Question

Hello it’s one question with 4 parts can you help me pleased. What is the average rate of growth between 2000 and 2015?

Answer 1

The future size of the city's population is modeled by the equation:

`P\left(t\right)=54000e^(0.018t)`

Where "t" represents the number of years after the initial year 2000 and 54,000 is the population size during the initial year.

a) To determine if the exponential function is decreasing or increasing, you have to look at the coefficient of the exponent:

`P\left(t\right)=a*e^(kt)`

If k is positive, the function is increasing its value.

If k is negative, the function is decreasing its value.

In this case, k= 0.018, the coefficient is positive, which means that the function is increasing.

b. You have to determine the size of the population by the year 2002, to calculate this value, the first step is to determine the value of t that corresponds to the year 2002

The initial value t=0 corresponds to the year 2000.

The variable "t" is measured in years, so each unit increase of "t" will represent one year, so that:

The year 2001 corresponds to t=1

The year 2002 corresponds to t=2

To calculate the size of the population you have to use t=2:

`\begin{gathered} P\left(t\right)=54000e^(0.018t) \\ P\left(2\right)=54000e^(0.018*2) \\ P\left(2\right)=54000e^(0.036) \\ P\left(2\right)=55979.41\cong55979 \end{gathered}`

By 2002 the population will be 55,979.

c. To determine the population by the year 2015, you have to use the value t= 15

`\begin{gathered} P\left(t\right)=54000e^(0.018t) \\ P\left(15\right)=54000e^(0.018*15) \\ P\left(15\right)=54000e^(0.27) \\ P\left(15\right)=70738.08\cong70738 \end{gathered}`

By 2015 the population will be 70,738.

d. To calculate the average growth rate between 2000 and 2015 you have to apply the following formula.

`(f\mleft(b\mright)-f\mleft(a\mright))/(b-a)`

Where (b,f(b)) and (a,f(a)) are the ordered pairs of the interval.

For the year 2015 you have to use the values obtained in part c (15,70738).

For the year 2000 you have to use the initial values of the exponential function (0,54000)

`\begin{gathered} (P\left(15\right)-P\left(0\right))/(15-0) \\ (70738-54000)/(15) \\ (16738)/(15)=1115.86\cong1116 \end{gathered}`

The average growth year between 2000 and 2015 is 1,116.