Suppose \(n\) houses are located at the distinct points \(\left(x_{1},
y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) .\) 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 \((0,0),(2,0),\) and (1,1)
b. Find the optimal location of the substation in the case that \(n=3\) and the
houses are located at distinct points \(\left(x_{1}, y_{1}\right)\)
\(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\)
c. Find the optimal location of the substation in the general case of \(n\)
houses located at distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2},
y_{2}\right), \ldots\) \(\left(x_{n}, y_{n}\right)\)
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 75 about Steiner's problem.)
