Centerville is the headquarters of Greedy Cablevision Inc. The
cable company is about to expand service to two nearby towns,
Springfield and Shelbyville. There needs to be cable connecting
Centerville to both towns. The idea is to save on the cost of
cable by arranging the cable in a Y-shaped configuation.
Centerville is located at
(8,0) in the xy -plane, Springfield is at (0,5) , and
Shelbyville is at (0,- 5) . The cable runs from Centerville
to some point (x,0) on the x -axis where it splits into two branches going to
Springfield and Shelbyville. Find the location (x,0)
that will minimize the amount of cable between the 3 towns and
compute the amount of cable needed. Justify your answer.

To solve this problem we need to minimize the following function
of

We find that

To verify that

Thus the minimum length of cable needed is

You can earn partial credit on this problem.