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.

Question

Centerville is located at (9,0) in the xy-plane, Springfield is at (0,4), and Shelbyville is at (0,-4). To save on the cost of cable, Greedy Cablevision wants to arrange the cable in a Y-shaped configuation, running cable from Centerville to some point (x,0) on the x-axis where it then splits into two branches, one going to Springfield and one to 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 conclusion by answering the following questions.

(a) To solve this problem we need to minimize the following function of x.
f(x) =

(b) We find that f(x) has a critical point at
x =

(c) To verify that f(x) has a minimum at this critical point we evaluate the second derivative f′′(x) at this point
f''(x)(critical point) is =

(d) Thus the minimum length of cable needed is

Reiny · Answer

I labelled the position of Centerville, Springfield, Shelbyville and (x,0) as
C, A, B, and P respectively

Length of cable = PC + AP + BP
clearly cost is dependent on the length

Cost = PC + AP + BP , for some unit of money
= 9-x + √(x^2 + (-4)^2) + √(x^2 + 4^2)
= 9 - x + √(x^2 + 16) + √(x^2 + 16) = 2√(x^2 + 16) + 9 - x
= 2(x^2 + 16)^(1/2) + 9 - x
d(Cost)/dx = (x^2 + 16)^(-1/2) (2x) - 1
= 2x/√(x^2 + 16) - 1
= 0 for a min of Cost
2x/√(x^2 + 16) = 1
2x = √(x^2 + 16)
square both sides
4x^2 = x^2 + 16
3x^2 = 16
x = ± 4/√3 , but x must be between 0 and 9
so the point is (4/√3, 0)

I will let you decide how all that fits in to your series of questions, everything
should be there.
The only thing I didn't do was to find the 2nd derivative to establish max or min, it
was obvious.
Here is what your cost graph looks like, just need to look at 0 < x < 9 , there is a min
https://www.wolframalpha.com/input/?i=plot+y+%3D+2%E2%88%9A%28x%5E2+%2B+16%29+%2B+9+-+x