College Algebra, Chapter 5, 5.5, Section 5.5, Problem 2

The function nt = $12e^{0.012t}$ models the number of Goldfish.
where:

t = measured in years

n(t) = measured in millions

a.) What is the relative rate of growth of the fish population?

b.) What will the fish population be after 5 years?

c.) After how many years will the number of fish reach 30 million?

d.) Sketch a graph of the fish population function $n(t)$.



a.) Recall the formula for growth rate

$n(t) = n_0 e^{rt}$

where

$n(t)$ = population at time $t$

$n_0$ = initial size of the population

$r$ = relative rate of growth

$t$ = time

By observation, the relative rate of growth $r = 0.012 \times 100 \% = 1.2 \%$

b.)


$
\begin{equation}
\begin{aligned}

\text{if } t =& 5 \text{ years, then}
\\
\\
n(5) =& 12 e^{0.012(5)}
\\
\\
=& 12.74

\end{aligned}
\end{equation}
$


After 5 years, the number of fish population will be $12.74$ million.

c.)


$
\begin{equation}
\begin{aligned}

\text{if } n(t) =& 30, \text{ then}
&&
\\
\\
30 =& 12 e^{0.012 (t)}
&& \text{Divide both sides by } 12
\\
\\
\frac{5}{2} =& e^{0.012 t}
&& \text{Take $\ln$ of each side}
\\
\\
\ln \left( \frac{5}{2} \right) =& 0.012 t
&& \text{Recall } \ln e = 1
\\
\\
t =& 76.36 \text{ years}
&& \text{Divide both sides by } 0.012


\end{aligned}
\end{equation}
$


d.)