Question

solve the problem related to population growth.A city had a population of 23,900 in 2007 and a population of 25,100 in 2012.(a) Find the exponential growth function for the city. Use t=0 to represent 2007. Round k to the five decimal places.Use the growth function to predict the population of the city in 2022. Round to the nearest hundred

Answer 1

a)

To find the exponential growth function, we apply the exponential growth formula which is

`N(t)=a(1+k)^t`

Where

`\begin{gathered} a\text{ is the }initial\text{ population} \\ k\text{ is the growth rate} \\ t\text{ is the number of time intervals} \end{gathered}`

Given that

`\begin{gathered} a=23,900 \\ N(5)=25,100 \\ t=2012-2007=5 \end{gathered}`

Substitute the variables into the exponential growth formula

`\begin{gathered} 25100=23900(1+k)^5 \\ \text{Divide both sides by 23,900} \\ (25100)/(23900)=(23900(1+k)^5)/(23900) \\ 1.05021=(1+k)^5^{} \\ \sqrt[5]{1.05021}=1+k \\ 1.00984=1+k \\ \text{Collect like terms} \\ k=1.00984-1 \\ k=0.00984 \end{gathered}`

Hence, the exponential growth function is

`\begin{gathered} N(t)=23900(1+0.00984)^t_{} \\ N(t)=23900(1.00984)^t \end{gathered}`

Hence, the exponential growth function is

`N(t)=23900(1.00984)^t`

b)

For the population of the city in 2022,

`\begin{gathered} t=2022-2007=15 \\ t=15 \end{gathered}`

Substitute for t into the exponential growth function

`\begin{gathered} N(t)=23900(1.00984)^t \\ N(15)=23900(1.00984)^(15) \\ N(15)=27700\text{ (nearest hundred)} \end{gathered}`

Hence, the population is 2022 is 27700 (nearest hundred)