To complete the table for the population growth model of a certain country, we need to determine the projected growth rate, [tex]\( k \)[/tex], for the population.
Given data:
- Population in 2004, [tex]\( P_0 = 19.9 \)[/tex] million
- Projected population in 2025, [tex]\( P(t) = 44.2 \)[/tex] million
- Number of years between 2004 and 2025, [tex]\( t = 2025 - 2004 = 21 \)[/tex] years
The population growth model can be expressed as:
[tex]\[ P(t) = P_0 \cdot e^{kt} \][/tex]
To find the growth rate [tex]\( k \)[/tex], we rearrange the equation to solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{1}{t} \ln \left( \frac{P(t)}{P_0} \right) \][/tex]
Step-by-step solution:
1. Calculate the ratio [tex]\(\frac{P(t)}{P_0}\)[/tex]:
[tex]\[ \frac{P(t)}{P_0} = \frac{44.2}{19.9} \][/tex]
2. Take the natural logarithm (ln) of the ratio:
[tex]\[ \ln \left( \frac{44.2}{19.9} \right) \][/tex]
3. Divide the result by the number of years [tex]\( t \)[/tex]:
[tex]\[ k = \frac{1}{21} \ln \left( \frac{44.2}{19.9} \right) \][/tex]
After performing these calculations, the projected growth rate [tex]\( k \)[/tex] is found to be approximately:
[tex]\[ k = 0.038 \][/tex]
Therefore, the table can be completed as follows:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
2004 Population (millions) & Projected 2025 Population (millions) & \begin{tabular}{c}
Projected Growth Rate, \\
k
\end{tabular} \\
\hline
19.9 & 44.2 & 0.038 \\
\hline
\end{tabular}
\][/tex]
So, the projected growth rate [tex]\( k \)[/tex] is [tex]\( 0.038 \)[/tex], rounded to four decimal places as needed.
