Question

complete the table shown to the right for the population growth model for a certain country
2006 population (millions) 19.5
projected 2033 population (millions) 33.8
projected growth rate, k
x =
(round to four decimal places as needed.)

Explanation:

Step1: Recall Population Growth Formula

The formula for exponential population growth is $P = P_0 e^{kt}$, where $P$ is the population at time $t$, $P_0$ is the initial population, $k$ is the growth rate, and $t$ is the time elapsed.

Step2: Determine Time Elapsed ($t$)

The initial year is 2006, and the projected year is 2033. So $t = 2033 - 2006 = 27$ years.

Step3: Substitute Known Values

We know $P_0 = 19.5$ (million), $P = 33.8$ (million), and $t = 27$. Substitute into the formula:
$$33.8 = 19.5 e^{27k}$$

Step4: Solve for $k$

First, divide both sides by 19.5:
$$\frac{33.8}{19.5} = e^{27k}$$
Calculate $\frac{33.8}{19.5} \approx 1.7333$. So:
$$1.7333 = e^{27k}$$
Take the natural logarithm of both sides:
$$\ln(1.7333) = 27k$$
Calculate $\ln(1.7333) \approx 0.5506$. Then:
$$k = \frac{0.5506}{27} \approx 0.0204$$

Answer:

$k \approx 0.0204$