Each night after work I leave through the door D shown in the diagram and walk across the quadrangle to my car. the most direct route is obviously the north gate corridor. Half of the time this gate is locked and, as I don't have a key, I have to go across the quadrangle again towards Y, through the south gate (which is never locked) and go around the long way to my car. I used to walk directly to X, from where I can see whether the gate is locked.
a. How far on average did I walk to reach my car? Show working out.
b. Recently I have walked from the door, parallel to the wall on my left, until I can just see V. Then I can tell weather the gate is locked. When it is locked I don't have so far to walk around the long way.
How far do I walk on average under this strategy? Show working out.
c. Today I realised, when standing at Y, that I was in a straight line with X and V. I am therefore able to see from Y whether or not the gate is locked.
With this information, find a new strategy which will give the least distance on average that I walk. State this least distance and explain how it is achieved.
(Diagram is not to scale, I drew it on MS paint)