Question

complete the table shown to the right for the population growth model for a certain country.

k =
(round to four decimal places as needed.)

Explanation:

Step1: Determine the time \( t \)

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

Step2: Solve for \( k \)

First, divide both sides by \( 40.5 \): \( \frac{59.8}{40.5}=e^{16k} \).
\( \frac{59.8}{40.5}\approx1.476543 \), so \( 1.476543=e^{16k} \).
Take the natural logarithm of both sides: \( \ln(1.476543)=\ln(e^{16k}) \).
Since \( \ln(e^{x}) = x \), we have \( \ln(1.476543)=16k \).
\( \ln(1.476543)\approx0.3892 \).
Then \( k=\frac{\ln(1.476543)}{16} \).
\( k=\frac{0.3892}{16}\approx0.0243 \).

Answer:

\( 0.0243 \)