Question

A city had a population of 23,800 in 2007 and a population of 26,700 in 2012. Find the exponential growth function for the city. Use t=0 to represent 2007. (Round k to five decimal places) Use the growth function to predict the population of the city in 2022. Round to the nearest hundred

Answer 1

Given city had a a population of 23,800 in 2007 and a population of 26,700 in 2012. At t=0 is for year 2007,

We take,

Exponential growth function as:

$N_0=N_0e^{kt}$

At t=0

$N_0=23,800$

At t = 5,

$N(5)=26700$

Now,

$\begin{gathered} \frac{26700}{23800}=\frac{23800e^{k\cdot5}}{23800} \\ \ln 1.12=\ln e^{k\cdot5} \\ \frac{\ln 1.12}{5}=\frac{k\cdot5}{5} \\ \frac{\ln 1.12}{5}=k \end{gathered}$

so,

$k=0.1133$

and

$N(t)=22300e^{0.1133t}$

b). At t=15,

$\begin{gathered} N(15)=23,800e^{0.1133(15)} \\ \text{ =23300}\times\text{5.}419 \\ =126273 \end{gathered}$

Hence, population of the city in 2022 will be 126273.