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Problem of the Day #194: Running Fast September 29, 2011

Posted by Arjun in : potd , trackback

Alex is running very fast on a circular track. The track has an inner radius of $4$ meters and an outer radius of $8$ meters, and allows running on painted lanes at $1$ meter intervals. Alex runs at $3 \pi$ m/s. Alex wants to run around the track as fast as possible, and decides to run as far inside as possible. However, some slow runners begin running at the same time as Alex at $\pi$ m/s, and Alex (being a very considerate fellow) does not want to end up within $\pi$ meters of them on the same lane, and will slow down to $\pi$ m/s running speed if he is at that distance. Alex can switch lanes at any time. What is the fastest possible time that Alex can complete $10$ laps around the track?


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