Chapter 6.1, Problem 56E

Where do they meet? Kelly started at noon (t = 0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15/(t + 1)² (decreasing because of fatigue). Sandy started at noon (t = 0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20/(t + 1)² (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.

1. a. Make a graph of Kelly’s distance from Niwot as a function of time.
2. b. Make a graph of Sandy’s distance from Berthoud as a function of time.
3. c. When do they meet? How far has each person traveled when they meet?
4. d. More generally, if the riders’ speeds are v(t) = A/(t + 1)² and u(t) = B/(t + 1)² and the distance between the towns is D, what conditions on A, B, and D must be met to ensure that the riders will pass each other?
5. e. Looking ahead: With the velocity functions given in part (d), make a conjecture about the maximum distance each person can ride (given unlimited time).