Sealing a Water Tank: Calculating Time with Inlet and Outlet Pipes
In this article, we will explore the mathematics behind a practical problem encountered in everyday life: filling a water tank using an inlet pipe while dealing with the challenge of two outlet pipes. This scenario not only tests your understanding of rates but also sharpens your ability to manage and combine different rates to solve real-world problems efficiently. Understanding these concepts is essential for a SEO experts, as it allows for the creation of content that is both engaging and informative for potential readers.
Understanding the Problem
The problem at hand is to determine the time it takes for an inlet pipe to fill a water tank if two outlet pipes are also open. To solve this, we need to break it down into several steps:
Determining the Rates of the Pipes
First, let's identify the rates at which the inlet pipe and the outlet pipes function. These are based on how long it takes each pipe to fill or drain the tank.
Inlet Pipe Rate
The inlet pipe can fill the tank in 3 hours. Thus, its rate is:
Inlet rate 1/(3) tanks per hour
Outlet Pipe Rate
One outlet pipe can drain the tank in 10 hours. Therefore, its rate is:
Outlet rate 1/(10) tanks per hour
Since we have two outlet pipes, their combined rate is:
Combined outlet rate 2 × (1/10) 2/10 1/5 tanks per hour
Calculating the Net Rate
Next, we need to find the net rate at which the tank is being filled when both the inlet pipe and the two outlet pipes are functioning.
The net rate is calculated by subtracting the combined outlet rate from the inlet rate:
Net rate Inlet rate - Combined outlet rate
Substituting the values, we get:
Net rate 1/3 - 1/5 (5/15) - (3/15) 2/15 tanks per hour
Calculating the Time to Fill the Tank
Now that we know the net rate, we can calculate the time it takes to fill the tank. The time t is given by the formula:
t 1 / (Net rate) 1 / (2/15) 15/2 7.5 hours
Therefore, it will take the inlet pipe 7.5 hours to fill the water tank when two outlet pipes are open.
Further Insights
For advanced understanding, let's explore a variation of this problem. Consider a scenario where you have an inlet pipe that fills the tank in 14 hours and two outlet pipes that drain the tank in 18 hours each. The same steps can be applied to solve this problem:
First, we calculate the rates:
Inlet rate 1/14 tanks per hour
Combined outlet rate:
Combined outlet rate 2 × (1/18) tanks per hour
Net rate:
Net rate 1/14 - 1/18
Combining the fractions:
Net rate (1/14) - (1/18) (9/126) - (7/126) 2/126 1/63 tanks per hour
The time to fill the tank is:
t 1 / (1/63) 63 hours
So, in this case, it would take 63 hours to fill the tank.
Conclusion
To summarize, the process of solving this problem involves determining the rates of the inlet and outlet pipes, calculating the net rate, and then using the net rate to find the time required to fill the tank. This problem is not only useful in practical scenarios but also serves as a valuable tool for enhancing problem-solving skills in mathematics.
Related Keywords
inlet pipe, outlet pipe, water tank, time calculation