91影视

The following data represent the population of the United States. An ecologist is interested in building a model that describes the population of the United States.

Let 1900 be the initial year, i.e. $t=0$. Therefore, $1910$is after 10 years $\left(fort=10\right)$

$$\begin{gathered} X_{min}=0 \\ {X}_{max}=120 \\ {X}_{scl}=10 \\ {Y}_{min}=50 \\ {Y}_{max}=310 \\ {Y}_{scl}=20 \end{gathered}$$

The logistic model is of the form

$y=\frac{c}{1+a{e}^{-bx}}$

Using the calculator and given values in the table we need to find the value of constant$a,b,c$

How do we draw $y=\frac{762,176,844.4}{1+8.7248e^{-00162x}}$graphing using a calculator?

Press $\mathit{Y}\mathbf{=}$

Write $Y_1=\frac{762,176,844.4}{1+8.7248e^{-00162x}}$

Press GRAPH

The logistic model is given by $y=\frac{c}{1+a{e}^{-bt}}$

where $c$represent carrying capacity.

Looking at the logistic model of the United States

$y=\frac{762,176,844.4}{1+8.7248e^{-0.0162x}}$

We conclude the carrying capacity is$c=762,176,844.4$

in 2012, $t=112$years passed since 1900.

We need to calculate $y\left(112\right).$

$$\begin{gathered} y\left(x\right)=\frac{762,144,844.4}{1+8.7248e^{-0.0162x}} \\ y\left(112\right)=\frac{762,144,844.4}{1+8.7248e^{-0.0162路112}} \\ y\left(112\right)=314,362,768 \end{gathered}$$

if $y\left(x\right)=350,000,000$

$$\begin{gathered} 350,000,000=\frac{762,176,844.4}{1+8.7248e^{-0.0162x}} \\ 1+8.7248e^{-0.0162x}=\frac{762,176,844.4}{350,000,000} \\ 8.7248e^{-0.0162x}=\frac{762,176,844.4}{350,000,000}-1 \\ 8.7248e^{-0.0162x}=\frac{412,176,844.4}{350,000,000脳8.7248} \\ {e}^{-0.0162x}=0.134699 \\ -0.0162x=\ln \left(0.134699\right) \\ x=\frac{\ln \left(0.134699\right)}{-0.0162} \\ x=123.75years \end{gathered}$$

The census is conducted every 10 years. Therefore, from the table we can observe only the data for 2010 and 2020.

We know that in 2010, there were 308,745,538 inhabitants on the census, and in 2020, there were 331,449,281 inhabitants.

In the e) exercise, we obtained that in 2012 there will be 314,362,768 inhabitants.

Since in 10 years (from 2010 to 2022 ) the population increased by about 23 million, and according to our model increased by 5.6 million in just 2 years, we conclude that according to the model, population growth is higher than in reality (but the model is still very good for

evaluation).

Our model predictions for 2022 model,$y\left(120\right)=\frac{762,176,844.4}{1+8.8248e^{-0.062路120}}=338,540,720$

7 million more than it was in 2020.

In exercise f) the model predicted that in 2023 the population in the U.S. would be 350,000,000 , and in 2020 there were 331,449,281 according to the census. According to the model, this means that in 3 years the population has grown by about million. Which is a little too much.