Question

19. The table below shows the population of Florida from 2010 to 2019.YearPopulation (millions)201018.7201119.1201219.3201319.6201419.9201520.2201620.6201721.0201821.2201921.5(a) Use a graphing calculator to build a logistic regression model that best fits this data, letting t=0 in 2010. Round each coefficient to two decimal places.Pt = (b) What does this model predict that the population of Florida will be in 2030? Round your answer to one decimal place. million people(c) When does this model predict that Florida's population will reach 23 million? Give your answer as a calendar year (ex: 2010).During the year (d) According to this model, what is the carrying capacity for Florida's population? million people

Answer 1

The formula for the logistic regression model that best fits the data is,

`y_1=\frac{a}{1+b\cdot e^{t\cdot x_(1)}}`

The graph, tables and details of the population data will be shown below

a) The equation that best fits the regression model is,

`\begin{gathered} P_t=y_1 \\ t=x_1 \\ a=93.2861\approx93.29(2\text{ decimal places)} \\ b=3.98291\approx3.98(2\text{ decimal places)} \\ t=-0.0198742\approx-0.02(2\text{ decimal places)} \end{gathered}`

Substitutes the data above into the equation

`P_t=(93.29)/(1+3.98\cdot e^(-0.02t))`

Hence,

`P_t=(93.29)/(1+3.98\cdot e^(-0.02t))`

b) In the year 2030, t = 20

`\begin{gathered} P_(20)=(93.29)/(1+3.98\cdot e^(-0.02*20))=(93.29)/(1+3.98\cdot e^(-0.4))=(93.29)/(1+3.98*0.67032) \\ P_(20)=(93.29)/(1+2.6678736)=(93.29)/(3.6678736)=25.43435521\approx25.4(1\text{ decimal place)} \\ P_(20)=25.4million\text{ people} \end{gathered}`

Hence, the answer is

`P_(20)=25.4\text{million people}`

c) Given that

`\begin{gathered} _{}P_t=23\text{million people} \\ 23=(93.29)/(1+3.98\cdot e^(-0.02t)) \end{gathered}`

Multiply both sides by 1+3.98e^{-0.02t}

`\begin{gathered} 23(1+3.98e^(-0.02t))=1+3.98e^(-0.02t)*(93.29)/(1+3.98\cdot e^(-0.02t)) \\ (23(1+3.98e^(-0.02t)))/(23)=(93.29)/(23) \\ 1+3.98e^(-0.02t)=4.056087 \end{gathered}`

Subtract 1 from both sides

`\begin{gathered} 1+3.98e^(-0.02t)-1=4.056087-1 \\ 3.98e^(-0.02t)=3.056087 \end{gathered}`

Divide both sides by 3.98

`\begin{gathered} (3.98e^(-0.02t))/(3.98)=(3.056087)/(3.98) \\ e^(-0.02t)=0.767861055 \end{gathered}`

Apply exponent rule

`\begin{gathered} -0.02t=\ln 0.767861055 \\ -0.02t=-0.264146479 \end{gathered}`

Divide both sides by -0.02

`\begin{gathered} (-0.02t)/(-0.02)=(-0.264146479)/(-0.02) \\ t=13.20732\approx13(nearest\text{ whole number)} \\ t=13 \end{gathered}`

Hence, the population will reach 23million in the year 2023.

d) The carrying capacity for Florida's population is equal to the value of a.

`\begin{gathered} \text{where,} \\ a=93.29\text{ million people} \end{gathered}`

Hence, the carrying capacity fof Florida's population is

`93.29\text{million people}`