Solving the Pipe Filling Problem: A Comprehensive Guide to Inverse Proportionality

The problem of determining the time required for a different number of pipes to fill a tank is a classic application of inverse proportionality. This article will guide you through the process of solving such a problem, ensuring that the solution is both accurate and easy to understand.

Understanding Inverse Proportionality in Fluid Dynamics

In fluid dynamics, the time taken to fill a tank is inversely proportional to the number of pipes used. This means that if you increase the number of pipes, the time required to fill the tank decreases, and vice versa. This relationship can be modeled using an inverse proportion equation. Mathematically, this is represented by the formula:

N1 * T1 N2 * T2

where N1 and T1 are the initial number of pipes and time, and N2 and T2 are the final number of pipes and time, respectively.

Problem Statement

Given that 8 pipes can fill a tank in 1 hour and 10 minutes, how long will it take for 5 pipes of the same type to fill the tank? First, we need to convert the time into a decimal format for easier calculations.

1 hour 10 minutes 1.17 hours

Solving the Problem

Using the inverse proportion relationship, we can set up the equation:

8 pipes * 1.17 hours 5 pipes * x hours

Solving for x, we get:

x (8 * 1.17) / 5

Calculating the value of x:

x 1.87 hours

Converting 1.87 hours back into minutes and hours, we get:

1 hour and 52 minutes

Alternative Solution

We can also solve this problem using direct calculations. First, we convert the time into minutes to simplify the calculations:

1 hour 10 minutes 70 minutes

8 pipes take 70 minutes to fill the tank, so 1 pipe will take:

70 * 8 560 minutes

Therefore, 5 pipes will take:

560 / 5 2800 minutes / 60 4 hours and 40 minutes

Conclusion

The inverse proportion relationship between the number of pipes and the time required to fill a tank is a fundamental concept in fluid dynamics. By understanding and applying this principle, one can solve a variety of related problems efficiently. Whether you are dealing with a theoretical scenario or a practical application, the inverse proportion method provides a robust solution.

Remember, the key to solving such problems lies in accurately setting up and solving the inverse proportion equation. With practice, you can master these calculations and apply them to different scenarios involving fluid flow and filling tanks.