Suppose there are 10,000 civilizations in the milky way galaxy. if the civilizations were randomly distributed throughout the disk of the galaxy, about how far (on average) would it be to the nearest civilization? (hint: start by finding the area of the milky way's disk, assuming that it is circular and 100,000 light-years in diameter. then find the average area per civilization, and use the distance across this area to estimate the distance between civilizations.)

The shape of the galaxy is a circular, with diameter 100, 000 lights year, 
so the radius is 50, 000 lights year, thus

 Area of the galaxy is  (lights year square)

The area per civilisation is 

from this result we can assume as a circle of Radius= lights year.

and assume that each civilisation lives in a planet at the center of these smaller circles.

Check the figure attached.

Planet A, and B are closest in distance. They are each at the center of a circle with radius 2500, and they are tangent. So the distance of A to B is 2500+2500=5000 (light years).

5000 light years

Hope i can you!