## Running Requirements

#### Problem

An official IAAF (International Amateur Athletics Federation) running track measures 400 m and is made up of a straight section measuring $84.39$ m and semi-circular curves with a radius of exactly $36.5$ m; the $400$ m distance is measured 30 cm from the inside edge of the track.

Tracks used to be marked out by using equal quadrant measures, which means that it was made up of $100$ m straight sections and $100$ m semi-circular curves. However, sport scientists have found that increasing the radius of the curves increases performance of athletes and reduces the chance of injury. In accordance with this, the IAAF has stipulated that the inside radius of the track must lie between $35$ m and $38$ m.

A school wishes to mark out a running track that satisfies the IAAF regulations. Show that it is necessary to have a non-equal quadrant measure track, and find the bounds of the straight sections to satisfy the requirements.