# 1. Optimal locations Suppose n houses are located at the distinct points 1 x 1 , y 1 2 , 1 x 2 , y 2

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1. Optimal locations Suppose
n
houses
are located at the distinct points 1×1, y12, 1×2, 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 1×1, y12, 1×2, y22, and
1×3, y32.

c. Find the optimal
location of the substation in the
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1. Optimal locations Suppose
n
houses
are located at the distinct points 1×1, y12, 1×2, 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 1×1, y12, 1×2, y22, and
1×3, y32.

c. Find the optimal
location of the substation in the general case of n houses
located at distinct points 1×1, y12, 1×2, 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 81 about Steiners problem.)

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