The population of a certain community is known to increase at a rate proportional to the number of people present at time $t$. Suppose it is known that the population of the community is $10,000$ after $3$ years. What was the initial population? What will be the population in $10$ years? How fast is the population growing at $t = 30$?
Here is what I have so far:
$P'(t)$ prop $P(t)$ therefore $P'(t) = kP(t)$
$P'(t)-kP(t) = 0$ and $\mu = e^{-kt}$
$P(t) = Ce^{kt}$
Plugging in $P(3) = 10,000$
$10000 = Ce^{3k}$
Therefore $C = 10,000e^{-3k}$
Thus $P(t) = 10,000e^{-3k}*e^{kt}$
How do I solve for $k$? It seems like this question is lacking some additional information.... $P_3 = 10,0000$ $P_0 = ?$
I really have no idea how to get here? Most everything else says the population increases at a rate $5$ times $P_0$ or something similar. Not this one.
Then there is $P(t) = P_0e^{rt}$ but $P_0$ must $= 0$ and that is not given. Or would I use this equation with $10,000 = P_0e^{3r}$ but that again leads me to I still do not have $r$.
****About 4 hours before this assignment was due the prof added the addendum: the population is 2,000 after 1 year; that is P(1) = 2,000.