Off to the Races

Part 1: Off to the Races! (20 points)

Suppose two bugs are crawling along linear paths.

Bug 1 begins a trek toward a point 70 inches from where he begins, traveling at a speed of 12 inches per hour.

Bug 2 travels at a speed of 18 inches per hour but leaves 1 hour after the other bug from a similar starting position on a parallel path.

Recall: distance = ratetime.

1. Given that T represents Bug 1’s travel time, what formulas represent the distance each bug travels over time? (4 points – 2 points for each bug.)

Distance (Bug 1) = ______Distance (Bug 2) = ______

2. Let’s watch the race! Below are the TI 84 setting to watch the bugs.

For TI-Nspire users, let’s watch the race on my TI-84 on the TV!

3. The Y equations are set to constants to show the bugs crawling across the screen. Why is there no T variable in the Y equation for each bug? Be sure to state your answer using at least one well constructed sentence. (1 point) 4. Graph the paths of the bugs in motion. Your parametric graph should include time and direction of each bug. Clearly label the path of each bug on your graph. Which bug wins the race? What needs to happen for a bug to win? Include a chart to approximate your answer.

5. At what time are the bugs the same distance from their starting points along their paths? In other words, when are the bugs alongside each other? Show an algebraic way to determine the solution and show a numeric way (using a table). (6 points – 2 points for the solution, 2 points for the algebraic work, 1 point for each bug’s table) Part 2: Scatter Bugs (20 points)

Now consider three bugs’ paths as the bugs run around on the xy-plane in time t seconds modeled by

Bug A Bug B Bug C

1. Eliminate the parameter for each set of equations and write the rectangular equation for each bug. 6 points – 2 for each bug

Bug A: Bug B: Bug C:

2. Which path on the graph belongs to Bug A? Which is the path of Bug B? Bug C’s path? Label each path on the picture with A, B, or C. Use at least one clear, well-constructed sentence to justify your choice for each bug. 6 points – 1 pt for each correctly identified path, 1 pt for each justification.

If all three bugs start moving from their initial positions at the same time:

3. What must happen for them to collide? (2 points)

4. Looking at the graphs in question 2, is there enough information to tell if the bugs collide? (2 points)

5. Do Bug A and Bug B collide? Use at least one well-constructed sentence to justify your conclusion. (2 points)

6. Do Bug A and Bug C collide? Use at least one well-constructed sentence to justify your conclusion. (2 points) Part 3: Catch Me If You Can (10 points)

1) A shark and a fish swim so that their positions in the ocean at time t (in seconds) are as follows:

shark fish x= 3t-4 x=4t+3 y= 2t-7 y= -3t+5

Will the shark catch the fish? If yes, at what point (position) and time? If no, explain why. 5 points – 1 for yes or no, 4 for position/time or for explanation.

2) Two children are playing tag. Their running patterns on the playground at time t (in minutes) are as follows:

"It" "Not It" x= -2t+5 x= 4t+1 y= -4t+10/3 y= -8t+6

Will "Not It" be tagged? If yes, at what point and time? If no, explain why. 5 points – 1 for yes or no, 4 for position/time or for explanation.