Running track and stagger · 3 marks

The figure depicts a 400 m athletics track. You can see two straight sections of length 84.39 m each, and two curved portions, which are semicircles with a common centre (points A and B); the innermost semicircle on each side has radius 36.5 m. The width of each lane is 1.22 m. Let an athlete make one complete circuit of the track. What is the total distance she runs?

Answer: Taking her path 0.3 m from the inner border: 2 × 84.39 + 2π × 36.8 = 168.78 + 231.22 = 400 m.

Step-by-step solution

Given: Straights: 84.39 m each; Inner radius of each semicircle: 36.5 m; She runs 0.3 m from the inner border (the usual assumption)
To find: Total distance for one complete circuit

Idea: One lap = two straights + two semicircles. Two semicircles of the same radius make one full circle, so use 2πr with the radius of her actual path, 36.5 + 0.3 = 36.8 m.

Straights: 2 × 84.39 = 168.78 m.½ mark
Her path on each bend is 0.3 m outside the inner border, so its radius is 36.5 + 0.3 = 36.8 m.½ mark
The two semicircles make one full circle: 2 × π × 36.8 = 2 × 3.1416 × 36.8 = 231.22 m.1 mark
Total = 168.78 + 231.22 = 400 m.1 mark

She runs 168.78 m + 231.22 m = 400 m in one complete circuit.

Check: 2 × 3.1416 × 36.8 = 231.222 m, and 168.78 + 231.22 = 400.00 ✓. This is exactly why the track is called a 400 m track.

Answer to write in the exam

Straights = 2 × 84.39 = 168.78 m

Radius of path on bends = 36.5 + 0.3 = 36.8 m

Two semicircles = 2πr = 2 × 3.1416 × 36.8 = 231.22 m

∴ Total distance = 168.78 + 231.22 = 400 m

Common mistakes that cost marks

Using radius 36.5 m: that gives about 398.1 m. A runner cannot run exactly on the border line.
Counting each semicircle as a full circle, which gives about 631 m.
Adding only one straight.

How this can come in the exam

Short answer (2 marks)

A track has two straights of 100 m and two semicircular ends. A runner’s path on the ends has radius 31.5 m. How far does she run in one lap? (Use π = 227.)

Show answer

Bends: 2 × 227 × 31.5 = 198 m (1 mark). Lap = 200 + 198 = 398 m (1 mark).

Try one yourself

On the track described above, a runner keeps 0.3 m outside the inner line of lane 2 (radius 36.5 + 1.22 + 0.3 = 38.02 m on the bends). How far does she run in one lap? (Use π = 3.1416.)