Talk:Cost as a Function of Land Pressure and Investment Count/@comment-98.225.44.66-20150925041706

A little exponential math. Using the facts that a^(b+c) = a^b * a^c, and a^(b*c) = (a^b)^c.

2^((R+I)*I) = 2^(I*(R+I)) = (2^I)^(R+I)

Define A = 2^I.

(2^I)^(R+I) = A^(R+I) = A^(I)*A^(R).

There's a hidden dependency on N in that R. So let's make that explicit.

R = F/L (footprint over land).

R = (F0 + I*N)/L. F0 is footprint from all buildings other than the one being increased. I*N is footprint from the building you are increasing.

A^(R) = A^((F0 + I*N)/L) = A^((1/L)*(F0 + I*N) = (A^(1/L))^(F0 + I*N)

Define B = 2^(I/L) = (2^I)^(1/L) = = A^(1/L).

A^((1/L)*(F0 + I*N) = B^(F0 + I*N) = B^(F0)*B^(I*N) = B^(F0)*(B^I)^N.

Define D = 2^(I^2/L) = (2^(I/L))^I = B^I

B^(F0)*(B^I)^N = (B^F0)*D^N

To recap, this establishes that

2^((R+I)*I)= (B^F0)*(D^N)

So now we're ready to put this back into the original equation.

C(2^((R+I)I))(1+ 0.00001*N^2) = C*(B^F0)*(D^N)*(1 + 0.00001*N^2)

As long as we're wildly defining terms to make our lives easier, let's define C' = C*(B^F0), and k = 0.00001. Now our equation simplifies to

C' * D^N * (1 + k*N^2).

C', D, and k are all constants.

You are free to try to integrate this by hand. I just asked Mathematica (note: where it says C, it should be C').

http://integrals.wolfram.com/index.jsp?expr=c*%28d^x%29*%281+%2B+k*x^2%29&random=false

Good news, our integral has an analytic solution. It's a little messy to write out; I'll just call it f(x). Finding the cost of going from, say, 100 structures to 150 structures is just f(150) - f(100).