You need to detect mines in an area having the shape of an equilateral triangle.
The working range of your detector equals half the triangle’s altitude.

Starting from a vertex of the triangle, what path shouid you follow to fully complete your mission while minimizing the distance you travel?

Lets coordinatize the triangle and make its sides length 1:
A=(0,0)
B=(√(3)/2,0)
C=(1,0)
The altitude is √(3)/2 so the range of the detector is √(3)/4.

An ideal path should head to the edge of a circle centered at B with radius √(3)/4 and then head towards C until it is within √(3)/4 of C.

It turns out the best path heads to the point on the circle where it intersects the altitude: (1/2, √(3)/4)

The distance to this point is √(7)/4 and the total distance is (2√(7)-√(3))/4 ≈ .88986

[I offer no proof that this is the best path, but it makes sense and Geometer's Sketchpad agrees with me.]

Posted by Jer on 2015-02-08 15:12:14