If every century the human population doubles. What is the annual growth rate?
Use differential equations.
My try:
Let $P$ the human population.
We know that $\frac{dP}{dt} = 2P\quad$ then $\quad\frac{dP}{2P} = dt\quad$
If we integrate from both sides:
$\quad\int\frac{dP}{2P} = \int dt\quad$
$\quad\frac{In|P|}{2} = t + c$
But since the population is always positive we got:
$\quad\frac{In(P)}{2} = t + c$
$\quad In(P) = 2(t + c)$
$\quad P = e^{2(t + c)}$
We want to know the change rate of $P$ with respect to $t$ so let's differentiate from both sides
$\quad \frac{dP}{dt} = e^{2(t + c)}dt$
I don't know if I'm on the right track. Any hints?
Best Answer
The population growth follows exponential model, $$P(t)=P(0)e^{rt}$$
You know that $$2P(0) = p(0)e^{100r}$$ that is $$e^{100r}=2$$ which implies $$r=ln(2)/100\approx 0.0069$$