## 39. Math Puzzle $8 / 68$

Six track and field athletes start a 400-m race. The straight and back straight sections are each $100 \mathrm{~m}$ long. The distance between the white lines marking each running lane is $2 \mathrm{~m}$. How large must the starting distance between the individual runners be so that all cover the same distance to the finish line?
For the solution, the difference of only two circumferences needs to be calculated. The sought lane differences are all the same.

The differences in the running distances are caused by the two $100 \mathrm{~m}$ long curves, which together form a full circle. The difference between two circumferences with radii $r_{1}$ and $r_{2}$ is

$$
\Delta u=u_{1}-u_{2}=2 \pi r_{1}-2 \pi r_{2}=2 \pi\left(r_{1}-r_{2}\right)
$$

Since the distance between the lane markings is $2 \mathrm{~m}$, it follows that $r_{1}-r_{2}=2 \mathrm{~m}$. This applies to all lanes. Therefore, the distance by which two adjacent runners must start is

$$
\Delta u=2 \pi \cdot 2 \approx 12.57 \mathrm{~m}
$$