Question

(1 point) 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 (9.0) in the xy-plane, Springfield is at (0,2), and Shelbyville is at (0-2). 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 nu Justify your answer. ble needed To solve this problem we need to minimize the following function of x: f(x) = (9-x) + 2'sqrt(x^2+4) We find that f(x) has a critical number at x = 2/sqrt3 To verify that f(x) has a minimum at this critical number we compute the second derivative /"(x) and find that its value at the critical number is . a positive number Thus the minimum length of cable needed is

Answer 1

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

f(0) = (9 - ) +2V:r2 + 4

Critical points of f(x) are given by

20 df (T) = 0 dac -1+0214 = 0 = =

Further

f"(1) = ਕਾ = ਆਤ

Thus at
I = 2/3
, we have
f" (2) = 3V3/8 > 0
​​​​​​

Hence f(x) has a minimum at the critical point and minimum value is given by

I = 2/3

f(2/3) = (9 – 2/3) +2 16/3 = 9+6/13 = 9+2V3

This is the desired minimum length of the cable.