91Ó°ÊÓ

When Jessica drives her car to a work-related conference, her employer reimburses her approximately 45 cents per mile to cover the cost of gas and the wear-and-tear on the vehicle. a) Using distance traveled d, in miles, as the independent variable and amount reimbursed A, in dollars, as the dependent variable, make a sketch of a coordinate system and label appropriately. Mark distance every 5 miles and amount reimbursed every \(\$ 0.45 .\) b) For traveling 0 miles, the reimbursement is 0 . This corresponds to the point \((0,0)\). Plot it on your coordinate system. c) For a trip that requires her to drive a total of 5 miles, she is reimbursed \(5(0.45)=\$ 2.25\). This corresponds to the point \((5, \$ 2.25)\). Plot it. d) For each 5 miles you go to the right, you must go up \(\$ 2.25\) and plot the point. Do this until you reach 20 miles. e) Keeping in mind that we are modeling this discrete situation continuously, draw a line through your data points. f) In March, Jessica attends a conference that is only 5 miles away. Counting round trip, she travels 10 total miles. Use the graph to determine how much she is reimbursed. g) In December, she attends a conference 10 miles away. How long is her trip in total? Use the graph to determine how much she will be reimbursed. h) For longer trips, such as 200 total miles, you will probably need to make a much larger graph. And what if she travels 400 miles? Or further? It is limitations such as these that make it useful to find an equation that describes what the graph shows. To find the equation, we start with a table that helps us to understand the relationship between the dependent and independent variables. Complete the table below. i) Use the table from part (h) to come up with an equation that relates \(\mathrm{d}\) and \(\mathrm{A}\). j) Now, use the equation to determine the reimbursement amounts for trips of 200 miles and 400 miles.

Step by step solution

01

Sketching the Coordinate System

First, set up a coordinate system with 'Distance Traveled (d)' on the horizontal axis (x-axis) and 'Amount Reimbursed (A)' on the vertical axis (y-axis). Label the x-axis with increments of 5 miles (5 miles per mark) and the y-axis with increments of $0.45 (0.45 per mark). This helps visualize the relationship between the miles driven and the reimbursement.

02

Plotting Initial Points

For 0 miles, the reimbursement is $0. Plot the point (0, 0) on your graph. This is the origin point indicating that if no miles are traveled, no reimbursement occurs.

03

Calculating and Plotting Incremental Points

Calculate the reimbursement for traveling 5 miles: \[A = 5 \times 0.45 = \\(2.25\]Then, plot the point (5, 2.25) on the graph. Repeat this process for 10, 15, and 20 miles by adding \)2.25 to each interval to get: (10, 4.50), (15, 6.75), and (20, 9.00).

04

Drawing the Line Through Data Points

Since the reimbursement relationship between distance and cost is linear, draw a straight line that passes through all plotted points. This line represents the continuous reimbursement model.

05

Determining Reimbursement for 10 Miles

Referencing the graph, look for the reimbursement value at 10 miles (counting the round trip distance). The reimbursement at point (10, 4.50) indicates Jessica is reimbursed $4.50 for a 10-mile round trip.

06

Determining Reimbursement for 20 Miles

For a conference 10 miles away (20 miles round trip), find the reimbursement on the graph at point (20, 9.00). Hence, Jessica will be reimbursed $9.00 for a 20-mile round trip.

07

Creating the Equation from the Table

From the table, note that the relationship between distance (d) and reimbursement (A) is linear with a slope of 0.45 (reimbursement per mile). The equation is A = 0.45d. This formula gives the reimbursement for any distance traveled.

08

Calculating Reimbursement for 200 and 400 Miles

Using the equation A = 0.45d,- For 200 miles: \[A = 0.45 \times 200 = \\(90.00\]- For 400 miles: \[A = 0.45 \times 400 = \\)180.00\]Jessica would be reimbursed \(90.00 for 200 miles and \)180.00 for 400 miles.

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Key Concepts

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

Understanding the Coordinate System

When dealing with linear equations, the coordinate system is a fundamental tool used to visualize data points and the relationship between variables. It's made up of two perpendicular axes: the horizontal axis called the x-axis and the vertical axis known as the y-axis. Each axis represents one of the variables under consideration.

In Jessica’s situation, the x-axis represents the independent variable, which is the distance traveled, noted as \(d\). This variable is within your control or the one you are testing. The y-axis, on the other hand, shows the dependent variable, which is the amount reimbursed, noted as \(A\). This is the outcome that depends on the independent variable. By plotting points and drawing lines on a coordinate system, one can easily see how, as Jess travels more miles, her reimbursement increases.
Particularly useful is noting points like \((0,0)\) where no travel means no reimbursement, providing a starting reference for calculations. This entire plotting system helps to visualize how changes in travel distance affect reimbursement.

Dependent and Independent Variables Simplified

Understanding the concepts of dependent and independent variables is crucial in analyzing linear equations. The independent variable is the factor you manipulate or consider the cause, while the dependent variable is what you measure or view as the effect.

• Independent Variable \( (d) \): In this scenario, the distance traveled by Jessica is our independent variable. It’s the variable you decide or plan, such as driving certain miles.


• Dependent Variable \( (A) \): This is the amount reimbursed for the distance traveled. It's dependent because it changes based on how many miles are driven. The more miles Jessica drives, the larger this value becomes.

By understanding which variable is which, one can accurately build and interpret models or equations that depict the relationship between them. In analytical graphs, the independent variable is always on the x-axis, whilst the dependent variable is placed on the y-axis. This structuring is important for creating accurate visualizations and solving linear equations effectively.

Understanding Slope in Linear Equations

The slope in a linear equation is a measure of how steep the line is on a graph, reflecting how much the dependent variable changes with a unit change in the independent variable.

In Jessica's case, the slope represents the reimbursement rate per mile. Mathematically, this is expressed as \(0.45\), which indicates that for every mile Jessica drives, her reimbursement increases by \(0.45\). This slope is constant and implies a linear relationship, meaning each additional mile yields the same additional reimbursement.

Determining the slope involves understanding these core components:

• Rise: The change in the dependent variable, in this case, the amount reimbursed.
• Run: The change in the independent variable, or the distance traveled.

The slope is calculated as the "rise over run" or \(\frac{\text{change in } A}{\text{change in } d} \). Such a consistent slope across distances showcases that Jessica's travel and reimbursement relationship is clearly linear and predictable.