1) To determine the length of the racetrack, can we use only one of the racers' curves to measure the length, or must we use both curves? Why? Your final justification must include at least one calculation of the racetrack length done in MATLAB, with the input code attached to or pasted within the post.

2) Who wins the race depicted in the GeoGebra application? Justify this using definite integrals.

3) In the moments where Torty and Harry meet, who has the greater speed? Who has a greater acceleration? (Hint: You can fairly intuitively determine where they meet by examining the similarities and differences between their functions. Then find a way to approximate the values of their speeds and accelerations)

4) What is the curvature difference between Torty and Harry's curves in the moments that they meet? Explain your result. (Hint: Setting up the curvature calculations in MATLAB will be useful, you will need to use a limit as part of these calculations when Torty's speed is 0).

Transcribed Image Text:

We began Math 241 by analyzing a footrace between Torty and Harry. One year after their first race, Harry decided that he could gain an advantage by hosting a new race. This new footrace would not transpire on flat ground, but instead along a hilly region in the mountains. The hilly region and track is visualized in the provided GeoGebra applet. Harry has become quite adjusted to racing along inclines and declines. His path in the provided GeoGebra applet is defined by the curve h(t) = (cos(t), sin(t), cos(t) sin (t)), for which two laps around the course takes 4 seconds. In their rematch, Torty is less adjusted to running along hilly regions, and tires after bursts of speed. His path in the provided GeoGebra applet is defined by the curve (t) = (cos(t + sin(t)), sin(t + sin(t)), cos(t + sin(t)) sin (t + sin(t))). You may alter the "t=" slider to view Torty and Harry's positions, velocities, and accelerations at any time during the race. You may also use the "Show Torty's Curve" and "Show Harry's Curve" checkboxes to view either of the racer's curve in isolation. t = 10.93 Show Hill Show Torty Show Harry Reset Zoom Zoom Torty Zoom Harry Torty O Harry We began Math 241 by analyzing a footrace between Torty and Harry. One year after their first race, Harry decided that he could gain an advantage by hosting a new race. This new footrace would not transpire on flat ground, but instead along a hilly region in the mountains. The hilly region and track is visualized in the provided GeoGebra applet. Harry has become quite adjusted to racing along inclines and declines. His path in the provided GeoGebra applet is defined by the curve h(t) = (cos(t), sin(t), cos(t) sin (t)), for which two laps around the course takes 4 seconds. In their rematch, Torty is less adjusted to running along hilly regions, and tires after bursts of speed. His path in the provided GeoGebra applet is defined by the curve (t) = (cos(t + sin(t)), sin(t + sin(t)), cos(t + sin(t)) sin (t + sin(t))). You may alter the "t=" slider to view Torty and Harry's positions, velocities, and accelerations at any time during the race. You may also use the "Show Torty's Curve" and "Show Harry's Curve" checkboxes to view either of the racer's curve in isolation. t = 10.93 Show Hill Show Torty Show Harry Reset Zoom Zoom Torty Zoom Harry Torty O Harry