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

Question

https://s3.postimg.org/q0pixg25f/olympic_track.png

In the Olympic track represented above 8 runners are going to compete to reach the finish line in the shortest amount of time. Between Tom in line 1 and Jack in line 2 there's a distance of 1,2 meters, the same as in the subsequent lanes. If each runner is to race onto their assigned line (and not in the middle of the lane), and Tom is to run on line 1 and Bob on line 8. Bob will have to start-position himself approximately how many meters in front of Tom for the two to run the same distance before reaching the finishing line?

A) 37
B) 51
C) 53
D) 64
E) 85

A) 37
B) 51
C) 53
D) 64
E) 85

DensetsuNo · Answer

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

aniketm.87@gmail.com · Answer

I did not understand why you subtracted those two distances ? Can you kindly explain. Thanks

DensetsuNo · Answer

Because you want to understand what the difference is in the 2 perimeters.

This is : first perimeter:402 meters, second perimeter:455, for the two of them to run 402 meters the second athlete will have to start 53 meters ahead.

Argo · Answer

Please change 1,2 meters to 1.2 meters. It's confusing.

DensetsuNo · Answer

Done, hope it's clearer now.

bumpbot · Answer

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

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