Math  /  Algebra

QuestionWrite Which representation of equal expressions-tables, graphs, equations, or verbal descriptions-do you prefer? Explain why you a table, a graph, an equi prefer this representation and provide an example. verbal descriptip represent a PP
Practice The Department of Transportation in each state is responsible for the improvements and repairs of that state's roads. One important job is to repaint the road lines that have worn away or faded. A painting crew is painting a 24 -mile stretch of road. They have already completed a total of 9.5 miles of the road. The crew has been painting at a rate of 0.25 mile per hour and continues to paint at the same rate.
1. Identify the two quantities that are changing in this situation, identify the independent and dependent quantities, and define the variables for these quantities. Then write an equation that relates the two quantities.
2. What is the unit rate of change in this situation? Explain.
3. How many total miles of the road will be completed if the crew works for another 2 hours?
4. How many more hours does the crew need to work to complete half of the job?
5. Complete the table and then construct a graph.

Quantities Units of Measure Variables \begin{tabular}{|c|c|} \hline \begin{tabular}{r} Independent \\ Quantity \end{tabular} & \begin{tabular}{c} Dependent \\ Quantity \end{tabular} \\ \hline 0 & \\ \hline 2 & \\ \hline 5 & \\ \hline 6.5 & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. The painting crew is working on a 24-mile stretch of road.
2. They have completed 9.5 miles.
3. The crew paints at a rate of 0.25 miles per hour.
4. We need to identify the independent and dependent quantities, define variables, and write an equation.
5. We need to calculate the unit rate of change.
6. We need to determine how many total miles will be completed after 2 more hours.
7. We need to calculate how many more hours are needed to complete half of the job.
8. We need to complete the table and construct a graph.

STEP 2

1. Identify and define variables for the independent and dependent quantities.
2. Write an equation relating the two quantities.
3. Determine the unit rate of change.
4. Calculate the total miles completed after 2 more hours.
5. Calculate the additional hours needed to complete half of the job.
6. Complete the table and construct a graph.

STEP 3

Identify the two quantities that are changing: - The time spent painting (in hours) is the independent quantity. - The total miles of road painted is the dependent quantity.
Define the variables: - Let t t represent the time in hours. - Let m m represent the miles of road painted.

STEP 4

Write an equation that relates the two quantities. Since the crew has already painted 9.5 miles, the equation is:
m=9.5+0.25t m = 9.5 + 0.25t

STEP 5

Determine the unit rate of change. The unit rate of change is the rate at which the crew paints, which is 0.25 miles per hour.

STEP 6

Calculate the total miles of the road completed if the crew works for another 2 hours:
m=9.5+0.25×2 m = 9.5 + 0.25 \times 2 m=9.5+0.5 m = 9.5 + 0.5 m=10 m = 10
So, 10 miles will be completed after 2 more hours.

STEP 7

Calculate how many more hours the crew needs to work to complete half of the job. Half of the 24-mile stretch is 12 miles. The crew has already completed 9.5 miles, so they need to complete 2.5 more miles:
2.5=0.25t 2.5 = 0.25t
Solve for t t :
t=2.50.25 t = \frac{2.5}{0.25} t=10 t = 10
The crew needs to work 10 more hours to complete half of the job.

STEP 8

Complete the table and construct a graph. We will calculate the miles painted for different times:
Time (hours)Miles Painted09.5210510.756.511.125\begin{array}{|c|c|} \hline \text{Time (hours)} & \text{Miles Painted} \\ \hline 0 & 9.5 \\ \hline 2 & 10 \\ \hline 5 & 10.75 \\ \hline 6.5 & 11.125 \\ \hline \end{array}
To construct the graph, plot the time on the x-axis and the miles painted on the y-axis. Connect the points to show the linear relationship.

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