Chapter 5.9, Problem 8AYU

(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 8AYU

The required graph is shown in Figure 1

Explanation of Solution

Given:

The given table is shown in table 1

Table 1

Year	Population in Billions
1993	5.531
1994	5.611
1995	5.691
1996	5.769
1997	5.847
1998	5.925
1999	6.003
2000	6.080
2001	6.157

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 8AYU

The logistic model is $y=\frac{10.05}{1+{0.84}^{-0.031x}}$ .

Explanation of Solution

From the given data the logistic model is,

  $y=\frac{10.05}{1+{0.84}^{-0.031x}}$

(c)

To determine

To find: The plot of the logistic model.

Answer to Problem 8AYU

The required plot is shown in Figure 2

Explanation of Solution

Given:

The given model is $y=\frac{10.05}{1+{0.84}^{-0.031x}}$ .

Calculation:

Consider the given model is,

  $y=\frac{10.05}{1+{0.84}^{-0.031x}}$

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 8AYU

The carrying capacity of the United Statesis $10.05\text{billions}$ .

Explanation of Solution

Given:

The given model is $y=\frac{10.05}{1+{0.84}^{-0.031x}}$ .

Calculation:

Consider the given model is,

  $y=\frac{10.05}{1+{0.84}^{-0.031x}}$

Then, the carrying capacity of the United States is,

  $c=10.05\text{billions}$

(e)

To determine

To find: The population of the united states in 2004.

Answer to Problem 8AYU

The population of the united states is $6.36\text{billions}$ .

Explanation of Solution

Given:

The given model is $y=\frac{10.05}{1+{0.84}^{-0.031x}}$ .

Calculation:

Consider the given model is,

  $y=\frac{10.05}{1+{0.84}^{-0.031x}}$

Then, the population of the united states in 2004 is,

  $\begin{array}{c}y=\frac{10.05}{1+{0.84}^{-0.031\left(12\right)}}\\ =6.36\text{billions}\end{array}$

(f)

To determine

To find: The year in which the population of the United Statesis 7 billion.

Answer to Problem 8AYU

The required population is after $21.17\text{years}$ .

Explanation of Solution

Given:

The given model is $y=\frac{10.05}{1+{0.84}^{-0.031x}}$ ..

Calculation:

Consider the given model is,

  $y=\frac{10.05}{1+{0.84}^{-0.031x}}$

Then, the population of the United States is 7 billion in,

  $\begin{array}{l}7\text{billion}=\frac{10.05}{1+{0.84}^{-0.031x}}\\ x=21.17\text{years}\end{array}$

(g)

To determine

To find: The comparison of the actual US census figure to the predicted one found in the part (e) and (f) and then discuss the differences.

Answer to Problem 8AYU

The population in year 2004 was 7.028 billion which differs 0.668 billion fom the predicted value of the population.

Explanation of Solution

The population in year 2004 was 7.028 billion which differs 0.668 billion fom the predicted value of the population.