Question

What is the exponential equation that models the population of the city Norfolk?Norfolk that had a population of 24, 100 in 2010.By the year 2014 the population grew to 29,015.(write your equation for p in terms of n-round your b value to 4 decimal places)Use your equation to predict the population in the year 2022The population of Norfolk in 2022 should beIn what year will the population reach 76,500 ?

Answer 1

Given:

Norfolk had a population of 24, 100 in 2010.

By the year 2014, the population grew to 29,015.

We will write the exponential equation that models the population of the city.

The general form of the equation will be as follows:

`p=a(b)^n`

Where: p is the population in terms of the number of years (n)

a is the initial population in the year 2010

n is the number of years from 2010

b is the rate of growth of the population

So, in 2010, n = 0, a = 24,100

And in 2014, n = 2014 - 2010 = 4, a = 24,100, p = 29,015

Substitute with the values of the year 2014 to find the value of (b)

`29015=24100\cdot b^4`

Solve the equation to find (b) as follows:

`\begin{gathered} (29015)/(24100)=b^4 \\ \\ \ln ((29015)/(24100))=4\ln b \\ \\ \ln b=(\ln ((29015)/(24100)))/(4)\approx0.0464 \\ \\ b=e^(0.0464)\approx1.047493612 \end{gathered}`

Rounding the value of (b) to 4 decimal places

So, the value of b = 1.0475

The equation of the population will be as follows:

`p=24,100(1.0475)^n`

Now, we will use the equation to predict the population in 2022.

So, n = 2022 - 2010 = 12

So, substitute with n = 12 into the equation of the population:

`p=24100(1.0475)^(12)\approx42,060`

So, the population in 2022 should be = 42,060

In what year will the population reach 76,500?

So, we will substitute with p = 76,500 and find the value of n

`\begin{gathered} 76500=24100(1.0475)^n \\ 1.0475^n=(76500)/(24100) \\ 1.0475^n=3.1743 \\ n\cdot\ln 1.0475=\ln 3.1743 \\ \\ n=(\ln 3.1743)/(\ln 1.0475)=24.89 \end{gathered}`

rounding to the nearest whole number n = 25

So, the year will be = 2010 + 25 = 2035

So, the answer will be the population reach 76,500 in 2035