In the 400 meter dash runners stay in their lanes the entire distance around the track. If the runners would all start at the same line (the finish line) only the runner in lane 1 would be running 400 meters. Everyone else would be running a greater distance. To rectify this, the starting lines are staggered for each lane so that each runner, when he/she crosses the finish line, will have run only 400 meters.
If the lanes are 1 meter wide and the lane 1 runner begins at the finish line, how far ahead is the next stagger line? Note, the starting line for each lane will be that far ahead of the previous lane’s start..
Use the following diagram for needed distances. Show your work and/or explain your reasoning.
each lane is 1 meter wide i.e. the radius increases by a meter for each lane. Distance on inside circle(lane one)is 100 meters on each end of the track
the circumference increases by 2pi meters for each extra meter of radius
To calculate how far ahead the next stagger line is, we need to consider the circumference of each lane.
Let's start with lane 1, which has a radius of 100 meters (since it is on the inside). The circumference of a circle is calculated by multiplying the radius by 2π (pi). Therefore, the circumference of lane 1 is:
C₁ = 2π * r₁ = 2π * 100 = 200π meters
Since each lane is 1 meter wide, the distance covered by the stagger line is equal to the circumference of lane 1 plus the width of all the other lanes. For the outer lanes, the radius will increase by 1 meter for each lane.
To find the distance covered by the stagger line, we add the width of all the lanes starting from lane 2 up to the desired lane. Let's say we want to find the distance covered by the stagger line for lane 2.
The radius for lane 2 would be 100 meters + 1 meter = 101 meters. Using the same formula as above, the circumference of lane 2 in this case would be:
C₂ = 2π * r₂ = 2π * 101 = 202π meters
Now, we need to add the width of lane 2 to the circumference of lane 1:
Distance covered by stagger line for lane 2 = C₁ + width of lane 2
= 200π + 1
Using the approximate value of π as 3.14, we can calculate the distance covered by the stagger line for lane 2:
Distance covered by stagger line for lane 2 ≈ 200 * 3.14 + 1 ≈ 628.64 + 1
≈ 629.64 meters
Therefore, the stagger line for lane 2 is approximately 629.64 meters ahead of the start line (finish line) for lane 1.