Suppose that we have many towns spread across the country and we are trying to connect them with a network of roads. If we would like to do so by laying as little road as possible, how do we do it? In this blog post, we will use Calculus to tackle a special case of this optimization problem.

The Case of Four Towns Arranged in a Square

Let us suppose that we have four towns arranged so that they form the four vertices of a square. Our task is then to connect them by the shortest network of roads possible so that each town is still connected to each other town. To make our problem more concrete, let us label our four towns A, B, C, and D, and assume that they are at the vertices of a square with side length 1 each.

Our first guess for such a road network might be to just join the four towns by a road network consisting of the four edges of the square. Four edges at length 1 each gives a network of total length 4.

We then quickly notice that one of our roads is superfluous. If we remove any one of our four roads, we get a road system of length 3 that still connects our four cities.

We might assume that we are done now, but a little thinking outside of the box can get us a more efficient road system. The key insight here is that we could add extra junctions that aren’t cities. For example, something like this:
Reasoning Our Way to the Ideal Configuration of Roads

It turns out that a similar pattern of roads will be the most efficient way of connecting our four cities. To see why, we could imagine that we had an optimal configuration of roads. Clearly, this set of roads should never venture outside of our original square as that would be suboptimal. There must be a path connecting city A to city C, and a path connecting city B to city D. These paths must cross each other at least once but may do so many times. Let us call the first time that they cross point P and the last time that they cross point Q. Then, our network must have just consisted of the straight segments AP, BP, PQ, QC, and QD, as any other such network would have been longer.

We can also convince ourselves that an optimal road system should be vertically and horizontally symmetric, so our optimal network of roads must look something like this.
Most Efficient Road System Using Calculus

We suppose that the amount of inset of the center joining road from each side of the square is x. Then, our goal is to find the x that minimizes the total length of the road.

## Step 3:

Voila! We have successfully used calculus to build the most efficient road network connecting our four towns.

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