A,B,C and D are four cities such that ABCD has the precise shape of a square with each side equal to 100 kilometers.

A railway network needs to be constructed connecting all the cities. Determine the minimum total length of this network. Prove that this is indeed the minimum network length.

Actually, there is a trade-off.

The simplest case is to connect the sides of the square. This gives a rail length of 400 and a round trip distance of 400, too.

We could improve this by connecting the corners of the square instead. This would give a rail length of around 282 miles but a round trip distance of 565 miles.

Last but not least, bisecting the square and angling in the side rails to take advantage of the central portion of the bisecting line seems to work best when the side rails are angled between the 2 pairs of adjacent towns at 120 degrees. This results in around 273 miles of rails (reduces construction costs by 32%) but an around 546 mile round trip (increases overhead by 36%).

 

Edited on April 5, 2013, 3:27 pm

Posted by broll on 2013-04-05 15:21:22