I will give you branilest! This size running track is usually called a 400-meter track. However, if a person ran as close to the "inside" as possible on the track, they would run less than 400 meters in one lap. How far away (in meters) from the inside border would someone have to run to make one lap equal exactly 400 meters?

Solution

The two straightaway sections of the track are each 84.39 meters. The two semi-circular sections can be joined to form a circle whose radius is 36.5 meters and so the diameter of this circle is 2×36.5=73 meters. The circumference of this circle will be π×73 meters and so the total perimeter of the track is

2×84.39+π×73≈398.12.

So the perimeter of the track is less than 400 meters.

For the first lane on the track, the straightaway sections are each 84.39 meters long. However, the curved sections form a circle whose radius is now 36.5+1.22=37.72 meters. The diameter of the circle will be 2×37.72=75.44 meters. So the perimeter of lane 1 is

2×84.39+π×75.44≈405.78.

So the perimeter of lane 1 on the track is more than 400 meters and is almost 8 meters more than the perimeter of the inside of the track.

Suppose we let x denote the distance from the inside of lane 1 which gives a perimeter of 400 meters. This perimeter will be consist of the two straight sections which contribute 2×84.39 meters to the perimeter. In addition, there will be two semi-circular sections of radius 36.5+x meters. Combining these gives a circle whose diameter is 2×(36.5+x) meters. So we want

2×84.39+π×2×(36.5+x)=400.

Rewriting this we find

2π×x=400−2×84.39−2π×36.5

Solving for x we find

x≈0.30.

Note that this value for x is not exact but approximate. It is accurate to within about two ten thousandths of a meter or a fraction of a millimeter. So approximately 30 centimeters from the inside of lane 1 the perimeter of the track is 400 meters.

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