Over the course of the 100m race, lasting at least 10 seconds, Torty’s average speed on any 5-second interval was less than Harry’s 5-second average speed, and yet Torty won the race!
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 is given in the provided GeoGebra applet.
Harry has become quite adjusted to racing along inclines and declines. Their path in the provided GeoGebra applet is defined by the curve
, for which two laps around the course takes seconds.
In their rematch, Torty is less adjusted to running along hilly regions, and tires after an initial burst of speed along the hilly regions. Their path in the provided GeoGebra applet is defined by the curve
Both Torty’s and Harry’s curve are depicted in the following GeoGebra application, as is the hilly region over which the track was built. You may alter the “=” 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.
Task one:
Answer the following questions about the race depicted in the GeoGebra applet:
1) How long is the race track? Are we forced to use either Torty’s curve or Harry’s curve to make the measurement? Why?
2) Who wins the race depicted in the GeoGebra application? Justify this
3) In the moments where Torty and Harry meet, who has the greater speed? Who has a greater acceleration?
4) What is the curvature difference between Torty and Harry’s curves in the moments that they meet? Explain this difference. (Note, you’ll need to use a limit to answer this question when Torty’s speed is 0).
Task two:
After the first race on this hilly track, Torty and Harry want a rematch. Define curves for Torty and Harry such that Torty’s average speed is less than Harry’s over any 5-second interval, but Torty wins the race. The race must last at least seconds, and for simplicity, assume that both Torty and Harry continue to run along the same curves for a whole lap even after the race is concluded.
Keep in mind that they need to run the exact same course, so the race track needs to remain unchanged.
Task three:
After Harry’s second defeat, they want to look for ways to make the course even more difficult for Torty. After talking to a landscaper, Harry determined that the terrain around the track (considering the center of the track to be the point (0, 0, 0) can be defined by the function f(x, y) = xy2. The race track is determined by taking the points (x, y, f(x, y)), where (x, y) follow the unit circle.
Harry suspects that making the course more difficult within the first quadrant (of the unit circle) will further hinder Torty’s ability to run effectively.
Using only partial derivatives, directional derivatives, and tangent planes, answer the following questions:
- Estimate the current height change along the course between the points and
- How does the current course height change between and compare to the estimated height change if the course had instead moved the same distance directly North or directly West? (Note: Consider the x-axis to be East and the y-axis to be North)
- Because of the topography around the course, Harry can only alter the course after the point by moving Northeast, Northwest, Southeast, or Southwest, and only by the same horizontal distance as the original change from and Compare the estimated height changes in each direction to determine the new course that would prove most difficult journey.
