Struggling with this Applied Optimization Problems

Question

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  in the -plane, Springfield is at , and Shelbyville is at . The cable runs from Centerville to some point  on the -axis where it splits into two branches going to Springfield and Shelbyville. Find the location  that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer.

1-Minimize f(x)

2-Find the critical points

3-find that its value at the critical number

4- Thus the minimum length of cable needed is

Answer 1

We should start by defining our f(x), which will represent the total amount of cabling required for some central point (x,0):

f(x) = 2√(x² + 9) + 8 - x

Here, x² + 9 is the distance from the central point to either Springfield or Shelbyville and 8 - x is the distance from the central point to Centerville.

To find the critical points, you would take the derivative of f(x) and find values of x that make f'(x) equal to 0. Can you do the rest of the problem yourself?