Efficient Tank Filling: Understanding the Roles of Two Pipes A and B

Understanding the efficiency of two pipes in filling a tank is a fundamental problem encountered in many practical scenarios, from domestic use to industrial applications. This article delves into the mathematical and practical aspects of piped systems, specifically focusing on how two pipes A and B, with capacities of 36 hours and 45 hours respectively, function in tandem.

Speed and Efficiency of Pipe A and B

Let's begin by examining the individual filling times of Pipe A and Pipe B:

Pipe A can fill a tank in 36 hours. Pipe B can fill the same tank in 45 hours.

When both pipes are opened simultaneously, the total capacity of water filling per hour by both pipes can be calculated. This collective rate is crucial for determining the total time to fill the tank efficiently.

The Combined Efficiency of Pipe A and B

Working together, the rate at which the tank is filled can be determined by the following calculations:

Mathematical Approach

The tank filling ratio is determined by the inverse of the individual times. This means that for Pipe A and Pipe B, the ratio of their filling rates is 36:45, which simplifies to 4:5.

To find the time taken to fill the tank when both pipes are opened simultaneously, we can use the formula for combined rates:

Combined Rate 1/36 1/45 185/1620

T 185/1620 27 hours (approximately)

This calculation gives us an estimate of how long it will take both pipes to fill the tank together. However, the problem requires a more detailed breakdown to verify the efficiency.

Detailed Breakdown

To elaborate, let's consider the detailed calculations:

- Pipe A fills the tank in 36 hours.

- Pipe B fills the tank in 45 hours.

- Combined, in one hour, Pipe A fills 1/36 of the tank, and Pipe B fills 1/45 of the tank.

The total part of the tank filled in one hour when both pipes are open:

(1/36) (1/45) 185/1620 1/8.78 (approximately)

Hence, both pipes can fill the tank in approximately 8.78 hours.

Practical Applications and Simplifications

For practical purposes, let's simplify the problem further:

1. Simplified Ratio Calculation

The problem states that the combined rate of filling the tank is split in the inverse ratio of their individual times, which is 30:28 (approximately 15:14).

- Ratio of A to B 30:28 15:14 (approximately).

- Let's calculate the combined time to fill the tank:

T 36/15 45/14 15/2928 6/2928 (approximately)

This gives an approximate time of 5.793 hours for both pipes to fill the tank together.

2. Partially Filled Tank

If the tank is already 3/5 full, only 2/5 remains to be filled. The time taken for A and B to fill these 2/5 of the tank is:

t 2/5 * (15/2928) 6/2928 (approximately 5.793 hours - 1.207 hours 4.586 hours)

This simplifies the problem to around 4.586 hours to fill the remaining part of the tank.

3. Direct Calculation Method

A more direct approach can be used to determine the time taken, such as:

- If Pipe A alone takes 20 minutes to fill the tank, Pipe B alone takes 90 minutes. Together, they fill 1/20 1/90 of the tank in one minute, which is 11/180.

- Therefore, the total time to fill the tank is 180/11, which is approximately 16.36 minutes or 16 minutes and 22 seconds.

Conclusion

The efficiency of the combined operation of two pipes can significantly reduce the time required to fill a tank. This article has provided a comprehensive analysis of the process, highlighting both the mathematical and practical approaches to determine the time taken to fill a tank with two pipes, A and B.

Whether you are planning a domestic water supply or an industrial application, understanding the efficiency of piped systems can save time and resources. By applying the principles discussed here, you can effectively manage and optimize the filling process in any setting.