Distance runners often run around an approximately circular trajectory. However, since some runners are running at a larger radius than others, they must be given an advantage in the beginning so that the race is fair.

If the path of the runner running at a greatest radius of 20 m is 100 m long, how far behind in m should the runner at the smallest radius start? The smallest radius is 15 m.

distance will be proportional to radius.

(perimeterofCircle=PI*2*radius)

difference in perimeters=2PI*(20-15)

so that distance is the how far behind should the inner radius be.

So, what is the answer?

I am not sure how you are calculating this problem since you wrote it in abbreviations so could you please explain again?

To determine how far behind the runner at the smallest radius should start, we first need to calculate the circumference of the circular trajectory for each runner.

The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.

For the runner with the greatest radius of 20 m, the circumference of their circular trajectory is:
C₁ = 2π(20) = 40π m

For the runner with the smallest radius of 15 m, the circumference of their circular trajectory is:
C₂ = 2π(15) = 30π m

Now, we need to find the difference in distance between the two runners. Since the path of the runner with the greatest radius is 100 m, we subtract the circumference of the circular trajectory for the smallest radius runner from that distance:
Difference in distance = 100 m - C₂

Substituting the value of C₂:
Difference in distance = 100 m - 30π m

To calculate the exact value, we can use an approximation for π, such as 3.14:
Difference in distance ≈ 100 m - (30 × 3.14) m
Difference in distance ≈ 100 m - 94.2 m
Difference in distance ≈ 5.8 m

Therefore, the runner at the smallest radius should start approximately 5.8 meters behind the runner at the greatest radius to make the race fair.