Expert: Scott A Wilson - 2/27/2011

Question
P(t)=c/((1-P_0/c) e^(-rt) )
Where
P_0=Initial Population  (271.31)
c = carrying capacity
P = Population (Final Population 1173.11)
t = time (110)
r = growth rate (0.0133)


I need to find what c is equal to. This is a very urgent request and im scraping at straws to try and figure it out.. Please help!

Answer
To make it look simpler, I'll rename P(t) as x, P_0 as y, and e^(-rt) as z
As I read it, the equation is x = c/((1 - y/c)z).

The 1st thing I'll do is get rid of the fractions involved, which involves a couple of steps.
Multiplying the equation by ((1 - y/c)z) gives x((1 - y/c)z) = c, which can be multilplied out xz - xzy/c = c.  To get rid of the last fraction, multiply all terms by c.  This gives
xzc - xzy = c².

Subtracting xzc - xyz from both sides gives the quadratic 0 = c² - xzc + xyz.

We both know the solution to a quadratic is c = (xz±√[(xz)²-4xyz])/2.
This can be rewritten as c = (xz±√[xz(xz-4y)])/2.

Either way, x, y, and z can be put back in.