I proposed this problem to my calculus students. It turned out to be more interesting than I thought (thanks, Ignacio and Noema).
The government intends to build a superhighway without speed limit so, therefore, without curves. It will start from city A, and should pass as close as possible to cities B and C.
In order to solve it you should start by stating what you mean by “as close as possible”. An option is to minimize the sum of the distances. But then, you get the following funny configuration:
Cities B and C are at the same distance from A, and make up a right angle. Intuition dicatates that the best route for the highway would be the bisector. Let d(A,C)=d(A,B)=1. Then, the sum of the distances from the cities to the highway is . But there is a better highway! You can just break the symmetry between B and C and make the road pass through C exactly. Then, the sum of the distances is just… 1.
If this was a real highway, the politician in charge would tell us that symmetry is also worth. Citizens of B may riot with un asymmetric solution… So, now let us change our target function. Let us minimize the sum of the squares of the distances. In that case, the bisector gives a square total distance of 1, same as the asymmetric road… Can you explain it?
Which is the best choice? Of course, it depends on the reason for which you’re fighting with the problem. If it is just to pass an exam, any of them will do…:) [Of course, there are many other alternatives. A student minimized the sum of the squares of the distances from the points to the straight line in the y direction. This makes sense in some cases, e.g.: when you’re fitting experimental points to a line.]
So, choosing the right function to minimize is crucial in practice… Kadanoff once explained in a talk that the government of the city of London has ordered a huge global study of traffic, taking into account both public and private transport, energetic and economic issues, taxes and prices, eeeeeverything. They made an enormous computer program that was running for days and, finally, told them the answer, how to optimize traffic in London. They had to remove all traffic lights. Why???? The program had many things into account… also the fact that in street car accidents, it’s normally old people who die. And old people do not pay taxes, they receive their pension money from the government. So it was convenient to remove the traffic lights… So, you see: garbage-in, garbage-out. Yes, maths is a nice girlfriend, she gives you more than you put in the relationship… but she can do no miracles. If you’re stupid, she can’t fix that…