please help me with this hw question Calculus hw ASAP thank youuu

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.

Answer 1

Use the diagram to write the total length of cable required as a function of x (the x-value on the x-axis where the cable splits into a "Y"). Then take the derivative of the length function, set it = 0, and solve:

Horizontal length: (9-x) Diagonal lengths: √(x²+16) by pythag thm

L(x) = 2·√(x²+16) + 9 - x

L^'(x) = 2x / √(x²+16) - 1 = 0

4x² = x² + 16

x² = 16/3 and x = 4/√3