Answer
a. $n(t)=10\cdot 2^{t/2}$
b. about 1.9 million
c. after about 20 hours
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. Given: n(0)=10 and n(1.5)=20 (double after 1.5 hours)
$20=10\cdot 2^{2/a}$
... solve for a, start by dividing both sides with 10...
$ 2=2^{2/a}\qquad$ ...apply $\log_{2}$ to both sides
$1=\displaystyle \frac{2}{a}\qquad /\times a$
$a=2$
The model is $n(t)=10\cdot 2^{t/2}$
b. t=35,
$ n(t)=10\cdot 2^{35/2}\approx$1,853,638.00047
about 1.9 million
c. Solve for t when n(t)=10,000
$10000=10\cdot 2^{t/2}\qquad /\div 10$
$ 1000=2^{t/2}\qquad$ ...apply $\log()$ to both sides
$3=\displaystyle \frac{t}{2}\log 2\qquad/\times\frac{2}{\log 2}$
$ t=\displaystyle \frac{6}{\log 2}\approx$19.9315685693
after about 20 hours
