Problem

Population growth is often modeled by the equation

where C is the carrying capacity of the population, and k and A are constants controlling details about the population. For example, in an environment which has a carrying capacity of 1000 fish, the population equation might be

Find

Solution

The graph of P(t) looks like this.

Using the rules of exponents, we can rewrite P(t) as

Now because e³ is less than 1, as t get very large this term will get very close to zero. Thus

Therefore as time continues, the population will tend toward its carrying capacity of 1000.