standard_description: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
problem_activity: Task
An important example of a model often used in biology or ecology to model
population growth is called the logistic growth model.  The general
form of the logistic equation is
$$
P(t) = \frac{KP_0e^{rt}}{K+P_0(e^{rt}-1)}.
$$
In this equation $t$ represents time, with $t = 0$ corresponding to when the population in question is first measured; $K,P_0$  and $r$ are all real numbers
with $K$ being called the ''carrying capacity'' while $r$ is a growth rate and
is normally a positive number.
Explain why the value $P_0$ represents
the  population when it is first measured.
Explain why, as time elapses, the population stabilizes, approaching the value $K$.
Explain how the behavior of $P$ changes if the growth rate $r$ is increased or decreased.
Below is the graph of a particular logistic function $P$, showing the growth of a
bacteria population.  Using the graph, identify $P_0$ and  $K$.
[IMG]
Using the values of $P_0$ and $K$ from the previous part, sketch the graph of the logistic function $Q$ given by
$$
Q(t) =  \frac{KP_0e^{2rt}}{K+P_0(e^{2rt}-1)}.
$$
Note that $Q$ is the same as $P$ except that the growth rate $r$ has been doubled.


aligns: no