Question

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 $a = 1183.1e^{0.008t}$ country c $a = 37.5e^{0.019t}$ country d $a = 125.7e^{-0.005t}$ country e $a = 141.5e^{-0.002t}$ 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.) \\(\bigcirc\\) a. country b and country c have the decreasing populations. the population of country b is decreasing by \\(\square\\)% and the population of country c is decreasing by \\(\square\\)% each year. \\(\bigcirc\\) b. country c and country d have the decreasing populations. the population of country c is decreasing by \\(\square\\)% and the population of country d is decreasing by \\(\square\\)% each year. \\(\bigcirc\\) c. country d and country e have the decreasing populations. the population of country d is decreasing by \\(\square\\)% and the population of country e is decreasing by \\(\square\\)% each year. \\(\bigcirc\\) d. 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 \\(\square\\)% each year.

Explanation:

Step1: Recall Exponential Decay Formula

The general form of an exponential model is \( A = A_0e^{kt} \). If \( k < 0 \), the population is decreasing. For a decreasing exponential model \( A = A_0e^{-rt} \) (where \( r>0 \)), we can rewrite it in the form \( A = A_0(1 - r)^t \) to find the percentage decrease. Using the property \( e^{-r} = 1 - r_{\text{decimal}} \), so \( r_{\text{decimal}}=1 - e^{-r} \), and the percentage decrease is \( r_{\text{decimal}}\times100\% \).

Step2: Analyze Country D

For Country D: \( A = 125.7e^{-0.005t} \). Here, \( r = 0.005 \). Calculate the percentage decrease:
First, find \( e^{-0.005} \approx 0.995012 \). Then, \( 1 - 0.995012 = 0.004988 \approx 0.005 \). So the percentage decrease is \( 0.005\times100\% = 0.5\% \).

Step3: Analyze Country E

For Country E: \( A = 141.5e^{-0.002t} \). Here, \( r = 0.002 \). Calculate the percentage decrease:
Find \( e^{-0.002} \approx 0.998002 \). Then, \( 1 - 0.998002 = 0.001998 \approx 0.002 \). So the percentage decrease is \( 0.002\times100\% = 0.2\% \).

Answer:

Country D and Country E have the decreasing populations. The population of Country D is decreasing by \( 0.5\% \) and the population of Country E is decreasing by \( 0.2\% \) each year. So the filled option is:

C. Country D and Country E have the decreasing populations. The population of Country D is decreasing by \( \boldsymbol{0.5}\% \) and the population of Country E is decreasing by \( \boldsymbol{0.2}\% \) each year.