Question

the exponential models describe the population of the indicated country, a, in millions, t years after 2006. which country has the greatest growth rate? by what percentage is the population of that country increasing each year?
country 1: a = 1096.6e^{0.016t}
country 2: a = 28.3e^{0.021t}
country 3: a = 134.3e^{0.008t}
country 4: a = 140.6e^{0.005t}
country has the greatest growth rate.

Explanation:

Step1: Recall Exponential Growth Formula

The exponential growth model is \( A = A_0 e^{rt} \), where \( r \) is the growth rate (in decimal). To find the percentage growth rate, we convert \( r \) to a percentage (\( r\times100\% \)). We need to identify \( r \) for each country from their \( A \) equations.

Step2: Identify \( r \) for Each Country

• Country 1: \( A = 1096.6e^{0.016t} \), so \( r_1 = 0.016 \). Percentage: \( 0.016\times100\% = 1.6\% \).
• Country 2: \( A = 28.3e^{0.021t} \), so \( r_2 = 0.021 \). Percentage: \( 0.021\times100\% = 2.1\% \).
• Country 3: \( A = 134.3e^{0.009t} \), so \( r_3 = 0.009 \). Percentage: \( 0.009\times100\% = 0.9\% \).
• Country 4: \( A = 140.6e^{0.005t} \), so \( r_4 = 0.005 \). Percentage: \( 0.005\times100\% = 0.5\% \).

Step3: Compare Growth Rates

Compare \( r_1 = 1.6\% \), \( r_2 = 2.1\% \), \( r_3 = 0.9\% \), \( r_4 = 0.5\% \). The largest \( r \) is for Country 2 (2.1%).

Answer:

Country 2 has the greatest growth rate, with a growth rate of \( 2.1\% \) per year.