Sealing the Tank with Two Inlet Pipes: A Comprehensive Guide

Understanding how to determine the time it takes to fill a tank using two inlet pipes with different rates is a common problem in engineering and practical applications. This article aims to provide a thorough explanation on how to solve such problems using the combined rate method.

Introduction to the Problem

Imagine you have two inlet pipes. Pipe A can fill a tank in 3 hours, while Pipe B can fill the same tank in 4 hours. If both pipes are opened simultaneously, how long will it take to fill the tank completely?

Step-by-Step Solution

Calculate the Filling Rates of Each Pipe

To begin solving this problem, we first need to determine the filling rate of each pipe. The filling rate is the amount of tank filled per unit time.

For Pipe A, which fills the tank in 3 hours:

Filling Rate of Pipe A (frac{1 text{ tank}}{3 text{ hours}} frac{1}{3} text{ tanks per hour})

For Pipe B, which fills the tank in 4 hours:

Filling Rate of Pipe B (frac{1 text{ tank}}{4 text{ hours}} frac{1}{4} text{ tanks per hour})

Combine the Filling Rates

Now, to find the combined rate at which the tank is filled when both pipes are opened simultaneously, we add their individual rates. To add these fractions, we need a common denominator. The least common multiple of 3 and 4 is 12:

(frac{1}{3} frac{4}{12}) (frac{1}{4} frac{3}{12})

Combined Rate (frac{1}{3} frac{1}{4} frac{4}{12} frac{3}{12} frac{7}{12} text{ tanks per hour})

Calculate the Time to Fill the Tank

The time (t) to fill 1 tank at the combined rate is given by the reciprocal of the combined rate:

(t frac{1}{text{Combined Rate}} frac{1}{frac{7}{12}} frac{12}{7} text{ hours})

Converting (frac{12}{7}) hours into hours and minutes:

(frac{12}{7} approx 1.714 text{ hours}) (1.714 times 60 approx 102.86 text{ minutes} approx 102 text{ minutes or} 1 text{ hour and} 43 text{ minutes})

Thus, when both pipes are opened at the same time, it will take approximately 1 hour and 43 minutes to fill the tank.

Additional Information

The same principle can be applied to other scenarios involving different filling times. For example, if Pipe A fills the tank in 4 hours and Pipe B in 6 hours:

Filling Rate of Pipe A (frac{1 text{ tank}}{4 text{ hours}} frac{1}{4} text{ tanks per hour})

Filling Rate of Pipe B (frac{1 text{ tank}}{6 text{ hours}} frac{1}{6} text{ tanks per hour})

Adding these rates with a common denominator of 12:

(frac{1}{4} frac{3}{12}) (frac{1}{6} frac{2}{12})

Combined Rate (frac{3}{12} frac{2}{12} frac{5}{12} text{ tanks per hour})

Time to fill the tank (frac{12}{5} 2.4 text{ hours} 2 text{ hours and} 24 text{ minutes})

Conclusion

This article has provided a clear and detailed explanation of how to determine the time it takes to fill a tank using two inlet pipes with different rates. By utilizing the combined rate method, we can solve similar problems efficiently and accurately. Understanding this concept is crucial in various engineering and practical scenarios.