Population Growth Question I am not sure how I am supposed to do this, but here is the problem and my attempt.

A farmer wants to produce chickens for the community. Let $P(t)$ represent the population of chickens at time $t$ in months. Assume the population of chickens reproduce at a rate proportional to its size. The farmer cultivates a small number of chickens for a trial run. In three months, the population of chickens increased from 75 to 300. The farmer wants to begin with an initial population of $P(0)$ chickens, and after a year, have $8000 + P(0)$ chickens, so that 8000 can be harvested, and $P(0)$ saved to start breeding again for the following year. What initial population $P(0)$ of chickens does the farmer need to start with?

Attempt:
First, since $P(t)$ represents the population at time $t$ in months, then the population growth equation can be written as: $$P(t) = P_0e^{kt}$$
Then I tried to find the relative growth rate. Since in three months, the population went from 75 to 300, then that means $$300 = 75e^{3k}$$
Or in other words, $k = \frac{\ln{4}}{3}$.
I am not exactly sure how to proceed from here, or if I am on the right track. I would like some assistance, or advice.

Excellent so far!
You have $e^{3k}=4$ and you want $$Pe^{12k}=P+8000.$$
Note that $e^{12k}=4^4$.
Therefore  $256P=P+8000$ and $P=32$ are required.
