So I had a question regarding bacteria growth where I was asked to find the growth rate of $500$ bacteria given this equation: $500\big(1 + (\frac{4t}{50+t^2})\big)$ where $t =$ time in hours and is equal to $2$.
I was supposed to use derivation to find the answer but I elected to use the continuous compounding function $A = Pe^{rt}$ where $A =$ final pop., $P =$ initial pop., $e$ is euler's number, $r =$ rate, and $t =$ time. I plugged $2$ into the original equation to get $574.074$. This should equal $A$. So I next did $574.074 = 500e^{2r}$, divided the $500$ out, took the natural $\log$ of both sides, and divided by $2$ to get finally $r$.
Now, my real questions: is the $r$ I found using $Pe^{rt}$ equal to the $r$ I would have found using derivation? If so why? How are the two functions equivalent? How can I transform the derivation so that its similar and equal to the growth function?