Understanding Water Tank Filling and Emptying with Pipes A and B
This article explores a real-world scenario involving two pipes, A and B, and how they affect the filling and emptying of a water tank. The problem involves understanding the rates at which each pipe operates and how these rates interact when both pipes are open.
Problem Analysis and Description
The water tank is initially 1/5 full, and Pipe A can fill the tank in 20 minutes. Meanwhile, Pipe B can empty the tank in 10 minutes. If both pipes are opened simultaneously, what will happen to the water in the tank? We will break down the mathematical reasoning behind the solution and provide a detailed step-by-step explanation.
Mathematical Reasoning
We start by analyzing the rates at which each pipe operates. To do this, we need to determine the filling and emptying rates in terms of the tank's volume per minute.
Fill Rate and Empty Rate
Pipe A's filling rate, denoted as R_A, is:
[ R_A frac{1}{20} text{ tank/min} ]
Pipe B's emptying rate, denoted as R_B, is:
[ R_B -frac{1}{15} text{ tank/min} text{ (negative sign indicates emptying)} ]
Total Change When Both Pipes Are Open
When both pipes are open, the net rate of change, Net Rate, is the sum of the individual rates:
[ text{Net Rate} R_A R_B frac{1}{20} - frac{1}{15} ]
To combine the fractions, we find a common denominator, which is 60:
[ frac{3}{60} - frac{4}{60} frac{-1}{60} text{ tank/min} ]
This means that, per minute, the tank is being emptied at a rate of 1/60 of the tank's capacity.
Time to Empty the Tank
Since the net rate of change is ( frac{-1}{60} ) tank per minute, the time ( t ) required to empty the tank is given by the volume of the initial water divided by the net emptying rate:
[ t frac{frac{1}{5} text{ tank}}{frac{1}{60} text{ tank/min}} frac{1}{5} times 60 12 text{ minutes} times 5 60 text{ minutes} ]
Hence, it will take 60 minutes to empty the tank completely.
Step-by-Step Solution
Step 1: Determine Individual Rates
Pipe A fills the tank at a rate of 1/20 of the tank per minute. Pipe B empties the tank at a rate of 1/15 of the tank per minute.Step 2: Calculate Net Rate of Change
[ text{Net Rate} frac{1}{20} - frac{1}{15} -frac{1}{60} ]
Step 3: Calculate Time to Empty the Tank
Given the net rate of emptying, the time to empty the tank (which is initially 1/5 full) is:
[ t frac{frac{1}{5}}{-frac{1}{60}} 60 text{ minutes} ]
Conclusion
In this scenario, when both Pipe A and Pipe B are open, the tank will be emptied in 60 minutes due to the net emptying rate of ( frac{1}{60} ) tank per minute. This result is consistent with our earlier calculations.