In the Olympic track represented above 8 runners are going to compete

Solution:

\(Ok\; the\; way\; to\; solve\; this\; problem\; is\; pretty\; simple,\;\)

\(we\; know\; from\; the\; graph\; that\; the\; track\; is\; a\; rectangle\; +\; a\; circle\; \left( 2\; semicircles \right).\)

\(The\; distance\; run\; by\; the\; first\; athlete\; will\; be\; 85*2=170\; of\; the\; rectangle,\; or\; straight\; segment\; of\; the\; track,\; and\; 2\pi r\; of\; the\; circle\;\)

\(where\; r=37,\; so\; 170\; +\; 2\pi r.\;\)

\(This\; is\; to\; be\; subtracted\; from\; the\; distance\; run\; by\; the\; athlete\; on\; the\; 8th\; line,\; this\; is\; 170\; +\; 2\pi \left( r+7*1,2 \right),\;\)
\(where\; 7*1,2\; is\; the\; distance\; between\; line\; 1\; and\; 8.\;\)

\(170\; +\; 2\pi r\; +\; 2\pi *7*1,2\; -\; 170\; -2\pi r\; =\; 2\pi \cdot 7\cdot 1,2\; =\; 6,28\cdot 8,4\; =\; 6,3\cdot 8,4\; =\; \frac{63\cdot 84}{100}\; =\; 52,92\; ≈\; 53\)

ps. The way I usually multiply strange things like 84*63 is to simplify them, I wanted to obtain 80 so I thought, 4/84≈ 5%, 5% of 63 is ≈ 3, so we can simply calculate 80*66 which makes finding 53 much easier and faster.

pps. The nice thing about this problem is that it represents the way the 400m work, runners will always have to start in different places to get to the same finish line having run the same distance. And also the measures of the field are realistic, the length of the straight part of the track is 84 and something, while the radius of the circumference is a little it less than 37 and the distance between the lines a bit more than 1,2 meters