Solved: Optimal locations Suppose n houses are located at

Chapter 12, Problem 69

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Optimal locations Suppose n houses are located at the distinct points 1x1, y12, 1x2, y22, c, 1xn, yn2. A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that n = 3 and the houses are located at 10, 02, 12, 02, and 11, 12. b. Find the optimal location of the substation in the case that n = 3 and the houses are located at distinct points 1x1, y12, 1x2, y22, and 1x3, y32. c. Find the optimal location of the substation in the general case of n houses located at distinct points 1x1, y12, 1x2, y22, c, 1xn, yn2. d. You might argue that the locations found in parts (a), (b), and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a). Then use a graphing utility to determine whether the optimal location is the same in the two cases. (Also see Exercise 77 about Steiners problem.)