Question

The population (in millions) of a country in 2007 and the expected continuous annual rate of change k of the population are given. † Country 2007 Population k Latvia 2.3 − 0.006 (a) Find the exponential growth model P = Cekt for the population by letting t = 0 correspond to 2000. (Round numerical values to three decimal places.) P = (b) Use the model to predict the population of the country in 2022. (Round your answer to two decimal places.) million

Answer 1

`a) P(t)=2.40 \cdot 10^6 \cdot e^(-0.006 t)`

Step-by-step explanation:

(a)

With the given information remember that the exponential model for the population of the country is given by

`P(t)=P_0 \cdot e^(kt)`

where
`P_0` is the initial population and
`k` is the proportionality constant for the problem. In our case we know that k=-0.006, but we don't know the value of
`P_0`. 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
`P(7)=2.3 \cdot 10^6`. Using the model, we get that

`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`

(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

`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`

arround 2.10 millions.