Section 2.4 Graphs, Tables, and Linear Functions: Analyzing a Race
Subsection 2.4.1 Overview
In this section you will continue with the Hare and the Tortoise tale and analyze and focus on their individual travels.
Student Page 2.4.2 Trodding Tortoise
In Trodding Tortoise, continue to model motion with graphs and consider the race between the Hare and the Tortoise in another way. This time, from knowledge of the tortoise's speed, create a table showing the tortoise's distance from the Starting Line at various times during the race. Plot points and consider the appearance of the graph as it relates to the tortoise's speed over several time intervals. Find an equation that models the data. Finally, consider the many ways that the Tortoise's speed arises in the table, graph, and equation. In the end, define slope.
During the race between the tortoise and the hare, the fox recorded the speed at which both the tortoise and the hare were traveling. The fox observed that the tortoise plodded along at a rate of 20 meters per minute throughout the 1000-meter race.
1.
(a)
Complete Table 2.4.2.1 indicating the distance traveled by the tortoise in the time elapsed.
| Time elapsed in minutes, \(t\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Distance traveled in meters, \(D\) | 0 | |||||||
| Time elapsed in minutes, \(t\) | 10 | 15 | 20 | 25 | 30 | 35 | 640 | \(t\) |
| Distance traveled in meters, \(D\) |
(b)
Which variable is the independent variable? Why do you think so?
2.
Use the graph in Figure 2.4.2.2. For each axis, choose a scale so the graph shows the tortoise's complete race, points are easy to plot, and you use as much of the graph as possible.
(a)
The horizontal axis represents the independent variable. Label it using the scale you chose.
(b)
The vertical axis represents the dependent variable. Label it using your chosen scale.
(c)
Label each axis with the variable it represents.
(d)
Sketch a graph that represents the tortoise's progress during the race using the data in Table 2.4.2.1.
3.
(a)
In one minute, how far did the tortoise travel? What does this number represent in terms of the tortoise's movement during the race?
(b)
How can you find the distance the tortoise traveled directly from the number of minutes that have elapsed since the start of the race?
(c)
Let \(t\) represent the elapsed time in minutes and \(D\) represent the distance in meters traveled by the tortoise. Write an equation showing the relationship between \(t\) and \(D\text{:}\)
(d)
Replace \(t\) in the equation in Task 2.4.2.3.c with some of the values in the table. What values did you get for \(D\) from the equation? Did the equation values for \(D\) match those in your table? Should they match?
(e)
How does your answer to Student Page Exercise 2.4.2.3 relate to the equation you found in Task 2.4.2.3.c?
4.
(a)
Complete Table 2.4.2.3 for the tortoise's travels for each of the time intervals indicated.
| Time Interval | Amount of time elapsed |
Change in distance during the time interval |
The tortoise's speed |
|---|---|---|---|
| 0 to 5 minutes | |||
| 5 to 10 minutes | |||
| 10 to 20 minutes |
(b)
How can you find the tortoise's speed over each time interval? Explain. Enter the speed in the table.
(c)
Compare the tortoise's speeds for the three time intervals in the table. What do you notice?
(d)
Should the tortoise's speeds for each time interval be the same? Why or why not?
(e)
The slope of a line is the steepness of the line. What is the slope of the line that models the tortoise's movement?
5.
How long does it take the tortoise to complete the race? How do you know?
Student Page 2.4.3 Hopping Hare
Extend what you learned through Trodding Tortoise to analyze the Hopping Hare's motion, data, and graph and relate these to equations that model each part of his race.
1.
The fox made the following observations about the hare's movement during the race with the tortoise:
The hare traveled at a constant rate of 250 meters per minute for the first 2 minutes.
The hare's nap started at exactly 2 minutes and lasted exactly 47.5 minutes.
The hare woke suddenly and traveled at a constant rate of 500 meters per minute during the last minute of the race.
(a)
Complete Table 2.4.3.1 indicating the distance traveled by the hare in the elapsed time.
| Time Interval | Amount of time elapsed |
Change in distance during the time interval |
The hare's speed |
|---|---|---|---|
| 0 to 2 minutes | 250 meters per minute | ||
| 2 to 49.5 minutes | |||
| 49.5 to 50.5 minutes | 500 meters per minute |
(b)
Use the axes you used to draw the tortoise's progress (Student Page Exercise 2.4.2.2) to draw an accurate graph to represent the hare's progress during the race from the information in the table and the fox's observations. What points will help you draw an accurate graph?
(c)
Draw an accurate graph for the hare's progress during the race.
2.
Write an equation relating elapsed time, \(t\text{,}\) and distance, \(D\text{,}\) traveled for the hare's first 2 minutes of travel.
3.
Use the equation you found in Student Page Exercise 2.4.3.2 to answer the following questions:
(a)
How far did the hare travel during the first 15 seconds (0.25 minutes) of the race?
(b)
If the hare had continued traveling at this rate, how long would it have taken him to complete the 1000-meter race? Would the hare have won?
(c)
How far could the hare have traveled if he continued at this rate for the entire 50.5 minutes?
4.
(a)
Find an equation relating the hare's elapsed time and distance traveled for 2 minutes to 49.5 minutes. Keep in mind that he is not sitting on the starting line at the beginning of this time interval.
(b)
What is the slope of the equation you found in Task 2.4.3.4.a? Why does this slope make sense?
5.
(a)
What is the slope of the line that shows the hare's movement during the last minute of the race?
(b)
Is the slope enough to determine an equation for the last minute of the race? Why or why not?
6.
Desmos allows you to plot equations based on time intervals.
(a)
Open a Desmos worksheet. Click on the “?” icon (Help) in the top right corner of the page. Click on the icon for Restrictions and learn how to use them.
(b)
Plot the equation for the tortoise.
(c)
Plot the two equations you found for the hare. For the hare's equations, restrict x to the intervals over which the fox observed him. What do you notice about how Desmos plots the hare's graph?
(d)
(Optional) You already know the slope for the last minute of the hare's travels. Play with a third equation for the final time interval of the hare's race to approximate the equation for this part of the hare's race.
Homework 2.4.4 Homework
1.
Consider your work on Trodding Tortoise and Hopping Hare. Write about your current understanding of slope and the relationships between slope, speed, rate of change in a table of values, and the change in distance divided by the change in time. In particular, how does speed show up in the table? How does it show up in the graph? How does it show up in an equation?
2.
Revisit The Eyes Have It. The relationships you investigated in The Eyes Have It are called proportional relationships. In linear proportional relationships, you get one quantity by multiplying the other by a constant. Are any of the Tortoise and Hare relationships proportional? See if you can find 2 or 3 examples.
3.
There are 8 sticks of gum in a small package.
(a)
Create a table to show the number of sticks of gum in 0, 1, 2, 3, 4, 5, 10, 15, and 22 packages if there are 8 sticks in each package.
(b)
Plot the data. Choose scales for both \(x\) and \(y\) so that the displayed graph uses most of the screen. Label each axis to show what quantities each represents.
(c)
What is the relationship between the number of packages and the number of sticks of gum?
(d)
Write the relationship in Task 2.4.4.3.c as an equation.
4.
Gasoline prices vary greatly over a week. Gas Buddy reported gasoline prices in the Allendale, MI area ranging from $2.03 to $2.69 per gallon one week in September 2018.
(a)
Complete Table 2.4.4.1 to find how much you would pay for the numbers of gallons of gas listed for each gasoline price. Round prices to the nearest penny (hundredth of a dollar).
(b)
Use Desmos to plot the data. Label scales and titles for each axis.
(c)
Write equations in the last row of the table that fit the data. Tell how you know your equations are correct.
(d)
How many gallons of gasoline can you purchase for $10? Show your work for one of the gasoline prices, accurate to 2 decimal places.
5.
One mile equals 2000 average steps while walking. If you walk at a rate of 3 miles per hour, you will average 100 steps per minute. 31 Revisit your table and graph from Exercise 2.2.7.2 in Homework 2.2.7.
(a)
Write an equation that will help you determine the number of steps you will walk in m minutes. Define your variables.
(b)
What is the slope of the graph? How do you know?
(c)
What does the slope mean in terms of the amount of time you walk and the number of steps you take?
6.
As of January 1, 2020, the Michigan minimum wage for non-tipped employees is $9.65 per hour. How much will you earn if you work \(h\) hours? (Revisit your table and graph from Exercise 2.2.7.3 in the Homework 2.2.7.)
(a)
Write an equation to help you determine the amount you will earn in \(h\) hours if you are paid minimum wage in Michigan. Define your variables.
(b)
What is the slope of the graph? How do you know?
(c)
What does the slope mean in terms of the number of hours worked and the amount of money earned?
7.
Practice using Desmos to graph the examples above. If you need more help, click on the “?” icon (Help) in the upper right corner of the Desmos graphing screen. Choose Tables for a tutorial on entering and graphing data. Choose Sliders to see what these do. Use a slider with the equation, \(y = mx\text{,}\) to fit your data. Explain why the equation you find makes sense for your data.
8.
Complete Slopes and Tree Trunks. Which of the equations represents a proportional relationship? How do you know?
Student Page 2.4.5 Slopes and Tree Trunks
Near the Calder Plaza in downtown Grand Rapids, Michigan a stand of trees is planted in a 5 by 5 grid as shown in Figure 2.4.5.1. Each green dot represents the location of a tree. The trunks of the trees are uniform in size. All of the trunks are straight and perpendicular to the ground.
1.
The line of sight from the origin to the trees with coordinates (1, 1), (2, 2), (3, 3), (4, 4), and (5, 5) is shown. Which of these tree trunks can you see if you are standing at the origin, (0, 0)? Why do you think so?
2.
Other than the tree trunks in Row 1 and Column 1, which tree trunks can you see if you are standing at the origin? Why do you think these tree trunks are visible?
3.
What is the slope of the line of sight from the origin (0, 0) to the tree trunk at point (4, 2)? What other tree trunks are along this line of sight? How do you know?
4.
At what point on one of the axes would you have to be standing in order for the tree trunks at points (2, 5) and (1, 3) to be on the same line of sight? What is the slope of this line of sight? What do you think the equation of this line of site is? Why do you think so?
5.
Draw a line of sight through point (2, 0) parallel to the line you found in Student Page Exercise 2.4.5.4. What trees are along this line of sight?
6.
Find the slope of the equation of the line in Student Page Exercise 2.4.5.5. What do you think is the equation of this line of site? Why do you think so?
7.
How are the slopes of the lines in Student Page Exercise 2.4.5.4 and Student Page Exercise 2.4.5.6 related?
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