Chapter 5.9, Problem 7AYU

(a)

To determine

To find: The scatter diagram for the data using years since 1900 as the independent variable and population as the dependent variable.

Answer to Problem 7AYU

The required graph is shown in Figure 1

Explanation of Solution

Given:

The given table is shown in table 1

Table 1

Year	Population
1900	76212168
1910	92228496
1920	106021567
1930	1232052624
1940	132164569
1950	151325798
1960	179323175
1970	203302031
1980	226542203
1990	248709873
2000	281421906

Calculation:

Consider the population as dependent variable.

The scatter plot for the functions in table 1 is shown in Figure 1

  

Figure 1

(b)

To determine

To find: The logistic model from the given data.

Answer to Problem 7AYU

The logistic model is $y=\frac{799475916.5}{1+9.1998e^{-0.016x}}$ .

Explanation of Solution

From the given data the logistic model is,

  $y=\frac{799475916.5}{1+9.1998e^{-0.016x}}$

(c)

To determine

To find: The plot of the logistic model.

Answer to Problem 7AYU

The required plot is shown in Figure 2

Explanation of Solution

Given:

The given model is $y=\frac{799475916.5}{1+9.1998e^{-0.016x}}$ ..

Calculation:

Consider the given model is,

  $y=\frac{799475916.5}{1+9.1998e^{-0.016x}}$

The required plot is shown in Figure 2

  

Figure 2

(d)

To determine

To find: The carrying capacity of the united states.

Answer to Problem 7AYU

The carrying capacity of the united states is $799475916.5$ .

Explanation of Solution

Given:

The given model is $y=\frac{799475916.5}{1+9.1998e^{-0.016x}}$ ..

Calculation:

Consider the given model is,

  $y=\frac{799475916.5}{1+9.1998e^{-0.016x}}$

Then, the carrying capacity of the United States is,

  $c=799475916.5$

(e)

To determine

To find: The population of the united states in 2012.

Answer to Problem 7AYU

The population of the united states is $291599733$ .

Explanation of Solution

Given:

The given model is $y=\frac{799475916.5}{1+9.1998e^{-0.016x}}$ ..

Calculation:

Consider the given model is,

  $y=\frac{799475916.5}{1+9.1998e^{-0.016x}}$

Then, the population of the united states in 2012 is,

  $\begin{array}{c}y=\frac{799475916.5}{1+9.1998e^{-0.016\left(104\right)}}\\ =291599733\end{array}$

(f)

To determine

To find: The year in which the population is 350000000.

Answer to Problem 7AYU

The required population is in the year $2006$ .

Explanation of Solution

Given:

The given model is $y=\frac{799475916.5}{1+9.1998e^{-0.016x}}$ ..

Calculation:

Consider the given model is,

  $y=\frac{799475916.5}{1+9.1998e^{-0.016x}}$

Then, the population of the United States is 350000000 in,

  $\begin{array}{l}350000000=\frac{799475916.5}{1+9.1998e^{-0.016x}}\\ x=106\end{array}$

Thus, the population is 3000000000 in the year

  $2006$ .