Let P(t) represent the number of wolves in a pop. at time t in years, when t is greater than or equal to 0. The pop. P(t) is increasing at a rate directly proportional to 800-P(t), where the constant of proportionality is k.

I just want to know if my setting up of the equation is correct.
800-P(t)=Ae^(kt).

You are probably a little ahead of the game. What you have proposed on the right hand side has something to do with the solution of the differential equation that you are attempting to set-up.

The problem statement is:
P(t)=number of wolves, P≥0
t=time in years
P(t) is increasing at a rate of k(800-P(t)), which means
d(P(t))/dt = k*(800-P(t)) ... (1)
where k>0
The solution of (1) will give an expression related to the right-hand side of your equation.

Your setup of the equation is almost correct, but there is a slight mistake. The correct equation to represent the population growth is:

P(t) = A * e^(kt)

Here's an explanation of the components of the equation:

- P(t): The number of wolves in the population at time t.
- A: The initial population size at t = 0. This is the value of P(0).
- e: The base of the natural logarithm, approximately equal to 2.71828.
- k: The constant of proportionality. It determines the rate at which the population grows. It should be a negative value in this case since the population is decreasing when P(t) is increasing.

In your equation, you substituted 800-P(t) for P(t), which is not necessary because P(t) represents the population itself. The equation should be written as P(t) = A * e^(kt), without any changes to P(t).

To determine the constant of proportionality, k, you would need additional information, such as the growth rate or the rate at which the population increases. From there, you could solve for k using given data or additional conditions related to the population growth.

No, your equation setup is incorrect.

The correct equation should be:

P'(t) = k(800 - P(t))

Where P'(t) represents the derivative of P(t) with respect to time t.

This equation represents the rate of change of the wolf population at time t, which is proportional to the difference between the carrying capacity (800) and the current population size (P(t)). The constant of proportionality is k.

It's important to note that the original problem states that the population is increasing, so the rate of change should be positive. If you use the negative exponential in your equation, it would imply a decreasing population.