A vistern can be filled by two supply pipes in 20 minutes and 30 minutes respectively. But a waste pipe can empty it in an hour. Both the spply pipes were opened together to fill the empty cistern, but by mistake the waste pipe was also open. Some time later the waste pipe was closed. How much later was the waste pipe closed, if the cistern is filled in 14 minutes?

Options

A. 6 minutes
B. 8 minutes
C. 10 minutes
D. 12 minutes

Answer

To solve the problem, we need to analyze the rates at which the supply pipes fill the cistern and the rate at which the waste pipe empties it. Here are the steps to find out how long the waste pipe was open before it was closed.

### Step-by-Step Solution:

1. **Determine the rates of the pipes:**
- Pipe A fills the cistern in 20 minutes. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1 \text{ cistern}}{20 \text{ minutes}} = \frac{3}{60} \text{ cisterns per minute} \]
- Pipe B fills the cistern in 30 minutes. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1 \text{ cistern}}{30 \text{ minutes}} = \frac{2}{60} \text{ cisterns per minute} \]
- The waste pipe (C) empties the cistern in 60 minutes. Therefore, its rate is:
\[ \text{Rate of C} = -\frac{1 \text{ cistern}}{60 \text{ minutes}} = -\frac{1}{60} \text{ cisterns per minute} \]

2. **Calculate the combined rate of filling when all pipes are open:**
- The combined rate when A, B, and C are all open is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} = \frac{3}{60} + \frac{2}{60} - \frac{1}{60} = \frac{4}{60} \text{ cisterns per minute} \]

3. **Calculate the total filling time:**
- Let \( t \) be the time in minutes that the waste pipe was open. The total time to fill the cistern is 14 minutes.
- Therefore, the waste pipe was closed for \( 14 - t \) minutes.
- During the time the waste pipe was open (t minutes), the amount filled is:
\[ \text{Amount filled in } t \text{ minutes} = \left(\frac{4}{60}\right) t \]
- During the time the waste pipe was closed (14 - t minutes), the amount filled is:
\[ \text{Amount filled in } (14 - t) \text{ minutes} = \left(\frac{5}{60}\right)(14 - t) \]
- The total amount filled must equal 1 cistern:
\[ \left(\frac{4}{60}\right)t + \left(\frac{5}{60}\right)(14 - t) = 1 \]

4. **Set up the equation:**
- Multiply through by 60 to eliminate the fractions:
\[ 4t + 5(14 - t) = 60 \]
- Expand and simplify:
\[ 4t + 70 - 5t = 60 \]
\[ -t + 70 = 60 \]
\[ -t = -10 \]
\[ t = 10 \]

5. **Conclusion:**
- The waste pipe was closed after **10 minutes**.