Solutions for Session 5, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6

Problem C1

 a. The equation A = 9h describes the distance in miles that Achilles travels, in terms of time measured in hours. "A" stands for the distance run by Achilles, while "h" stands for hours. b. For this problem, h = 1 1/2, so A = 9(1 1/2) = 13 1/2 miles. c. The graph should be a line through the origin, since the relationship is a proportional one. d. The graph is a line through the origin. You should find that the line has a constant slope, or rate of change, of 9 miles per hour.

Problem C2

 a. The equation is T = h + 32. There is an invisible "1" in front of h, since the tortoise runs at 1 mile per hour. b. As in Problem C1, you should find a constant rate; this time, the rate is 1 mile per hour. c. A comparison of the spreadsheets finds that after exactly 4 hours, both Achilles and the tortoise are 36 miles from the start.

 Problem C3 According to the graphs, the intersection of the two lines occurs at the point (4, 36). Note that the independent variable (on the horizontal axis) represents time, and the dependent variable (on the vertical axis) represents distance.

 Problem C4 At the time h = 4, the distance for both Achilles and the tortoise is 36 miles. Since the graph represents both travelers' positions over time, the two are at the same point at h = 4. It's at this time that Achilles overtakes the tortoise.

 Problem C5 Achilles' graph is the proportional relationship because it is the graph of a line passing through (0, 0).

 Problem C6 Because the two people are traveling at the same speed, the person with the 25 mile lead will keep that lead, at exactly 25 miles, for the entire race. The two distance graphs will never intersect, no matter how long the race is, which suggests that the graphs will be parallel. So, linear graphs with the same rate of change will be parallel.

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