Question: In 2012, the population of a city was 6.08 million. The exponential growth rate was 1.77% per year…

In 2012, the population of a city was 6.08 million. The exponential growth rate was 1.77% per year.

a) Find the exponential growth function.

b) Estimate the population of the city in 2018.

c) When will the population of the city be 8 million?

d) Find the doubling time.

a) The exponential growth function is P(t) = 6.08 * e^(0.0177t), where t is in terms of the number of years since 2012 and P(t) is the population in millions.

b) The population of the city in 2018 is 6.8 million.

(Round to one decimal place as needed.)

c) The population of the city will be 8 million in about years after 2012.

(Round to one decimal place as needed.)

Solution

In 2012, the population of a city was 6.08 million. The exponential growth rate was 1.77% per year. a) Find the exponential growth function. To find the exponential growth function, we use the formula: \[ P(t) = P_0 \cdot e^{rt} \] where \( P_0 = 6.08 \) million is the initial population, \( r = 0.0177 \) is the growth rate, and \( t \) is the time in years since 2012. The function is: \[ P(t) = 6.08 \cdot e^{0.0177t} \] b) Estimate the population of the city in 2018. For 2018, \( t = 2018 - 2012 = 6 \). Substitute \( t = 6 \) into the growth function to find the population: \[ P(6) = 6.08 \cdot e^{0.0177 \times 6} \] \[ P(6) = 6.08 \cdot e^{0.1062} \] Calculate \( e^{0.1062} \): \[ e^{0.1062} \approx 1.1119 \] \[ P(6) = 6.08 \cdot 1.1119 \approx 6.76 \] So, the population in 2018 is approximately 6.8 million. c) When will the population of the city be 8 million? We solve the equation \( P(t) = 8 \) for \( t \). \[ 8 = 6.08 \cdot e^{0.0177t} \] Divide both sides by 6.08: \[ \frac{8}{6.08} = e^{0.0177t} \] \[ 1.3164 = e^{0.0177t} \] Take the natural logarithm of both sides: \[ \ln(1.3164) = 0.0177t \] \[ 0.274 = 0.0177t \] Solve for \( t \): \[ t = \frac{0.274}{0.0177} \approx 15.48 \] So, the population will be 8 million approximately 15.5 years after 2012. d) Find the doubling time. Doubling time can be calculated using the formula: \[ T = \frac{\ln(2)}{r} \] \[ T = \frac{\ln(2)}{0.0177} \] Calculate \( \ln(2) \): \[ \ln(2) \approx 0.693 \] \[ T = \frac{0.693}{0.0177} \approx 39.16 \] The doubling time is approximately 39.2 years.