Problem 2.

Suppose a runner running at 10 miles per hour and a bicyclist riding 20 miles per hour are headed toward each other. At the instant they are 15 miles apart a humming bird passes the runner flying in the direction of the bicyclist. The humming bird is flying 30 miles per hour. When it reaches the bicyclist, without slowing down the humming bird reverses direction and flies back toward the runner. When the humming bird reaches the runner, it again reverses direction without any slowing and heads back toward the bicyclist. The hummingbird continues this pattern, going back and forth between the runner and the bicyclist, until the distance between the runner and the bicyclist has closed to zero. What is the total distance traveled by the humming bird from the time the runner and bicyclist are 15 miles apart until they meet?

This problem is mentioned in the biography of Nobel prize winner John Forbes Nash. In the early 1950's the problem was presented to John von Neumann, a professor at Princeton and the most famous mathematician in the world at that time. Amazingly von Neumann was able to solve the problem as an infinite sum in his head (without pencil or paper) within a few seconds! Nash, who was a 22 year old graduate student at Princeton also solved the problem in his head within a few seconds. However, Nash's solution was much simpler than von Neumann's.

[Problem submitted by Vin Lee, LACC Associate Professor of Mathematics. Source: A Beautiful Mind, by Sylvia Nasar; Touchstone Books, 1999.]


Since the runner is going 10 mph and the bicyclist 20 mph, the gap is being closed at a rate of 10 mph + 20 mph = 30 mph. The runner and bicyclist are 15 miles apart.

distance = speed x time

15 miles = 30 mph x time

time = 1/2 hour

So the total time the humming bird is flying is 1/2 hour. It's speed is 30 mph.

distance = speed x time

distance = 30 mph x 1/2 hour

distance = 15 miles

So, the total distance flown by the hummingbird is 15 miles.

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