipe A can fill the tank in 36 minutes.
- Pipe B can fill the tank in 48 minutes.
- Pipe C can empty the tank in 18 minutes.

2. **Calculate the Filling and Emptying Rates:**
- The rate of Pipe A (filling) = 1 tank / 36 minutes = \( \frac{1}{36} \) tanks per minute.
- The rate of Pipe B (filling) = 1 tank / 48 minutes = \( \frac{1}{48} \) tanks per minute.
- The rate of Pipe C (emptying) = 1 tank / 18 minutes = \( \frac{1}{18} \) tanks per minute.

3. **Find a Common Capacity for Calculation:**
- To simplify calculations, we can assume the capacity of the tank is the least common multiple (LCM) of the times taken by the pipes.
- LCM of 36, 48, and 18 is 144 liters. Therefore, we assume the tank capacity is 144 liters.

4. **Calculate the Efficiency of Each Pipe:**
- Efficiency of Pipe A = \( \frac{144 \text{ liters}}{36 \text{ minutes}} = 4 \text{ liters/minute} \).
- Efficiency of Pipe B = \( \frac{144 \text{ liters}}{48 \text{ minutes}} = 3 \text{ liters/minute} \).
- Efficiency of Pipe C = \( -\frac{144 \text{ liters}}{18 \text{ minutes}} = -8 \text{ liters/minute} \) (negative because it is emptying).

5. **Calculate the Work Done by Pipes A and B in 16 Minutes:**
- Combined efficiency of Pipes A and B = \( 4 + 3 = 7 \text{ liters/minute} \).
- Work done in 16 minutes = \( 7 \text{ liters/minute} \times 16 \text{ minutes} = 112 \text{ liters} \).

6. **Determine Remaining Capacity After 16 Minutes:**
- Remaining capacity of the tank = Total capacity - Work done = \( 144 - 112 = 32 \text{ liters} \).

7. **Calculate the Combined Efficiency When All Three Pipes are Open:**
- Combined efficiency of Pipes A, B, and C = \( 4 + 3 - 8 = -1 \text{ liter/minute} \).

8. **Determine the Time Taken to Empty the Remaining 32 Liters:**
- Since the combined efficiency is -1 liter/minute, it means the tank is emptying at a rate of 1 liter per minute.
- Time to empty 32 liters = \( \frac{32 \text{ liters}}{1 \text{ liter/minute}} = 32 \text{ minutes} \).

9. **Total Time Taken to Empty the Tank:**
- Total time = Time with A and B open + Time with A, B, and C open = 16 minutes + 32 minutes = 48 minutes.

### Final Answer:
The time taken to empty the tank after the third pipe is opened is **48 minutes**.