A Tub Of Water Is Emptied At A Rate Of 3 Gallons Per Minute. The Equation Y − 12 = − 3 ( X − 1 Y-12=-3(x-1 Y − 12 = − 3 ( X − 1 ] Models The Amount Of Water Remaining, Where X X X Is Time (in Minutes) And Y Y Y Is The Amount Of Water Left (in Gallons). Analyze The Work

Mar 12, 2025 by ADMIN 270 views

A Tub of Water is Emptied: Analyzing the Work with Mathematics

Introduction

In this article, we will delve into the world of mathematics and analyze a real-world scenario involving the emptying of a tub of water. The equation $y-12=-3(x-1)$ models the amount of water remaining, where $x$ is time (in minutes) and $y$ is the amount of water left (in gallons). We will break down the equation, understand its components, and explore the implications of the given rate of emptying.

Understanding the Equation

The equation $y-12=-3(x-1)$ can be rewritten as $y=-3x+15$ . This is a linear equation in the slope-intercept form, where the slope is $-3$ and the y-intercept is $15$ . The equation represents a straight line with a negative slope, indicating that the amount of water remaining decreases as time increases.

Slope and Rate of Emptying

The slope of the equation, $-3$ , represents the rate at which the tub is being emptied. In this case, the tub is being emptied at a rate of $3$ gallons per minute. This means that for every minute that passes, $3$ gallons of water are removed from the tub.

Y-Intercept and Initial Amount of Water

The y-intercept of the equation, $15$ , represents the initial amount of water in the tub. This means that when $x=0$ , the amount of water remaining is $15$ gallons. In other words, the tub initially contains $15$ gallons of water.

Analyzing the Work

To analyze the work, we need to understand the implications of the given rate of emptying. If the tub is being emptied at a rate of $3$ gallons per minute, it will take $5$ minutes to empty the tub completely. This is because the initial amount of water in the tub is $15$ gallons, and the rate of emptying is $3$ gallons per minute.

Time and Amount of Water Remaining

We can use the equation to determine the amount of water remaining at any given time. For example, if we want to know the amount of water remaining after $2$ minutes, we can plug in $x=2$ into the equation:

$y=-3(2)+15$ $y=-6+15$ $y=9$

This means that after $2$ minutes, there will be $9$ gallons of water remaining in the tub.

Graphing the Equation

We can graph the equation to visualize the relationship between time and the amount of water remaining. The graph will be a straight line with a negative slope, indicating that the amount of water remaining decreases as time increases.

Conclusion

In conclusion, the equation $y-12=-3(x-1)$ models the amount of water remaining in a tub that is being emptied at a rate of $3$ gallons per minute. The equation represents a straight line with a negative slope, indicating that the amount of water remaining decreases as time increases. We can use the equation to determine the amount of water remaining at any given time and to analyze the implications of the given rate of emptying.

Applications of the Equation

The equation has several applications in real-world scenarios. For example, it can be used to model the amount of water remaining in a swimming pool that is being drained, or the amount of fuel remaining in a tank that is being depleted.

Real-World Scenarios

The equation can be applied to various real-world scenarios, such as:

Modeling the amount of water remaining in a reservoir that is being drained
Determining the amount of fuel remaining in a tank that is being depleted
Calculating the amount of time it will take to empty a container that is being drained at a given rate

Future Work

Future work can involve exploring other real-world scenarios where the equation can be applied, such as modeling the amount of water remaining in a lake that is being drained, or determining the amount of time it will take to empty a container that is being drained at a given rate.

Extensions of the Equation

The equation can be extended to model more complex scenarios, such as:

Modeling the amount of water remaining in a container that is being drained at a variable rate
Determining the amount of time it will take to empty a container that is being drained at a variable rate

Conclusion

Introduction

In our previous article, we analyzed the equation $y-12=-3(x-1)$ , which models the amount of water remaining in a tub that is being emptied at a rate of $3$ gallons per minute. In this article, we will answer some frequently asked questions related to the equation and its applications.

Q&A

Q: What is the initial amount of water in the tub?

A: The initial amount of water in the tub is $15$ gallons, which is represented by the y-intercept of the equation, $15$ .

Q: How long will it take to empty the tub completely?

A: It will take $5$ minutes to empty the tub completely, since the rate of emptying is $3$ gallons per minute and the initial amount of water is $15$ gallons.

Q: What is the rate of emptying of the tub?

A: The rate of emptying of the tub is $3$ gallons per minute, which is represented by the slope of the equation, $-3$ .

Q: Can the equation be used to model other real-world scenarios?

A: Yes, the equation can be used to model other real-world scenarios, such as modeling the amount of water remaining in a reservoir that is being drained, or determining the amount of fuel remaining in a tank that is being depleted.

Q: How can the equation be extended to model more complex scenarios?

A: The equation can be extended to model more complex scenarios, such as modeling the amount of water remaining in a container that is being drained at a variable rate, or determining the amount of time it will take to empty a container that is being drained at a variable rate.

Q: What are some real-world applications of the equation?

A: Some real-world applications of the equation include:

Modeling the amount of water remaining in a swimming pool that is being drained
Determining the amount of fuel remaining in a tank that is being depleted
Calculating the amount of time it will take to empty a container that is being drained at a given rate

Q: Can the equation be used to solve problems involving time and rate?

A: Yes, the equation can be used to solve problems involving time and rate, such as determining the amount of time it will take to complete a task at a given rate, or calculating the rate at which a task is being completed.

Q: How can the equation be used to model the behavior of a system over time?

A: The equation can be used to model the behavior of a system over time, such as modeling the amount of water remaining in a reservoir that is being drained over time, or determining the amount of fuel remaining in a tank that is being depleted over time.

Conclusion

Additional Resources

For more information on the equation and its applications, please refer to the following resources:

Final Thoughts

The equation $y-12=-3(x-1)$ is a simple yet powerful tool for modeling the behavior of a system over time. By understanding the equation and its applications, we can gain a deeper understanding of the world around us and make more informed decisions in our personal and professional lives.