## 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.

#### Solution

Although the inside of the track has radius $R$, the $400$ m is measured $30$ cm from this edge. So the radius of the curve that will trace the $400$ m distance will have radius $R + 0.3$, as shown in the diagram below.

It can be seen that the two semi-circles combine to form a circle with a diameter of $2R + 0.6$.

If the school built an equal quadrant measure track, $\pi(2R + 0.6) = 200$.

$\begin{align}\therefore \pi(R + 0.3) &= 100\\R + 0.3 &= \dfrac{100}{\pi}\\\therefore R &= \dfrac{100}{\pi} - 0.3 \approx 31.53 m\end{align}$

As this falls outside the IAAF requirements, the school must employ a non-equal quadrant measure track.

$\begin{align}\therefore \pi(2R + 0.6) + 2L &= 400\\\pi(R + 0.3) + L &= 200\\\therefore L &= 200 - \pi(R + 0.3)\end{align}$

As $35 \le R \le 38$, we get the approximate bounds, $79.68 \le L \le 89.10$ m.

The IAAF requires that each lane be $1.22 \pm 0.01$ m wide. If a $400$ m race is to be run then how far must the runner in lane two be ahead of the runner in lane one?