Question

complete the table shown to the right for the population growth model for a certain country.
2005 population (millions) projected 2021 population (millions) projected growth rate, k
40.5 59.8
k = □
(round to four decimal places as needed.)

Explanation:

Step1: Determine the time \( t \)

The year 2005 is the initial year (\( t = 0 \)), and for 2021, \( t=2021 - 2005=16 \) years.
The population growth model is \( P(t)=P_0e^{kt} \), where \( P_0 = 40.5 \) (population in 2005) and \( P(16)=59.8 \) (population in 2021).
Substitute into the formula: \( 59.8 = 40.5e^{16k} \).

Step2: Solve for \( k \)

First, divide both sides by \( 40.5 \):
\(\frac{59.8}{40.5}=e^{16k}\).
\(1.476543 = e^{16k}\).
Take the natural logarithm of both sides: \( \ln(1.476543)=\ln(e^{16k})\).
Since \( \ln(e^{x})=x \), we have \( \ln(1.476543)=16k \).
We know that \( \ln(1.476543)\approx0.3892 \).
Then \( k=\frac{\ln(1.476543)}{16} \).
\(k=\frac{0.3892}{16}\).
\(k = 0.0243\).

Answer:

\(0.0243\)