Solving the Tank Filling and Emptying Problem with Multiple Taps

When dealing with real-world problems involving multiple taps or sources, understanding the rates at which they fill or empty a tank is crucial. This article will guide you through solving such problems, using a specific example: a tank that can be filled in 15 minutes and emptied in 8 minutes. We will explore how to determine the time it takes to empty the tank if it is half full and both taps are opened simultaneously.

Problem Understanding and Setup

Imagine a tank with a filling tap that can fill it in 15 minutes and an emptying tap that can empty it in 8 minutes. If the tank is currently half full, we need to determine how long it will take for the tank to be emptied when both taps are open.

Calculating Individual Rates

Let's begin by calculating the rates of both taps.

Filling Tap Rate

The filling tap can fill the whole tank in 15 minutes. Therefore, the rate at which the tank is being filled is:

Rate of Filling Tap 1 tank / 15 minutes 1/15 tanks per minute

Emptying Tap Rate

The emptying tap can empty the whole tank in 8 minutes. Therefore, the rate at which the tank is being emptied is:

Rate of Emptying Tap 1 tank / 8 minutes 1/8 tanks per minute

Net Rate Calculation

When both taps are open simultaneously, the net rate of change in the tank volume can be calculated by subtracting the emptying rate from the filling rate.

Net Rate (1/15 - 1/8) tanks per minute

Finding a Common Denominator

To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 8 is 120. Therefore, we convert the rates to have the same denominator:

1/15 8/120 and 1/8 15/120

Subtracting the rates:

Net Rate (8/120 - 15/120) (-7/120) tanks per minute

The negative value indicates that the tank is being emptied at a rate of 7/120 tanks per minute.

Determining the Time to Empty the Tank

Now that we know the net emptying rate, we need to determine how long it will take to empty the tank if it is half full.

The current state of the tank is half full, which is equivalent to 1/2 of the tank's capacity. With the net emptying rate of -7/120 tanks per minute, we can calculate the time required to empty the tank:

Time |Amount of tank / Net rate| (1/2) / (7/120) (1/2) * (120/7) 60/7 minutes

Converting this to a decimal:

Time ≈ 8.57 minutes

Conclusion

When both taps are opened together, the tank will be emptied in approximately 8.57 minutes, provided it is half full.

By understanding and calculating the rates of individual taps and then combining them, we can effectively solve complex problems involving multiple sources or sinks. This method can be adapted to various real-world scenarios, making it a valuable skill for engineers, designers, and problem solvers.