Problem:

The roads that connect six towns in the county all have bridges that are in desperate need of repair. The county transportation department has limited funds so it can only repair five bridges and will close the roads of the unsafe bridges until more funds are available.   The distance between all towns is given on the graph below.  The county transportation department needs to find the shortest route that will connect all towns and will repair the bridges only on those roads.  Find the shortest route that will allow every town to be connected to every other town.