Any club in a town of 20,000 with a lack of industry needs to be able to draw members from surrounding towns, and that job is exponentially easier for a club located within 20 miles of a city of over 100,000 residents than it is for a club over 30 miles from such a city.
Not a reliable mathematical statement. Maybe with a base marginally greater than one. More likely linear until the mileage becomes a bit larger. The entire function would likely be piecewise.
Doubt it. I would hypothesize that with most local private clubs, plotting the number of members living at any X distance from the club along any given axis will generally follow a normal distribution, except where special causes (such as a large population center like the city of Dayton) exist in close enough proximity to the club to throw off that distribution.
Let's choose a simple North-South axis. I suspect that Piqua CC's membership generally reflected a normal distribution, as follows (simplified for brevity):
Living >20 miles south of the club: 5% of members
Living 10-20 miles south of the club: 15% of members
Living 0-10 miles south of the club: 35% of members
Living 0-10 miles north of the club: 30% of members
Living 10-20 miles north of the club: 12% of members
Living >20 miles north of the club: 3% of members
On that same North-South axis, I suspect that Troy CC's membership shows a more pronounced skewing to the south, as its closer proximity to the city of Dayton gives it a greater number of members from that city and region relative to Piqua
Living >20 miles south of the club: 5% of members
Living 10-20 miles south of the club: 25% of members
Living 0-10 miles south of the club: 40% of members
Living 0-10 miles north of the club: 20% of members
Living 10-20 miles north of the club: 7% of members
Living >20 miles north of the club: 3% of members
Of course, neither of us have any real data, so we're all full of horsepiss. But customer distribution within any given brick-and-mortar business tends to follow a normal distribution prone to being skewed by inconsistencies in general population distribution, and I see no reason why country clubs would be any different.