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

In this article, we will delve into the world of mathematics and explore the relationship between time and water drainage in a clogged bathtub. The problem presented to us is that Raj's bathtub is clogged and is draining at a rate of 1.5 gallons of water per minute. We are given a table that 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. Our goal is to understand the mathematical relationship between these two variables and make predictions about the amount of water remaining in the bathtub at any given time.

The Problem

The table provided shows the amount of water remaining in the bathtub at different times. For example, after 1 minute, there are 90 gallons of water remaining, after 2 minutes, there are 75 gallons of water remaining, and so on. We are asked to find the amount of water remaining in the bathtub as a function of time.

Analyzing the Data

To understand the relationship between time and water drainage, we need to analyze the data provided in the table. We can start by looking at the pattern of the data. As we can see, the amount of water remaining in the bathtub decreases by 15 gallons every minute. This suggests that the rate of water drainage is constant at 1.5 gallons per minute.

Mathematical Modeling

To model the relationship between time and water drainage, we can use a linear equation. The general form of a linear equation is:

y = mx + b

where y is the amount of water remaining in the bathtub, x is the time in minutes, m is the slope of the line, and b is the y-intercept.

In this case, we know that the rate of water drainage is 1.5 gallons per minute, so the slope of the line is -1.5. We also know that the amount of water remaining in the bathtub is 90 gallons after 1 minute, so we can use this point to find the y-intercept.

Finding the Y-Intercept

To find the y-intercept, we can substitute the values of x and y into the equation and solve for b.

90 = -1.5(1) + b

Simplifying the equation, we get:

90 = -1.5 + b

Adding 1.5 to both sides, we get:

91.5 = b

So, the y-intercept is 91.5.

The Equation

Now that we have found the slope and y-intercept, we can write the equation that models the relationship between time and water drainage.

y = -1.5x + 91.5

This equation tells us that the amount of water remaining in the bathtub, y, is equal to -1.5 times the time in minutes, x, plus 91.5.

Interpreting the Equation

To interpret the equation, we can look at the slope and y-intercept. The slope of -1.5 tells us that the rate of water drainage is 1.5 gallons per minute. The y-intercept of 91.5 tells us that the amount of water remaining in the bathtub is 91.5 gallons when the time is 0 minutes.

Making Predictions

Now that we have the equation, we can make predictions about the amount of water remaining in the bathtub at any given time. For example, if we want to know the amount of water remaining in the bathtub after 10 minutes, we can substitute x = 10 into the equation.

y = -1.5(10) + 91.5

Simplifying the equation, we get:

y = -15 + 91.5

y = 76.5

So, after 10 minutes, there will be 76.5 gallons of water remaining in the bathtub.

Conclusion

In this article, we have explored the relationship between time and water drainage in a clogged bathtub. We have analyzed the data provided in the table, modeled the relationship using a linear equation, and made predictions about the amount of water remaining in the bathtub at any given time. The equation y = -1.5x + 91.5 models the relationship between time and water drainage, and we can use it to make predictions about the amount of water remaining in the bathtub.

Future Work

In the future, we can use this equation to make predictions about the amount of water remaining in the bathtub at different times. We can also use this equation to model the relationship between time and water drainage in other situations, such as a clogged sink or a overflowing pool.

References

• [1] "Linear Equations" by Math Open Reference
• [2] "Mathematics for the Nonmathematician" by Morris Kline

Appendix

The following is the R code used to create the table and plot the data.

# Create the table
time <- c(1, 2, 3, 4, 5)
water_remaining <- c(90, 75, 60, 45, 30)

# Create the plot
plot(time, water_remaining, main = "Water Remaining in the Bathtub", xlab = "Time (minutes)", ylab = "Amount of Water Remaining (gallons)")
```<br/>
**Q&A: Understanding the Relationship Between Time and Water Drainage in a Clogged Bathtub**
====================================================================================

**Introduction**
---------------

In our previous article, we explored the relationship between time and water drainage in a clogged bathtub. We analyzed the data provided in the table, modeled the relationship using a linear equation, and made predictions about the amount of water remaining in the bathtub at any given time. In this article, we will answer some frequently asked questions about the relationship between time and water drainage in a clogged bathtub.

**Q: What is the rate of water drainage in a clogged bathtub?**
---------------------------------------------------------

A: The rate of water drainage in a clogged bathtub is 1.5 gallons per minute.

**Q: How can I model the relationship between time and water drainage in a clogged bathtub?**
-----------------------------------------------------------------------------------------

A: You can model the relationship between time and water drainage in a clogged bathtub using a linear equation. The general form of a linear equation is:

y = mx + b

where y is the amount of water remaining in the bathtub, x is the time in minutes, m is the slope of the line, and b is the y-intercept.

**Q: How can I find the slope and y-intercept of the linear equation?**
-------------------------------------------------------------------------

A: You can find the slope and y-intercept of the linear equation by analyzing the data provided in the table. The slope of the line is equal to the rate of water drainage, which is 1.5 gallons per minute. The y-intercept of the line is equal to the amount of water remaining in the bathtub when the time is 0 minutes, which is 91.5 gallons.

**Q: How can I make predictions about the amount of water remaining in the bathtub at any given time?**
-----------------------------------------------------------------------------------------

A: You can make predictions about the amount of water remaining in the bathtub at any given time by substituting the time into the linear equation. For example, if you want to know the amount of water remaining in the bathtub after 10 minutes, you can substitute x = 10 into the equation.

**Q: What are some real-world applications of the relationship between time and water drainage in a clogged bathtub?**
-----------------------------------------------------------------------------------------

A: Some real-world applications of the relationship between time and water drainage in a clogged bathtub include:

* Modeling the relationship between time and water drainage in a clogged sink or overflowing pool
* Predicting the amount of water remaining in a bathtub or sink at any given time
* Designing systems for draining water from bathtubs or sinks

**Q: What are some limitations of the linear equation model?**
---------------------------------------------------------

A: Some limitations of the linear equation model include:

* The model assumes that the rate of water drainage is constant, which may not be the case in all situations
* The model assumes that the amount of water remaining in the bathtub is a linear function of time, which may not be the case in all situations
* The model may not be accurate for very large or very small values of time

**Q: How can I improve the accuracy of the linear equation model?**
-------------------------------------------------------------------------

A: You can improve the accuracy of the linear equation model by:

* Using more data points to model the relationship between time and water drainage
* Using a more complex model, such as a quadratic or cubic equation
* Using techniques such as regression analysis to improve the fit of the model to the data

**Conclusion**
--------------

In this article, we have answered some frequently asked questions about the relationship between time and water drainage in a clogged bathtub. We have discussed the rate of water drainage, how to model the relationship using a linear equation, and how to make predictions about the amount of water remaining in the bathtub at any given time. We have also discussed some real-world applications of the relationship and some limitations of the linear equation model.