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Chapter 12: Problem 12

Use the following information to answer the next three questions. Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is \(y=12,000 x\) . What are the independent and dependent variables?

Short Answer

Expert verified
The independent variable is \(x\), and the dependent variable is \(y\).

Step by step solution

01

Understand the Variables

In the equation \(y = 12,000x\), the variables represent different quantities. Generally, the independent variable is the one you change or control, while the dependent variable is what you measure or observe as the effect.
02

Identify the Independent Variable

In the given equation, the independent variable is \(x\). It typically represents the number of years, or time, because it is what we increase incrementally to observe change.
03

Identify the Dependent Variable

The dependent variable is \(y\), which represents the total amount of soil lost per year. This variable depends on the changes or different values of \(x\), as the loss of soil increases as the number of years increases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Variable
In a linear equation like \( y = 12,000x \), the independent variable is a crucial component. It is the variable that you can completely control or modify to see how it influences other variables. In our example of soil erosion, the independent variable is \( x \). This variable represents time, specifically the number of years that have passed.

Consider these key points about the independent variable:
  • It is often the input in an equation.
  • You can choose different values for it, like 1, 2, or 3 years, to see how it impacts the outcome.
  • The independent variable is foundational for experimentation and modeling, as it initiates the cause-effect cycle.
In this scenario, increasing \( x \) by one year will directly impact the dependent variable, which is essential in mathematical modeling.
Dependent Variable
Identifying the dependent variable is essential to understand how variables relate to each other. In the equation \( y = 12,000x \), the dependent variable is \( y \). This variable measures something specific: the total amount of soil lost each year.

Let's explore more about dependent variables through these highlights:
  • It is the output of the equation, changing based on the independent variable's value.
  • The dependent variable is what allows us to observe and measure the effects of changes.
  • This variable relies entirely on another input, highlighting crucial cause-effect relationships in various contexts.
In the shoreline example, \( y \) becomes larger as \( x \), the number of years, increases, demonstrating the direct relationship facilitated by the equation.
Mathematical Modeling
Mathematical modeling is a powerful tool that helps illustrate real-world problems with equations. In this context, it simplifies complex processes like soil erosion into an easy-to-understand format. The equation \( y = 12,000x \) neatly represents how the soil loss progresses with time.

Here鈥檚 what is pivotal about mathematical modeling:
  • It transforms complex phenomena into manageable mathematical expressions.
  • This process allows predictions and problem-solving with numerical representations.
  • The model provides a clear, visual way to understand relationships and dynamics in data.
Through the linear model \( y = 12,000x \), it becomes clear how much soil is lost after a given number of years, illustrating the power and usefulness of mathematical modeling in environmental sciences and beyond.

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Most popular questions from this chapter

The maximum discount value of the Entertainment庐 card for the 鈥淔ine Dining鈥� section, Edition ten, for various pages is given in Table 12.21 $$\begin{array}{|c|c|}\hline \text { Page number } & {\text { Maximum value (s) }} \\ \hline 4 & {16} \\ \hline 14 & {19} \\ \hline 25 & {19} \\\ \hline 25 & {17} \\ \hline 43 & {17} \\ \hline 42 & {15} \\ \hline 72 & {15} \\ \hline 85 & {17} \\ \hline 90 & {17} \\ \hline\end{array}$$ a. Decide which variable should be the independent variable and which should be the dependent variable. b. Draw a scatter plot of the ordered pairs. c. Calculate the least-squares line. Put the equation in the form of: \(\hat{y}=a+b x\) d. Find the correlation coefficient. Is it significant? e. Find the estimated maximum values for the restaurants on page ten and on page 70 . f. Does it appear that the restaurants giving the maximum value are placed in the beginning of the 鈥淔ine Dining鈥� section? How did you arrive at your answer? g. Suppose that there were 200 pages of restaurants. What do you estimate to be the maximum value for a restaurant listed on page 200? h. Is the least squares line valid for page 200? Why or why not? i. What is the slope of the least-squares (best-fit) line? Interpret the slope.

Use the following information to answer the next five exercises. A random sample of ten professional athletes produced the following data where \(x\) is the number of endorsements the player has and \(y\) is the amount of money made (in millions of dollars). $$\begin{array}{|c|c|c|c|}\hline x & {y} & {x} & {y} \\ \hline 0 & {2} & {5} & {12} \\ \hline 3 & {8} & {4} & {9} \\ \hline 2 & {7} & {3} & {9} \\\ \hline 1 & {3} & {0} & {3} \\ \hline 5 & {13} & {4} & {10} \\\ \hline\end{array}$$ Table 12.13 What is the \(y\)-intercept of the line of best fit? What does it represent?

Use the following information to answer the next two exercises. The cost of a leading liquid laundry detergent in different sizes is given in Table 12.31. $$\begin{array}{|c|c|}\hline \text { Size (ounces) } & {\text { cost }(\mathrm{s})} & {\text { cost per ounce }} \\ \hline 16 & {3.99} \\ \hline 32 & {4.99} \\ \hline 64 & {5.99} \\ \hline 600 & {10.99} \\\ \hline\end{array} $$ a. Using 鈥渟ize鈥� as the independent variable and 鈥渃ost鈥� as the dependent variable, draw a scatter plot. b. Does it appear from inspection that there is a relationship between the variables? Why or why not? c. Calculate the least-squares line. Put the equation in the form of: \(\hat{y}=a+b x\) d. Find the correlation coefficient. Is it significant? e. If the laundry detergent were sold in a 40-ounce size, find the estimated cost. f. If the laundry detergent were sold in a 90-ounce size, find the estimated cost. g. Does it appear that a line is the best way to fit the data? Why or why not? h. Are there any outliers in the given data? i. Is the least-squares line valid for predicting what a 300-ounce size of the laundry detergent would you cost? Why or why not? j. What is the slope of the least-squares (best-fit) line? Interpret the slope.

For each of the following situations, state the independent variable and the dependent variable. a. A study is done to determine if elderly drivers are involved in more motor vehicle fatalities than other drivers. The number of fatalities per 100,000 drivers is compared to the age of drivers. b. A study is done to determine if the weekly grocery bill changes based on the number of family members. c. Insurance companies base life insurance premiums partially on the age of the applicant. d. Utility bills vary according to power consumption. e. A study is done to determine if a higher education reduces the crime rate in a population.

Use the following information to answer the next two exercises. An electronics retailer used regression to find a simple model to predict sales growth in the first quarter of the new year (January through March). The model is good for 90 days, where x is the day. The model can be written as follows: \(\hat{y}=101.32+2.48 x\) where \(\hat{y}\) is in thousands of dollars. What would you predict the sales to be on day 90?
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