Roger Burton West wrote:
If anyone else would like to check my maths, please feel free!
I think I would set up the math slightly differently, but I get similar
results. Mostly, I would measure D in sites per unit area, and thus I
wind up with a factor of 2pi in the equations.
Let's assume D is the total sites per unit area, x is the production per
site, y is the cost per unit distance, and r is distance from the center.
The total from any given site is equal to x-ry.
Now, if you look at all sites within a given radius R, the formula for
the total number of sites is equal to 2pi * D * integral (0-R) r, or pi
* D * R^2. The formula for the total energy, however, is equal to 2pi *
D * integral(0-R) (rx - r^2y) = 2pi * D * (1/2R^2x - 1/3R^3y).
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