in 2012, the population of a city was 5.78 million. the exponential growth rate was 2.52% per year.\na) find…

in 2012, the population of a city was 5.78 million. the exponential growth rate was 2.52% per year.\na) find the exponential growth function.\nb) estimate the population of the city in 2018.\nc) when will the population of the city be 9 million?\nd) find the doubling time.\na) the exponential growth function is ( p(t)=5.78e^{0.0252t} ), where ( t ) is in terms of the number of years since 2\n(type exponential notation with positive exponents. do not simplify. use integers or decimals for any number\n\n\n\n\n\nb) the population of the city in 2018 is 6.7 million.\n(round to one decimal place as needed.)\n\nc) the population of the city will be 9 million in about ( square ) years after 2012.\n(round to one decimal place as needed.)

in 2012, the population of a city was 5.78 million. the exponential growth rate was 2.52% per year.\na) find the exponential growth function.\nb) estimate the population of the city in 2018.\nc) when will the population of the city be 9 million?\nd) find the doubling time.\na) the exponential growth function is ( p(t)=5.78e^{0.0252t} ), where ( t ) is in terms of the number of years since 2\n(type exponential notation with positive exponents. do not simplify. use integers or decimals for any number\n\n\n\n\n\nb) the population of the city in 2018 is 6.7 million.\n(round to one decimal place as needed.)\n\nc) the population of the city will be 9 million in about ( square ) years after 2012.\n(round to one decimal place as needed.)

Answer

Explanation:

Step1: Set up the equation

We know the growth function (P(t)=5.78e^{0.0252t}). We want to find (t) when (P(t) = 9). So we set up the equation (9=5.78e^{0.0252t}).

Step2: Solve for (t)

First, divide both sides by (5.78): (\frac{9}{5.78}=e^{0.0252t}). Then take the natural logarithm of both sides: (\ln(\frac{9}{5.78})=\ln(e^{0.0252t})). Since (\ln(e^{x}) = x), we have (\ln(\frac{9}{5.78})=0.0252t). Now, calculate (\ln(\frac{9}{5.78})\approx\ln(1.5571)\approx0.442). Then (t=\frac{0.442}{0.0252}).

Answer:

(t\approx17.5)