Answer:
[tex]a) P(t)=2.40 \cdot 10^6 \cdot e^{-0.006 t}[/tex]
Step-by-step explanation:
(a)
With the given information remember that the exponential model for the population of the country is given by
[tex]P(t)=P_0 \cdot e^{kt}[/tex]
where [tex]P_0[/tex] is the initial population and [tex]k[/tex] is the proportionality constant for the problem. In our case we know that k=-0.006, but we don't know the value of [tex]P_0[/tex]. To find the initial population, we must take into account that we are letting t=0 correspond to the year 2000, hence t=7 correspnds to the year 2007, and we know that [tex]P(7)=2.3 \cdot 10^6[/tex]. Using the model, we get that
[tex]2.3\cdot 10^6 = P_0 \cdot e^{-0.006 \cdot 7}\\\\\Rightarrow P_0 = 2.3 \cdot 10^{6} \cdot e^{0.006 \cdot 7} \approx 2.398 \cdot 10^6 \approx 2.40 \cdot 10^6[/tex]
(b) To predict the population of the country in the year 2022, note that t=22 corresponds to this year. Hence, the population of the country in 2022 would be
[tex]P(22)=P_0 e^{-0.006 \cdot 7}=2.40 \cdot 10^6\cdot e^{-0.006 \cdot 7} \approx 2.10 \cdot 10^4[/tex]
arround 2.10 millions.
