Tunnel Vision

In the traditional trigonometry course, no problem is more time-honored than the tunnel problem. What is it? A competent authority is trying to build a road, or a railroad, and some mountains are in the way. The decision has been made to build a tunnel, and the crews who are ready to start drilling at each end need to know the precise direction they should drill so that they will meet in the middle. In the classic problem, a surveyor is placed at some point A from which both ends B and C of the projected tunnel are visible. The surveyor then measures the distances AB and AC and the angle BAC. These measurements completely determine the triangle ABC, and therefore the measures of the angles BCA and CBA. These angles represent the directions in which the drilling teams should proceed.

In the past, it has been my habit to make fun of this problem. The pictures in the book always seem to be drawn in such a way that there is a nice plane region in front of the mountains where I can place my surveyor. In that case, why in the world are we building a tunnel? Let's just go around the mountains! On the present occasion, however, I shall ignore this perhaps questionable assumption of the model, and look at a question which is not usually asked: If I do have a choice in the location of A, where should I put it? What would be my goal, my optimization criterion in selecting the site for the surveyor?