The track coach plans to make 8 lanes each 1-meter wide on the running track. The track measures y meters along the inside curb. The track has straight parallel sides and semicircular ends. If the runners in a race lined up at the same spot and stayed in their own lanes throughout the race, they would not run the same distance. To make the race even, by how much should each lane be staggered?
Without knowing the radius of the semicircular ends, this can't be solved. However, given that the radius is r on the inside lane, the circular part of the lane has circumference 2πr. The straight part is the same length for each lane.
So, the radii for the lanes are
r, r+1, r+2 ... r+7
Thus each lane is 2π meters longer than the one inside it.
To determine how much each lane should be staggered, we first need to calculate the distance each lane covers.
Let's break down the problem into parts and find the distance covered by each lane separately:
1. Straight Parts:
- The track has two straight parallel sides, so each straight part will be y meters long.
- Since there are 8 lanes, each lane will cover y meters.
2. Curved Parts:
- The track has two semicircular ends.
- The innermost lane will run along the inside curb and cover the entire circumference of the semicircle.
- The outer lanes will have wider curves since they cover a larger circumference.
- In this case, the outermost lane will run along the outside curb and cover the entire circumference of the larger semicircle.
To calculate the distances covered by each lane in the curved parts, we need to find the circumference of the semicircular ends:
The circumference of a circle is given by the formula: C = 2πr, where r is the radius.
Since we have a semicircle, the radius will be half of the width of the lane.
3. Calculation:
- Each lane is 1 meter wide, so the radius of the innermost lane is 0.5 meters.
- The circumference of the semicircle for the innermost lane is C1 = 2π(0.5) = π meters.
- Similarly, the radius of the outermost lane will be (0.5 + 7) meters because 7 lanes separate the outermost lane from the innermost lane.
- The circumference of the semicircle for the outermost lane is C8 = 2π(0.5 + 7) = 16π meters.
Now we can calculate the distances covered by each lane in the curved parts:
- The distance covered by the innermost lane in the curved parts is π meters.
- The distance covered by the outermost lane in the curved parts is 16π meters.
To make the race even, we want each runner to cover the same distance. Since the innermost lane covers π meters and the outermost lane covers 16π meters in the curved parts, the difference is 16π - π = 15π meters.
Therefore, each lane should be staggered by 15π meters to make the race even.
Note: The calculation assumes the track is perfectly circular, and the measurements provided are accurate. Depending on the actual dimensions of the track, the values might vary slightly.