Avg Distance Truck Travels in Corn Field of Area n

[jaxnnux]

Picture this: Corn is planted in a large circle of area n. A truck drives from a negligibly small processing plant in the center of the field to collect the corn and bring it back to the plant.

What is the average distance that the truck carrying the corn would have to travel in this case?

 

[vsage]

Assuming they stop at one point, fill it up and move back to the plant, you can break up the field using a circular cut (centered on the processing plant) so that half the corn lies on the inside of the cut and half the corn lies on the outside. I think it's pretty straightforward to see that the radius of this cut is the average distance a truck has to travel to the corn.