Chapter 9.3, Problem 86E

a.

To determine

To find: A geometric sequence that models the data from 2004 through 2010.

Answer to Problem 86E

  $a_n=1272.23{\left(1.005\right)}^{n-1}$

Explanation of Solution

Given: The table shows the mid-year populations $a_n$ (in millions) of china from 2004 through 2010.

  

Since $n=4$ corresponding to 2004.

  $\begin{array}{l}\therefore a_4=1291.0,a_5=1297.8,a_6=1304.3,a_7=1310.6\\ a_8=1317.1,a_9=1323.6,a_{10}=1330.1\end{array}$

First we draw the graph using exponential regression.

  

Using exponential regression graphing utility the geometric sequence is $a_n=1272.23{\left(1.005\right)}^{n-1}$

b.

To determine

To find: The rate of population of china is growing.

Answer to Problem 86E

0.5%

Explanation of Solution

Given: The geometric sequence of the data is $a_n=1272.23{\left(1.005\right)}^{n-1}$

The model of data, $a_n=1272.23{\left(1.005\right)}^{n-1}$

Rate $=1.005-1=.005$

Hence, the rate is 0.5% per year.

c.

To determine

To find: The population of china in 2017.

Answer to Problem 86E

5.81 million

Explanation of Solution

Given: The geometric sequence of the data is $a_n=1272.23{\left(1.005\right)}^{n-1}$

Since $n=4$ corresponding to 2004.

Therefore, 2017 will $n=17$ .

Put $n=17$ in $a_n=1272.23{\left(1.005\right)}^{n-1}$

  $\begin{array}{l}{a}_{17}=1272.23{\left(1.005\right)}^{17-1}\\ a_{17}=1377.91\end{array}$

Using geometric sequence model the population of China in 2017 will be 1377.91 million.

But the U.S. Census Bureau predicts the population of China will be 1372.1 million in 2017.

U.S. Census Bureau predicts lesser population as compare to model.

Difference $=1377.91-1372.1=5.81$

Hence, U.S. Census Bureau prediction is 5.81 million population lesser.

d.

To determine

To find: The year when population of china will be 1.4 billion.

Answer to Problem 86E

2020 year

Explanation of Solution

Given: The geometric sequence of the data is $a_n=1272.23{\left(1.005\right)}^{n-1}$

Since $n=4$ corresponding to 2004.

The population of China will be 1.4 billion.

1 billion = 1000 million

Therefore, 1.4 billion = 1400 million

Put $a_n=1400$ in $a_n=1272.23{\left(1.005\right)}^{n-1}$

  $\begin{array}{c}1400=1272.23{\left(1.005\right)}^{n-1}\\ {\left(1.005\right)}^{n-1}=\frac{1400}{1272.23}=1.1004\\ \left(n-1\right)\ln 1.005=\ln 1.1004\\ n-1=19.2\\ n=20.2\\ n\approx 20\end{array}$

Hence, the population of China will be 1.4 billion in 2020.