Exam 2 bonus (+10%)
(from P. Yasskin and A. Belmonte)
Upon moving to a new city, you want to find an apartment which is
conveniently located relative to your school (point S in 2-dimensional space),
your place of work (point W ) and the shopping mall (point M ). You want to
minimize the sum of distances from your apartment A = (x, y) to the above
~ + |AW
~ | + |AM
~ |. Find and prove the geometric
three points: f (x, y) = |AS|
~ AW
~ and AM
~ which characterizes the
condition on the three vectors AS,
minimum of f . You may assume that A, W and M form an acute triangle.
Hints:
~ expressing your answer in terms of the
1. Compute the gradient of |AS|
~ In particular, how are their directions related and what can
vector AS.
you say about the magnitude of the gradient?
2. Write the condition for the critical point in the vector notation.
~
3. The final answer is a condition on the angles between the vectors AS,
~ and AM
~ .
AW
4. (hard, +10% extra) Discuss how the answer would change if one of
the angles in the triangle SW M were large (how large exactly?) and
why does the above proof fail (why differentiation fails to locate the
critical point).
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