I've been studying a little bit of basic math with a friend of mine that has just started engineering school, and we were reading from the algebra and trigonometry book from Swokowski (12ve edition). On the fifth chapter, they introduce the law of Growth (and decay), which states that, if $q_{0}$ is the value of a quantity $q$ at the time $t=0$, and such quantity grows instantaneously at a rate proportional to its current value, then the amount of quantity at a time $t$ is:
$$ q(t) = q_{0}e^{rt} $$
Where r is the growth rate (in case of it being positive).
Then, my friend suggested an example he came up with. Imagine that there exists a group of 10 rabbits at a time $t=0$, that double their number each month, so, their growth rate would be 100%, which makes $r=1$. The model would then be:
$$ N(t)= 10e^{t}$$
But, this model does not hold at all, since at t=1, when the first month has passed, there should be 20 rabbits, but this model predicts around 27. And, it escalates even worse, when three months have passed (t=3), there should be 80 rabbits, but this model predicts 200.
Perhaps my approach to this problem is totally wrong, but I really fail to see where.
How did you come up with $r=1$?
Note that $$ q=10e^{rt}$$ along with $q(1)=20$ implies, $$20=10e^r$$ which implies $$ e^r=2 $$ that is $r=\ln 2$
The equation is $ q=10e^{t\ln 2}$ or $$ q(t)= 10(2^t)$$