Calculas 1 Minimize qustion

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 (11,0) in the xy-plane, Springfield is at (0,3), and Shelbyville is at (0,−3). 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.

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

1)

f(x) = 2√(x² + 3²) + (11-x)

The first term represents the lengths of the two slanted branches and the second

term is the length of the horizontal branch.

2)

f'(x) = 2x/√(x² + 3²) - 1

2x/√(x² + 3²) - 1 = 0

x = ±√3

Several things to notice:

The critical points of f(x) are independent of the position of Centerville.

Even if Centerville were very far away, √3 would still be the optimal position of the split.

The position of Centerville would play a role if it were closer to the origin than √3.

In that case placing the split at √3 would not result in a Y shape, or in a optimal wiring.

The optimal wiring would not be Y-shaped, but V-shaped, which is not represented by f(x).

The position of Centerville also plays a role in choosing one of the two critical points.

We would chose -√3 if Centerville had a negative x-coordinate; but given that it has

positive x-coordinate, the optimal wiring corresponds to x = +√3.

3)

f(√3) = 2√((√3)² + 3²) + (11-√3) = 11 + 3√3

4)

The minimal length is 11 + 3√3