Chapter 3.2, Problem 58E

Solution

a.

To determine: The population of Mexico City surpass the population of United States by using the exponential growth model and fined the year, if surpassed.

The population surpasses in the year 2133.

Given Information:

The table is defined as,

Year	Population
1900	$13.6$
1950	$25.8$
1960	$34.9$
1970	$48.2$
1980	$66.8$
1990	$88.1$
2001	$101.9$
2011	115
2016	$123.2$

Calculation:

Consider the given polynomial,

Using a graphing calculator, enter the year values starting from $t=0$ for 1900 on L1 and the population in millions on L2 (Table 3.9).

Step 1. Press the tables’ option and enter the values as mentioned above.

  

Step 2. Use the Logistic feature to find the exponential regression model.

Display values are following as,

  

Thus, the function can be written as following,

  $\begin{array}{c}y=81.921\cdot {1.0124}^x\\ P\left(t\right)=81.921\cdot {1.0124}^t\left[\text{Population model}\right]\end{array}$

Similarly, find the exponential model by using the calculator.

  

Now, use Logistic function to find the exponential regression model.

  

Thus, the function can be written as following,

  $\begin{array}{c}y\approx 11.7132\cdot {1.0209}^x\\ P\left(t\right)=11.7132\cdot {1.0209}^t\left[\text{Population model}\right]\end{array}$

Now, draw both the function o graphing calculator. Set the window of the calculator as following,

  

Now, draw both the obtained result.

  

This implies that graph the exponential models on the same viewing window which shows that they intersect at the point $\left(233,1447\right)$ . Thus, Mexico's population will surpass the United States' population in the year 2133.

Therefore, the population surpass in the year 2133.

b.

To determine: The population of Mexico City surpass the population of United States by using the logistic model and fined the year, if surpassed.

The population do not surpass in logistic model.

Given Information:

The table is defined as,

Year	Population
1900	$13.6$
1950	$25.8$
1960	$34.9$
1970	$48.2$
1980	$66.8$
1990	$88.1$
2001	$101.9$
2011	115
2016	$123.2$

Explanation:

Consider the given information,

Using a graphing calculator, enter the year values starting from $t=0$ for 1900 on L1 and the population in millions on L2 (Table 3.9).

Step 1. Press the tables’ option and enter the values as mentioned above.

  

Step 2. Use the Logistic feature to find the exponential regression model.

Display values are following as,

  

Thus, the function can be written as following,

  $\begin{array}{c}y=\frac{702.31723}{1+8.026e^{-0.0166615x}}\\ P\left(t\right)=\frac{702.31723}{1+8.026e^{-0.0166615t}}\left[\text{Population model}\right]\end{array}$

Similarly, find the exponential model by using the calculator.

  

Now, use Logistic function to find the exponential regression model.

  

Thus, the function can be written as following,

  $\begin{array}{c}y\approx \frac{160.2524}{1+42.9809e^{-0.0427561x}}\\ P\left(t\right)=\frac{160.2524}{1+42.9809e^{-0.0427561t}}\left[\text{Population model}\right]\end{array}$

Now, draw both the function o graphing calculator. Set the window of the calculator as following,

  

Now, draw both the obtained result.

  

Graph the logistic models on the same viewing window which shows that they never intersect. Thus, Mexico's population will never surpass the United States' population.

Therefore, the population do not surpass in logistic model.

c.

To determine: The maximum sustainable population for the both countries.

The population is $702.32\text{ million}$ for the United States and about $160.25\text{ million}$ for Mexico.

Given Information:

The table is defined as,

Year	Population
1900	$13.6$
1950	$25.8$
1960	$34.9$
1970	$48.2$
1980	$66.8$
1990	$88.1$
2001	$101.9$
2011	115
2016	$123.2$

Explanation:

Consider the given information,

Refer the both model form the previous part (b).

The united state population model is defined as,

  $P\left(t\right)=\frac{702.31723}{1+8.026e^{-0.0166615t}}$

And the Mexico city population model is defined as,

  $P\left(t\right)=\frac{160.2524}{1+42.9809e^{-0.0427561t}}$

By taking the c -values of the both models (numerator values and denominator as 1), the maximum sustainable populations are about 702.32 million for the United States and about 160.25 million for Mexico.

Therefore, the population is $702.32\text{ million}$ for the United States and about $160.25\text{ million}$ for Mexico.

d.

To determine: The model from both type is more valid in the given case and explain your answer.

The required model is logistic model.

Given Information:

The table is defined as,

Year	Population
1900	$13.6$
1950	$25.8$
1960	$34.9$
1970	$48.2$
1980	$66.8$
1990	$88.1$
2001	$101.9$
2011	115
2016	$123.2$

Explanation:

Consider the given information,

Refer the both model form the previous part (b).

The united state population model is defined as,

  $P\left(t\right)=\frac{702.31723}{1+8.026e^{-0.0166615t}}$

And,

  $\begin{array}{c}y=81.921\cdot {1.0124}^x\\ P\left(t\right)=81.921\cdot {1.0124}^t\left[\text{Population model}\right]\end{array}$

And the Mexico City population model is defined as,

  $P\left(t\right)=\frac{160.2524}{1+42.9809e^{-0.0427561t}}$

And,

  $\begin{array}{c}y\approx 11.7132\cdot {1.0209}^x\\ P\left(t\right)=11.7132\cdot {1.0209}^t\left[\text{Population model}\right]\end{array}$

Therefore, the logistic models seem to be more valid as the growth rate of Mexico is smaller than the United States.