Final answer:
The total time for Alice to run the race, with x as her speed for the first half, is represented by T = (10/x) + (10/(x - 2)). For a speed of 10 km/h in the first half, the total time taken to run the race to the nearest tenth is 2.3 hours.
Step-by-step explanation:
The student's question involves determining the total time Alice takes to run a 20-km race with different average speeds in each half. To answer part a) of the question, we use the variable 'x' to represent Alice's speed for the first half of the race. Therefore, her speed in the second half will be x - 2 km/h. Since the race is 20 km in total, we can calculate the time taken for each half by dividing the distance (10 km) by the speed (time equals distance over speed).
For the first half of the race:
Time1 = 10 km / x
For the second half of the race:
Time2 = 10 km / (x - 2)
The total time T needed to run the race is the sum of Time1 and Time2:
T = Time1 + Time2 = (10/x) + (10/(x - 2))
For part b), we substitute x with 10 km/h, giving us:
Time1 = 10 km / 10 km/h = 1 hour
Time2 = 10 km / (10 km/h - 2 km/h) = 10 km / 8 km/h = 1.25 hours
The total time T Alice takes to run the race at 10 km/h for the first half is:
T = 1 hour + 1.25 hours = 2.25 hours, or to the nearest tenth, 2.3 hours.