Chapter 9.3, Problem 84E

(a)

To determine

To Find: The nth term of the geometric sequence that models the data.

Answer to Problem 84E

The required model is $a_n=37.43{\left(1.01\right)}^n$ .

Explanation of Solution

Given:

The required table is shown in Table 1

Table 1

Year	Population
2009	40.9
2010	41.3
2011	41.8
2012	42.2
2013	42.6
2014	43.0
2015	43.4

Calculation:

Input the year on L and the population on L2 in the graphing calculator as shown in Figure 1

  

Figure 1

From above the nth term of the sequence is,

  $a_n=37.43{\left(1.01\right)}^n$

(b)

To determine

To Find: The rate at which the population of Argentina is growing.

Answer to Problem 84E

The growth rate is $1\%$ .

Explanation of Solution

Consider the gravitational growth model is,

  $y=a{\left(1+r\right)}^x$

Then, the growth rate is r and compare it to the model of part (a), then,

  $\begin{array}{c}r=0.01\\ =1\%\end{array}$

(c)

To determine

To Find: The prediction for the population of the Argentina in 2025 and compare it to the value that population of Argentina will be 47.2 million in 2025.

Answer to Problem 84E

The prediction $48\text{million}$ is $2.8\text{million}$ less by $47.2\text{million}$ mark.

Explanation of Solution

Consider the required model is,

  $a_n=37.43{\left(1.01\right)}^n$

Then, the population in 2025 is,

  $\begin{array}{c}{a}_{25}=37.43{\left(1.01\right)}^{25}\\ =48.0\text{million}\end{array}$

The above prediction is $2.8\text{million}$ less by $47.2\text{million}$ mark.

(d)

To determine

To Find: The sequence from part (a) to predict that when will the population of Argentina will reach 50.0 million.

Answer to Problem 84E

The required population is achieved in year 2029.

Explanation of Solution

Consider the required model is,

  $a_n=37.43{\left(1.01\right)}^n$

Then,

  $\begin{array}{l}50=37.43{\left(1.01\right)}^n\\ \ln \left(\frac{50}{37.43}\right)=\ln {\left(1.01\right)}^n\\ \ln \left(\frac{50}{37.43}\right)=n\ln \left(1.01\right)\\ n=29\end{array}$

Thus, the required population is achieved in year 2029.