CMO 1974 Problem 7

A bus route consists of a circular road of circumference 10 miles and a straight road of length 1 mile which runs from a terminus to the point $Q$ on the circular road (see diagram).

It is served by two buses, each of which requires 20 minutes for the round trip. Bus No. 1, upon leaving the terminus, travels along the straight road, once around the circle clockwise and returns along the straight road to the terminus. Bus No. 2, reaching the terminus 10 minutes after Bus No. 1, has a similar route except that it proceeds counterclockwise around the circle. Both buses run continuously and do not wait at any point on the route except for a negligible amount of time to pick up and discharge passengers. A man plans to wait at a point $P$ which is $x$ miles ( $0\le x < 12$ ) from the terminus along the route of Bus No. 1 and travel to the terminus on one of the buses.

Assuming that he chooses to board that bus which will bring him to his destination at the earliest moment, there is a maximum time $w(x)$ that his journey (waiting plus travel time) could take.

Find $w(2)$ ; find $w(4)$ .

For what value of $x$ will the time $w(x)$ be the longest?

Sketch a graph of $y = w(x)$ for $0\le x < 12$ .