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} (26700)/(23800)=(23800e^(k\cdot5))/(23800) \\ \ln 1.12=\ln e^(k\cdot5) \\ (\ln 1.12)/(5)=(k\cdot5)/(5) \\ (\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}*\text{5.}419 \\ =126273 \end{gathered}`

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