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The exponential models describe the population of the indicated country, $A$, in millions, $t$ years after 2010. Which countries have a decreasing population? By what percentage is the population of these countries decreasing each year?
Country B $\quad A=1183.1 e^{0.006 t}$
Country $\mathrm{C} \quad \mathrm{A}=37.5 e^{0.015 t}$
Country D $A=123.3 e^{-0.0051}$
Country E $A=141.7 e^{-0.002 t}$
Select the correct choice below and fill in the answer boxes to complete your choice.
(Simplify your answers. Type integers or decimals rounded to the nearest tenth as needed.)
A. Country B and Country D have the decreasing populations. The population of Country B is decreasing by $\square$ and the population of Country D is decreasing by $\%$ each year.
B. Country $\mathrm{C}$ and Country $\mathrm{D}$ have the decreasing populations. The population of Country $\mathrm{C}$ is decreasing by $\%$ and the population of Country $\mathrm{D}$ is decreasing by $\%$ each year.
C. Country $\mathrm{D}$ and Country $\mathrm{E}$ have the decreasing populations. The population of Country D is decreasing by $\%$ and the population of Country $\mathrm{E}$ is decreasing by $\%$ each year.
D. Country $\mathrm{B}$ and Country $\mathrm{C}$ have the decreasing populations. The population of Country $\mathrm{B}$ is decreasing by $\%$ and the population of Country $\mathrm{C}$ is decreasing by $\%$ each year.
Answer
Final Answer: The correct choice is C. Country D and Country E have the decreasing populations. The population of Country D is decreasing by \(\boxed{0.51\%}\) each year and the population of Country E is decreasing by \(\boxed{0.2\%}\) each year.
Step 1 ：The exponential models given in the question are of the form \(A = P e^{rt}\), where \(P\) is the initial population, \(r\) is the rate of growth or decay, and \(t\) is the time in years. If \(r\) is positive, the population is increasing; if \(r\) is negative, the population is decreasing. The rate of decrease is given by \(-r\) as a percentage. So, we need to identify the countries with negative \(r\) values and calculate the percentage decrease for each.
Step 2 ：The rates for each country are as follows: Country B: 0.006, Country C: 0.015, Country D: -0.0051, Country E: -0.002.
Step 3 ：The countries with decreasing populations are those with negative rates. These are Country D and Country E.
Step 4 ：The rate of decrease for each country is given by \(-r\) as a percentage. Therefore, the population of Country D is decreasing by 0.51% each year and the population of Country E is decreasing by 0.2% each year.
Step 5 ：Final Answer: The correct choice is C. Country D and Country E have the decreasing populations. The population of Country D is decreasing by \(\boxed{0.51\%}\) each year and the population of Country E is decreasing by \(\boxed{0.2\%}\) each year.
