complete the table shown to the right for the population growth model for a certain country.\n2005…

complete the table shown to the right for the population growth model for a certain country.\n2005 population (millions) projected 2021 population (millions) projected growth rate, k\n40.5 59.8\nk = □\n(round to four decimal places as needed.)

complete the table shown to the right for the population growth model for a certain country.\n2005 population (millions) projected 2021 population (millions) projected growth rate, k\n40.5 59.8\nk = □\n(round to four decimal places as needed.)

Answer

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)