Raj's Bathtub Is Clogged And Is Draining At A Rate Of 1.5 Gallons Of Water Per Minute. The Table Shows That The Amount Of Water Remaining In The Bathtub, $y$, Is A Function Of The Time In Minutes, $x$, That It Has Been
Introduction
Raj's bathtub is clogged, and water is draining at a rate of 1.5 gallons per minute. The table below shows the amount of water remaining in the bathtub, denoted as $y$, as a function of the time in minutes, denoted as $x$, that it has been draining.
| Time (minutes) | Water Remaining (gallons) |
|---|---|
| 0 | 120 |
| 1 | 108 |
| 2 | 96 |
| 3 | 84 |
| 4 | 72 |
| 5 | 60 |
| 6 | 48 |
| 7 | 36 |
| 8 | 24 |
| 9 | 12 |
| 10 | 0 |
Understanding the Problem
The problem presents a classic example of a linear function, where the amount of water remaining in the bathtub decreases at a constant rate. The table shows that for every minute that passes, the amount of water remaining decreases by 12 gallons. This indicates a linear relationship between the time and the amount of water remaining.
Linear Functions
A linear function is a polynomial function of degree one, which can be written in the form:
where $m$ is the slope of the line, and $b$ is the y-intercept. In this case, the slope represents the rate at which the amount of water remaining decreases, and the y-intercept represents the initial amount of water in the bathtub.
Finding the Slope
To find the slope of the line, we can use the formula:
where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line. Using the data from the table, we can choose any two points to find the slope. Let's choose the points $(0, 120)$ and $(1, 108)$.
Finding the Y-Intercept
Now that we have the slope, we can find the y-intercept by plugging in the values of $m$ and $x$ into the equation:
Using the point $(0, 120)$, we get:
Writing the Linear Function
Now that we have the slope and the y-intercept, we can write the linear function:
Interpreting the Results
The linear function $y = -12x + 120$ represents the amount of water remaining in the bathtub as a function of the time in minutes that it has been draining. The slope of the line, $-12$, represents the rate at which the amount of water remaining decreases, and the y-intercept, $120$, represents the initial amount of water in the bathtub.
Conclusion
In conclusion, the problem of Raj's clogged bathtub can be analyzed using linear functions. By finding the slope and the y-intercept, we can write a linear function that represents the amount of water remaining in the bathtub as a function of the time in minutes that it has been draining. This analysis provides a mathematical understanding of the problem and can be used to make predictions about the amount of water remaining in the bathtub at any given time.
Real-World Applications
The concept of linear functions has many real-world applications, including:
- Physics: Linear functions are used to describe the motion of objects, such as the trajectory of a projectile or the velocity of a car.
- Economics: Linear functions are used to model the relationship between variables, such as the demand for a product and its price.
- Biology: Linear functions are used to model the growth of populations, such as the growth of a bacteria culture.
Future Research Directions
Future research directions in this area could include:
- Non-Linear Functions: Investigating the use of non-linear functions to model more complex relationships between variables.
- Real-World Data: Collecting and analyzing real-world data to test the accuracy of linear functions in different contexts.
- Mathematical Modeling: Developing mathematical models to describe complex systems, such as the behavior of fluids or the growth of populations.
References
- [1] "Linear Functions" by Khan Academy
- [2] "Mathematics for the Nonmathematician" by Morris Kline
- [3] "Calculus" by Michael Spivak
Appendix
The following is a list of mathematical formulas and concepts used in this article:
- Linear Function: A polynomial function of degree one, written in the form $y = mx + b$.
- Slope: The rate at which the amount of water remaining decreases, represented by the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
- Y-Intercept: The initial amount of water in the bathtub, represented by the formula $b = y_1$.
- Mathematical Modeling: The use of mathematical equations and formulas to describe complex systems and relationships.
Raj's Bathtub Clog: A Q&A Session =====================================
Q: What is the problem with Raj's bathtub?
A: Raj's bathtub is clogged, and water is draining at a rate of 1.5 gallons per minute.
Q: How can we analyze the problem mathematically?
A: We can analyze the problem using linear functions, which describe the relationship between the amount of water remaining in the bathtub and the time in minutes that it has been draining.
Q: What is a linear function?
A: A linear function is a polynomial function of degree one, written in the form $y = mx + b$, where $m$ is the slope of the line, and $b$ is the y-intercept.
Q: How do we find the slope of the line?
A: We can find the slope of the line using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.
Q: What is the y-intercept?
A: The y-intercept is the initial amount of water in the bathtub, represented by the formula $b = y_1$.
Q: How do we write the linear function?
A: We can write the linear function by plugging in the values of $m$ and $x$ into the equation $y = mx + b$.
Q: What does the linear function represent?
A: The linear function represents the amount of water remaining in the bathtub as a function of the time in minutes that it has been draining.
Q: What are some real-world applications of linear functions?
A: Linear functions have many real-world applications, including physics, economics, and biology.
Q: What are some future research directions in this area?
A: Some future research directions in this area could include investigating the use of non-linear functions, collecting and analyzing real-world data, and developing mathematical models to describe complex systems.
Q: What are some references for further reading?
A: Some references for further reading include "Linear Functions" by Khan Academy, "Mathematics for the Nonmathematician" by Morris Kline, and "Calculus" by Michael Spivak.
Q: What is the appendix?
A: The appendix is a list of mathematical formulas and concepts used in this article.
Q: What are some common mistakes to avoid when working with linear functions?
A: Some common mistakes to avoid when working with linear functions include:
- Not checking the units: Make sure to check the units of the variables and the slope to ensure that they are consistent.
- Not using the correct formula: Make sure to use the correct formula for finding the slope and the y-intercept.
- Not plugging in the correct values: Make sure to plug in the correct values of $m$ and $x$ into the equation $y = mx + b$.
Q: What are some tips for working with linear functions?
A: Some tips for working with linear functions include:
- Start with a clear problem statement: Make sure to clearly define the problem and the variables involved.
- Use a consistent notation: Use a consistent notation for the variables and the slope to avoid confusion.
- Check your work: Make sure to check your work to ensure that the linear function is correct.
Q: What are some resources for further learning?
A: Some resources for further learning include:
- Online tutorials: Websites such as Khan Academy and Coursera offer online tutorials and courses on linear functions.
- Textbooks: Textbooks such as "Mathematics for the Nonmathematician" by Morris Kline and "Calculus" by Michael Spivak provide a comprehensive introduction to linear functions.
- Practice problems: Practice problems and exercises can be found in textbooks and online resources to help reinforce understanding of linear functions.