1. **Problem statement:** We have a 400-meter Olympic track composed of two straight sections and two semicircular ends. The straight sections are each 84.39 m long, and the radius of the inside lane semicircles is 36.8 m. 2. **Show that the inside lane runner runs 400 m:** The total length of the track is the sum of the lengths of the two straight sections plus the lengths of the two semicircular ends. Formula for the length of the track: $$\text{Total length} = 2 \times \text{straight length} + 2 \times \text{semicircle length}$$ The length of a semicircle is half the circumference of a full circle: $$\text{semicircle length} = \pi \times r$$ Calculate the total length: $$2 \times 84.39 + 2 \times (\pi \times 36.8) = 168.78 + 2 \times 115.56 = 168.78 + 231.12 = 399.9 \approx 400$$ So, the inside lane length is approximately 400 meters. 3. **Find the stagger for lane 2:** The radius of lane 2 is 37.92 m. The difference in radius between lane 2 and lane 1 is: $$37.92 - 36.8 = 1.12$$ The extra distance lane 2 runner must run is the difference in the lengths of the two semicircular ends: $$\text{extra length} = 2 \times \pi \times 1.12 = 2 \times 3.1416 \times 1.12 = 7.04$$ Therefore, the lane 2 runner should start approximately 7.04 meters ahead to compensate for the longer distance. **Final answers:** - Inside lane length: approximately 400 meters. - Lane 2 stagger (head start): approximately 7.04 meters.