Final Answer:
(a) You run a total of 20 miles.
(b) It takes you 3 hours in total.
(c) At 45 minutes, you are 3/4 of the way around the circular track from your starting point.
Step-by-step explanation:
(a) To find the total distance run, we sum the distances covered during each lap. The first lap covers 1/4 hour at 10 mi/hr, resulting in 1/4 * 10 = 2.5 miles. Subsequent laps cover decreasing distances as the speed decreases by 2 mi/hr after each lap. The total distance is given by the infinite geometric series formula S = a / (1 - r), where a is the first term and r is the common ratio. In this case, a = 2.5 and r = 8/10, yielding S = 20 miles.
(b) To find the total time, we sum the times taken for each lap. The first lap takes 1/4 hour, and the subsequent laps form an infinite geometric series with a = 1/4 and r = 2/5. Using the sum formula S = a / (1 - r), we find S = 1/3 hours. The total time is 1/4 + 1/3 = 7/12 hours, which is equivalent to 3 hours.
(c) To determine your position at 45 minutes, we calculate the fraction of the track covered by that time. At 45 minutes (3/4 of an hour), you are 3/4 of the way around the circular track from your starting point. This corresponds to an angle of 3/4 * 360° = 270°, expressing your position as an angle taken from the starting point.