To complete the table for the population growth model, we follow the steps to find the value of the growth rate [tex]\( k \)[/tex]:
1. Given Data:
- Initial population in 2004, [tex]\( P_0 \)[/tex]: 53.8 million
- Projected population in 2022, [tex]\( P(t) \)[/tex]: 46.3 million
- Time period, [tex]\( t \)[/tex]: 2022 - 2004 = 18 years
2. Population Growth Model:
The population growth model can be represented by the equation:
[tex]\[
P(t) = P_0 \cdot e^{kt}
\][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population at year [tex]\( t \)[/tex]
- [tex]\( P_0 \)[/tex] is the initial population
- [tex]\( k \)[/tex] is the growth rate
- [tex]\( t \)[/tex] is the time in years
3. Rearranging the Formula:
To find [tex]\( k \)[/tex], we need to rearrange the formula:
[tex]\[
\frac{P(t)}{P_0} = e^{kt}
\][/tex]
Taking the natural logarithm of both sides:
[tex]\[
\ln\left(\frac{P(t)}{P_0}\right) = kt
\][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{\ln\left(\frac{P(t)}{P_0}\right)}{t}
\][/tex]
4. Substituting the Given Values:
[tex]\[
k = \frac{\ln\left(\frac{46.3}{53.8}\right)}{18}
\][/tex]
5. Calculation:
[tex]\[
\ln\left(\frac{46.3}{53.8}\right) \approx \ln(0.860415) \approx -0.150130
\][/tex]
[tex]\[
k = \frac{-0.150130}{18} \approx -0.008340639
\][/tex]
6. Rounding:
Rounding the value of [tex]\( k \)[/tex] to four decimal places, we get:
[tex]\[
k \approx -0.0083
\][/tex]
Therefore, the completed table looks like this:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline 2004 Population (millions) & Projected 2022 Population (millions) & \begin{tabular}{c} Projected Growth $R$ \\ $k$ \end{tabular} \\
\hline 53.8 & 46.3 & -0.0083 \\
\hline
\end{tabular}
\][/tex]
Hence, the growth rate [tex]\( k \)[/tex] rounded to four decimal places is [tex]\(-0.0083\)[/tex].
