Monday, March 27, 2017

An Efficient Connector

Airport authorities would like to build a connector road (dashed line) so that maintenance vehicles can drive from one runway to the other without having to go all the way back to the airport terminal.

Given that the connector will be vertical, how far from the terminal should it be?

In my approach to the problem, I neglect the startup cost of building the connector road. My goal instead is to minimize the average distance that a maintenance vehicle would have to travel in order to get from the top runway to the bottom runway.

For example, with reference to the next figure, suppose that the vehicle is initially on the top runway, at the point indicated by the gray circle. To get to the indicated point on the bottom runway, the vehicle must drive along the magenta path to the other gray circle.

One way to begin the problem is to create a formula for the distance between a given initial location and a given final location. Then, one can use integration to average the distances over all initial and final locations. Because the average will depend on the location of the connector, the possibility arises of choosing the location of the connector that minimizes the average.

I did the problem for the dimensions shown:

In case anyone wants to solve it for themselves, I'll post my answer at a later time.

If you don't know calculus, you can try using your intuition to place the connector in an efficient location. Do you think that the most efficient connector will be halfway across, or less than halfway, or more than halfway?

Follow-up questions: Was it safe to assume that the most efficient connector is vertical? What if the runways are asymmetrical, as in this configuration?

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