Final answer:
Eileen's average running speed back home was 8 km/h, calculated by dividing the distance of 6 km she covered while walking (at 4 km/h for 90 minutes) by the 45 minutes it took her to run back.
Step-by-step explanation:
To find Eileen's average speed on her run back home, we need to calculate the distance she ran and the time it took her to run that distance. Eileen leaves at 6 pm and returns at 8:30 pm, including a 15-minute rest. The total time spent walking, resting, and running is 2 hours and 30 minutes, or 150 minutes. We subtract the 90 minutes of walking and 15 minutes of resting to find that Eileen spent 45 minutes on her run back home.
Eileen walked for 90 minutes at an average speed of 4 km/h. The distance covered is given by the formula: distance = speed × time. Thus, she walked 6 km (4 km/h × 1.5 h). Since she ran back home along the same path, her running distance is also 6 km.
To find the average speed of Eileen's run, we use the formula: average speed = total distance / total time. Since Eileen ran 6 km in 45 minutes (or 0.75 hours), her average speed is 8 km/h (6 km / 0.75 h).