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) = \boxed{}, where t is in terms of the number of years since 2012 and P(t) is the population in millions.

(Type exponential notation with positive exponents. Do not simplify. Use integers or decimals for any numbers in the equation.)

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. The formula for exponential growth is: \[ P(t) = P_0 \cdot e^{rt} \] where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is time in years. Given: - \(P_0 = 6.08\) million - \(r = 0.0177\) The exponential growth function is: \[ P(t) = 6.08 \cdot e^{0.0177t} \] b) Estimate the population of the city in 2018. 2018 is 6 years after 2012, so \(t = 6\). \[ P(6) = 6.08 \cdot e^{0.0177 \times 6} \] Calculate: \[ P(6) = 6.08 \cdot e^{0.1062} \] \[ P(6) \approx 6.08 \cdot 1.1119 \approx 6.756512 \] The estimated population in 2018 is approximately 6.76 million. c) When will the population of the city be 8 million? Set \(P(t) = 8\) and solve for \(t\): \[ 8 = 6.08 \cdot e^{0.0177t} \] Divide both sides by 6.08: \[ \frac{8}{6.08} = e^{0.0177t} \] \[ 1.3158 = e^{0.0177t} \] Take the natural logarithm of both sides: \[ \ln(1.3158) = 0.0177t \] \[ 0.2757 = 0.0177t \] Solve for \(t\): \[ t = \frac{0.2757}{0.0177} \approx 15.57 \] The population will be 8 million in approximately 16 years after 2012, so around 2028. d) Find the doubling time. The formula for doubling time is: \[ t_d = \frac{\ln(2)}{r} \] \[ t_d = \frac{\ln(2)}{0.0177} \] \[ t_d \approx \frac{0.6931}{0.0177} \approx 39.16 \] The doubling time is approximately 39 years.