Answer
a. $\quad n(t)=n_{0}e^{\frac{t\cdot\ln 2}{a}}$
b. $\quad n(t)=n_{0}e^{rt}$
Work Step by Step
Exponential Growth Model (p. 373):
A population experiences exponential growth if it can be modeled by the exponential function
$n(t)=n_{0}e^{rt}$
where $n(t)$ is the population at time $t,$
$n_{0}$ is the initial population (at time $t=0$),
and $r$ is the relative growth rate
(expressed as a proportion of the population).
-----------
a.
If we know that the initial population doubles for time a,
$n(a)=2n_{0}=n_{0}e^{ra},$
we find the growth rate (solve for r):
$2n_{0}=n_{0}e^{ra}\quad/\div n_{0}$
$2=e^{ra}\qquad/$ ... apply ln( ) to both sides
$\ln 2=ra\quad/\div a$
$r=\displaystyle \frac{\ln 2}{a}$
so, the model is
$n(t)=n_{0}e^{\frac{t\cdot\ln 2}{a}}$
b.
In terms of r, $n(t)=n_{0}e^{rt}$
