Flying trains

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Here is the classic not-really-a-calculus puzzle.


Towns A and B are connected by a single railroad track, exactly 210 miles long. One fateful day, at exactly 1:00pm, a red train leaves town A traveling to town B at 40 miles per hour. At the same time, a bright blue train leaves town B traveling to town A at 30 miles per hour. As the red train starts to move, a brave fly takes off of the windshield and flies at 55 miles per hour towards town B. As soon as the fly reaches the blue train, he immediately changes direction and flies back towards town A, again, traveling at 55 miles per hour. When he gets to the red train, he changes direction again. The fly continues to fly back and forth between the two, ever nearing trains until he is smashed to bits when the trains sadly collide.

How far did the fly between 1:00pm and his all-to-early death?


You certainly could try to set up some sort of infinite sum, but there is an easier way. First answer this: how long did the fly travel?
165 miles.
Given the velocities of the trains, the distance between them is decreasing at a rate of 70 miles per hour. Thus the trains will collide in exactly 3 hours. The fly will be traveling at 55 miles per hour for this entire 3 hour period, which comes to 165 miles traveled.

See also

Girl, boy and dog