Simultaneous Efforts: Understanding the Emptying Rate of a Tank by Two Pipes
In many practical scenarios, we encounter situations where two or more operations occur simultaneously. One classic example is the emptying of a tank using two pipes, each with its own efficiency. This article will delve into the solution and calculations involved, providing valuable insights for SEO optimization.
Understanding the Problem
The problem at hand involves two pipes, each capable of emptying a tank, but at different rates. The first pipe can empty the tank in 20 minutes, while the second can do so in 30 minutes. This article will explore how to determine the time it takes to empty the tank when both pipes are operated simultaneously.
Calculating Individual Rates
To solve the problem, we first need to calculate the rate at which each pipe can empty the tank individually. The rate at which the first pipe can empty the tank is 1/20 of the tank per minute, and the rate at which the second pipe can empty the tank is 1/30 of the tank per minute.
Simultaneous Operation
When both pipes are opened simultaneously, the total rate at which the tank is emptied is the sum of their individual rates. We calculate this as follows:
Rate of first pipe: 1/20 tank/minute
Rate of second pipe: 1/30 tank/minute
Total rate: 1/20 1/30 3/60 2/60 5/60 tank/minute
The total rate of emptying the tank is 5/60 of the tank per minute. This fraction simplifies to 1/12 of the tank per minute. Therefore, if the total capacity of the tank is 1 (or 100%), the time taken to empty the tank completely is:
Time 1 tank / (1/12 tank/minute) 12 minutes
Additional Examples and Calculations
Alternative Calculation Method
For a more complex scenario, consider pipes where the rates are expressed in different time units. For instance, pipe 1 pours fluid into the tank at a rate of 1/20 tank per minute, while pipe 2 pours fluid out at a rate of -1/15 tank per minute (negative because it's emptying the tank).
The combined rate when both pipes are operating simultaneously is:
1/20 (-1/15) (3 - 4)/60 -1/60 tank per minute
This results in a negative rate of -1/60 tank per minute, meaning the tank is being emptied at a rate of 1/60 of the tank per minute. Given that the tank is full, it will take 60 minutes to completely empty it.
Scenario Analysis
Consider another example where the tank starts empty. If both pipes are opened simultaneously, the filling rate is 1/15 tank per minute, while the emptying rate is 1/20 tank per minute. The net result is:
1/15 - 1/20 (4/60) - (3/60) 1/60 tank per minute
This indicates that the tank is being filled at a rate of 1/60 tank per minute. Since the filling rate is less than the emptying rate, the tank can never be filled to capacity when both pipes are open simultaneously.
Conclusion
In conclusion, understanding the simultaneous operations of two pipes can be crucial in various real-world applications. By calculating their rates and combining them, we can determine whether a tank will be filled or emptied over time. Whether it's a tank emptying example or a more complex fluid dynamics problem, the fundamental approach remains the same.
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