Calculus Applied Optimization Question

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 (9,0) in the xy-plane, Springfield is at (0,5), and Shelbyville is at (0,-5). 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.

Part 1: To solve this problem we need to minimize the following function of x in the format f(x) =

Part 2: Find the critical number of f(x)

Part 3: To verify that f(x) has a minimum at this critical number, compute the second derivative f′′(x) and find that its value at the critical number is  __, a positive number.

Part 4: The minimum length of cable needed is __.

Answer 1

Let x = distance from (0,0) along the x-axis to the point, P, where the cables to Shelbyville and Springfield meet. Then, 9 - x = length of cable from P to Centerville.

By the Pythagorean Theorem, the length of the cable from P to Shelbyville (which is the same as the length of the cable from P to Springfield) is √(x²+25).

Total length of cable = f(x) = (9 - x) + 2√(x²+25), where 0 < x < 9.

f'(x) = -1 + 2x / √(x²+ 25) = [-√(x²+25) + 2x)] / √(x²+25)

f'(x) = 0 when √(x²+25) = 2x.

x² + 25 = 4x² So, x = 5 / √3 ≈ 2.89

Show that f is minimized when x = 5 / √3.

Answer 2

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