A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and φ(t), respectively, where 0 ≤ t ≤ 4 and t is measured in minutes (see figure). These angles are measured in radians, where θ = φ = 0 represent the starting position and θ = φ = 2π represent the finish position. The angular velocities of the runners are θ′(t) and φ′(t).
- a. Compare in words the angular velocity of the two runners and the progress of the race.
- b. Which runner has the greater average angular velocity?
- c. Who wins the race?
- d. Jean’s position is given by θ(t) = πt2/8. What is her angular velocity at t = 2 and at what time is her angular velocity the greatest?
- e. Juan’s position is given by φ(t) = πt(8 − t)/8. What is his angular velocity at t = 2 and at what time is his angular velocity the greatest?
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