91Ó°ÊÓ

One day Lydia walked from Allentown to Brassville at a constant rate of 4 kilometers per hour. The towns are 30 kilometers apart. a. Write an equation for the relationship between the distance Lydia traveled, \(d,\) and the hours she walked, \(h\). b. Graph your equation to show the relationship between hours walked and distance traveled. Put distance traveled on the vertical axis. c. How many hours did it take Lydia to reach Brassville? d. Now write an equation for the relationship between the hours walked, \(h\), and the distance remaining to complete the trip, \(r\). e. Graph the equation you wrote for Part d on the same set of axes you used for Part b. Label the vertical axis for both \(d\) and \(r\). f. How can you use your graph from Part e to determine how many hours it took Lydia to reach Brassville?

Step by step solution

01

- Write an equation for distance traveled

The distance traveled by Lydia can be modeled by the equation: \[ d = 4h \] where \(d\) is the distance in kilometers and \(h\) is the time in hours.

02

- Graph the equation

To graph the equation \( d = 4h \), plot \(d\) (distance traveled) on the vertical axis and \(h\) (hours walked) on the horizontal axis. Plot points such as (0,0), (1,4), (2,8), up to (7.5, 30). Draw a straight line through these points.

03

- Calculate hours to reach Brassville

To find how many hours it took Lydia to reach Brassville, solve for \(h\) when \(d = 30\): \[ 30 = 4h \] Divide both sides by 4: \[ h = \frac{30}{4} = 7.5 \] So, it took Lydia 7.5 hours to reach Brassville.

04

- Write an equation for the distance remaining

The distance remaining \(r\) can be found by subtracting the distance traveled \(d\) from the total distance (30 kilometers): \[ r = 30 - d \] Substitute \(d = 4h\) into the equation: \[ r = 30 - 4h \]

05

- Graph the equation for distance remaining

To graph \( r = 30 - 4h \) on the same set of axes, plot \(r\) (remaining distance) on the vertical axis. Use points such as (0, 30), (1, 26), (2, 22), up to (7.5, 0). Draw a straight line through these points.

06

- Determine hours to reach Brassville using the graph

The graph for \( r = 30 - 4h \) intersects the horizontal axis where the remaining distance \(r\) is 0. This intersection occurs at \(h = 7.5\). This means it took Lydia 7.5 hours to reach Brassville, confirming the result from the previous calculation.

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Key Concepts

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

Linear Equations

Linear equations are expressions that depict a straight-line relationship between two variables. In Lydia's walk from Allentown to Brassville, the equation given was \( d = 4h \). Here, \( d \) represents the distance in kilometers and \( h \) stands for hours walked. This equation is called linear because the variables \( d \) and \( h \) increase at a constant rate. For every hour that Lydia walks, she covers 4 additional kilometers. This relationship always forms a straight line when graphed. The simplicity of linear equations makes them useful in many real-world scenarios like calculating distances, predicting expenses, and determining speed.

Graphing Equations

Graphing equations involves plotting points on a coordinate system to visualize the relationship between two variables. For the equation \( d = 4h \), we plot distance \( d \) on the vertical axis and time \( h \) on the horizontal axis. Points like (0,0), (1,4), (2,8), etc., are plotted, and connecting these points forms a straight line. This line helps us quickly see how distance increases with time. Additionally, for the equation \( r = 30 - 4h \) (distance remaining to complete the trip), we plot points like (0,30), (1,26), (2,22), etc. By graphing both equations on the same set of axes, we can easily compare the distance Lydia has traveled to the distance she has left to cover. This visual representation simplifies understanding their relationship and how they change over time.

Problem-Solving

Problem-solving involves breaking down a task into smaller steps and systematically addressing each part. In this exercise, we first wrote an equation to represent Lydia's walk. Using \( d = 4h \), we calculated the time taken for a 30-kilometer trip. Solving \( 30 = 4h \) for \( h \), we found \( h = 7.5 \) hours. Then, we constructed another equation \( r = 30 - 4h \) to show the remaining distance \( r \), considering time. Graphing both equations, we visualized how distance and time interrelate, confirming our solution methodically. Approaching problems this way, with clear steps and graphical solutions, makes complex problems more manageable and understandable.