A pipe can fill a cistern in 15 min and another pipe can fill the same cistern in 60 min. a third pipe can empty it in 10 min. the first two pipes are kept open for 10 min in the beginning and then the third pipe is also opened, what is the time taken to empty the cistern?

Options

A. A)45 min
B. B)60 min
C. C)50 min
D. D)48 min

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 filled in the first 10 minutes, and finally find out how long it takes to empty the cistern after the third pipe is opened.

### Step 1: Determine the filling and emptying rates of the pipes.
- **Pipe A** fills the cistern in 15 minutes.
- Rate of Pipe A = 60 liters / 15 minutes = 4 liters/minute.

- **Pipe B** fills the cistern in 60 minutes.
- Rate of Pipe B = 60 liters / 60 minutes = 1 liter/minute.

- **Pipe C** empties the cistern in 10 minutes.
- Rate of Pipe C = 60 liters / 10 minutes = 6 liters/minute (but since it is emptying, we consider this as -6 liters/minute).

### Step 2: Calculate the total filling rate of pipes A and B when both are open.
- Combined rate of A and B = Rate of A + Rate of B = 4 liters/minute + 1 liter/minute = 5 liters/minute.

### Step 3: Calculate the amount of water filled in the first 10 minutes.
- Amount filled in 10 minutes = Combined rate × Time = 5 liters/minute × 10 minutes = 50 liters.

### Step 4: Set up the equation for the time taken to empty the cistern after opening pipe C.
- After 10 minutes, the cistern has 50 liters of water.
- When pipe C is opened, the combined rate of A, B, and C becomes:
- Combined rate = Rate of A + Rate of B - Rate of C = 4 + 1 - 6 = -1 liter/minute.

This means that the cistern is emptying at a rate of 1 liter per minute.

### Step 5: Calculate the time taken to empty the remaining water in the cistern.
- Since there are 50 liters of water left and the rate of emptying is 1 liter/minute, the time taken to empty the cistern is:
- Time = Amount of water / Rate of emptying = 50 liters / 1 liter/minute = 50 minutes.

### Final Answer:
The total time taken to empty the cistern after the third pipe is opened is **50 minutes**.

---