Yesterday, Matt and Rachel jogged for a half hour around a circular track with a radius of 100 meters. In that time, Matt ran 10 laps and Rachel ran 7 laps. Each started and stopped in the same location and ran at a constant speed in the same direction around the track. Unfortunately, Matt discovered today that he had inadvertently placed a highly radioactive rock into his running vest right before the run. Rachel's doctor is trying to assess her risk of radiation poisoning based on her distance across the track from Matt during the run. On average, how many meters apart (in a straight line) were Matt and Rachel during the 30 minute run? Round to the nearest integer.
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4$\begingroup$ We should be averaging the inverse square of the distance between them to get the total dose received by Rachel. You need to set a minimum distance (rather larger than the size of the source) between them when they pass or the calculation will blow up. $\endgroup$– Ross MillikanCommented Jul 8 at 2:33
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What matters is the relative distance between the two people. So assume Rachel stayed still and Matt ran 3 laps. The average should be the same in each 10 minute interval so we can consider only the first 10 minutes, where Matt ran 1 lap. The distance between Matt and Rachel after $x$ minutes is $200\sin\frac{\pi x}{10}$. So we need $(\int_0^{10}200\sin{\frac{\pi x}{10}}dx)/10=400/\pi\approx127$. The average distance is 127m (rounded to the nearest metre).
