Calculating the Time to Fill a Tank with Multiple Pipes
In fluid dynamics and water management, understanding the combined effect of multiple pipes can be crucial for determining efficient water distribution. This article will guide you through the process of calculating the time it takes to fill a tank when using multiple pipes with varying rates of filling and emptying.
Introduction to the Problem
We have three pipes, Pipe X, Pipe Y, and Pipe Z, with different rates of filling and emptying a tank. Pipe X can fill the tank in 5 hours, Pipe Y in 6 hours, and Pipe Z can empty the tank in 8 hours. When all three pipes are opened simultaneously, the question is: how long will it take to fill the tank?
Calculating Individual Pipe Rates
First, let's determine the individual rates of the pipes in terms of the amount of tank they can fill (or empty) per hour.
Pipe X
Pipe X fills the tank in 5 hours, so its rate is:
```text Rate of X 1/5 tank/hour ```Pipe Y
Pipe Y fills the tank in 6 hours, so its rate is:
```text Rate of Y 1/6 tank/hour ```Pipe Z
Pipe Z empties the tank in 8 hours, thus its rate is negative:
```text Rate of Z -1/8 tank/hour ```Combining the Rates
When all three pipes are opened together, we need to find the combined rate at which the tank is being filled or emptied.
```text Combined Rate Rate of X Rate of Y Rate of Z ```Substituting the rates we calculated:
```text Combined Rate 1/5 1/6 - 1/8 ```Finding a Common Denominator
We need a common denominator to combine these fractions. The least common multiple of 5, 6, and 8 is 120. Converting each fraction to this common denominator:
```text 1/5 24/120 1/6 20/120 -1/8 -15/120 ```Now we can combine the fractions:
```text Combined Rate 24/120 20/120 - 15/120 (24 20 - 15) / 120 29/120 tank/hour ```Calculating the Time to Fill the Tank
Finally, to find the time it takes to fill 1 tank with the combined rate, we take the reciprocal of the combined rate:
```text Time 1 tank / (29/120 tank/hour) 120/29 hours ```Performing the final calculation:
```text 120/29 ≈ 4.138 hours ```Converting 4.138 hours to hours and minutes:
4 hours 0.138 * 60 minutes ≈ 4 hours and 8 minutes
Therefore, when all three pipes are opened together, the tank will be filled in approximately 4.14 hours, or about 4 hours and 8 minutes.
Conclusion and Additional Examples
The problem of determining the time to fill a tank with multiple pipes is a classic example in fluid mechanics. Understanding how to calculate the combined rate of pipes that both fill and empty a tank is crucial for various real-world applications, such as wastewater management, irrigation systems, and industrial fluid handling.
Here are a couple of additional examples for further practice:
Example 1
Compute the filling time when A, B, and C operate together, with rates 1/5, 1/6, and 1/10, respectively.
```text 1/5 1/6 - 1/10 8/30 1 / (8/30) 30/8 ≈ 3.75 hours or 3 hours and 45 minutes ```The tank will be filled in approximately 3 hours and 45 minutes.
Example 2
If we have three new rates: A (1/5), B (1/6), and C (1/10), the combined rate is:
```text 6/30 5/30 - 3/30 8/30 1 / (8/30) 30/8 ≈ 3.75 hours or 3 hours and 45 minutes ```The tank will again be filled in approximately 3 hours and 45 minutes.