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 (12, 0) in the zy-plane, Springfield is at (0,4), and Shelbyville is at (0, -4). The cable runs from Centerville to some point (z, 0) on the x-axis where it splits into two branches going to Springfield and Shelbyville. Find the location (z, 0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer. To solve this problem we need to minimize the following function of 2: f(x) = We find that f(x) has a critical number at x = To verify that f(x) has a minimum at this critical number we compute the second derivative f''(z) and find that its value at the critical number is , a positive number. Thus the minimum length of cable needed is
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 (12, 0) in the zy-plane, Springfield is at (0,4), and Shelbyville is at (0, -4). The cable runs from Centerville to some point (z, 0) on the x-axis where it splits into two branches going to Springfield and Shelbyville. Find the location (z, 0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer. To solve this problem we need to minimize the following function of 2: f(x) = We find that f(x) has a critical number at x = To verify that f(x) has a minimum at this critical number we compute the second derivative f''(z) and find that its value at the critical number is , a positive number. Thus the minimum length of cable needed is
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1a)
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 (12, 0) in the xy-plane, Springfield is at (0, 4), and
Shelbyville is at (0,4). The cable runs from Centerville to some point (z, 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.
To solve this problem we need to minimize the following function of a:
f(x) =
We find that f(x) has a critical number at a
To verify that f(x) has a minimum at this critical number we compute the second
derivative f''(z) and find that its value at the critical number is
, a positive number.
Thus the minimum length of cable needed is
1b) Evaluate the limit using L'Hospital's rule
8xe7
8x
lim
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