Alvin, Simon, and Theodore are running around a 1000 meter circular track starting at different positions. Alvin is running in the opposite direction of Simon and Theodore. He is also the fastest, running twice as fast as Simon and three times as fast as Theodore. If Alvin meets Simon for the first time after running 312 meters, and Simon meets Theodore for the first time after running 2526 meters, how far apart along the track (shorter distance) did Alvin and Theodore meet?
wlog, Alvin begins at A = 0 and runs ccw at speed 6v. Define the ccw direction as positive.
Simon and Theodore start at S and T and run cw at speeds -3v and -2v, respectively. (A, S, T are in meters and v is in meters/second).
(note that Steven Lord and I used different definitions of location 0 so although our answers look different, we found the same solution)
Alvin meets Simon at position 312, so Simon began at position 468. Alvin runs at speed 312x, Simon at -156x, and Theodore at -104x.
Simon starts at 468 and meets Theodore catching up to him after running 2526 meters at position: 468 - 2526 at location 942.
In the time Simon ran 2526 meters, Theodore would only run 2/3 of that distance: 1684 meters.
1684 meters ccw from location 942 is location 626.
So the starting locations are:
Alvin 0 goes ccw
Simon 468 goes cw
Theodore 626 goes cw
Alvin met Simon at location 312 (given).
Simon met Theodore at location 942.
Since Alvin is 3 times as fast as Theodore, they meet 3/4 of their initial starting distance. 626*3/4 = 469.5
This is 1.5 Meters clockwise from Simon's initial starting position.
This matches Steven Lord's solution of 1000 - 1.5 = 998.5
He defined location 0 as Simon's start whereas I defined it as Alvin's start.
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Posted by Larry
on 2024-11-09 06:34:16 |
Solution
