Points to Remember:
- Pipes A and B fill the tank, while pipe C empties it.
- Individual filling/emptying rates need to be calculated.
- The net filling/emptying rate when multiple pipes are open needs to be determined.
- The problem involves calculating the remaining work after A and B operate for 3 minutes.
Introduction:
This is a problem in time and work, a common application of rates and proportions. We are given the individual rates at which three pipes (A, B, and C) fill or empty a tank. The question requires us to calculate the combined effect of these pipes operating sequentially and determine the total time taken to empty the tank. We will approach this using a factual and analytical method, focusing on calculating the work done by each pipe per unit time.
Body:
1. Individual Pipe Rates:
- Pipe A fills the tank in 10 minutes, so its rate is 1/10 tank per minute.
- Pipe B fills the tank in 15 minutes, so its rate is 1/15 tank per minute.
- Pipe C empties the tank in 5 minutes, so its rate is 1/5 tank per minute.
2. Combined Filling Rate of A and B:
When pipes A and B are open together, their combined filling rate is:
1/10 + 1/15 = (3 + 2) / 30 = 5/30 = 1/6 tank per minute.
3. Work Done by A and B in 3 Minutes:
In 3 minutes, pipes A and B fill:
(1/6 tank/minute) * 3 minutes = 1/2 tank.
4. Remaining Portion of the Tank:
After 3 minutes, half the tank is filled. Therefore, 1/2 tank remains to be emptied.
5. Net Emptying Rate with A, B, and C Open:
When all three pipes are open, the net emptying rate is:
1/5 (emptying rate of C) – 1/6 (filling rate of A and B) = (6 – 5) / 30 = 1/30 tank per minute.
6. Time Taken to Empty the Remaining Half Tank:
To empty 1/2 tank at a rate of 1/30 tank per minute, the time taken is:
(1/2 tank) / (1/30 tank/minute) = 15 minutes.
Conclusion:
In summary, pipes A and B fill half the tank in 3 minutes. When pipe C is opened along with A and B, the net effect is emptying at a rate of 1/30 of the tank per minute. It takes an additional 15 minutes to empty the remaining half tank. Therefore, the total time taken to empty the tank from the point when pipe C is opened is 15 minutes. This problem highlights the importance of understanding individual rates and their combined effects when dealing with simultaneous processes. A clear understanding of rates and proportions is crucial for solving such problems efficiently. This approach emphasizes a methodical and analytical solution, ensuring accuracy and clarity in problem-solving.