Beginning Algebra With Applications, Chapter 5, 5.1, Section 5.1, Problem 30

The table below shows the population of Colonial America for each decade from 1610 to 1780. These figures are estimates, because this period was prior to the establishment of the U.S Census in 1790.

$\textbf{Population of Colonial America}$

$\begin{array}{|c|c|} \hline\\ \textit{Year} & \textit{Population} \\ \hline\\ 1610 & 350 \\ 1620 & 2,300 \\ 1630 & 4,600 \\ 1640 & 26,600 \\ 1650 & 50,400 \\ 1660 & 75,100 \\ 1670 & 111,900 \\ 1680 & 151,500 \\ 1690 & 210,440 \\ 1700 & 250,900 \\ 1710 & 331,700 \\ 1720 & 466,200 \\ 1730 & 629,400 \\ 1740 & 905,600 \\ 1750 & 1,170,800 \\ 1760 & 1,593,600 \\ 1770 & 2,148,100 \\ 1780 & 2,780,400 \\ \hline \end{array}$

a. Determine the average annual rate of change in the population of the Colonies from 1650 to 1750.


$\begin{equation} \begin{aligned} \text{average annual rate of change of population} =& \frac{\text{population in 1750 - population in 1650}}{1750-1650} \\ \\ =& \frac{1,170,800-50,400}{1750-1650} \\ \\ =& 11,204 \text{ people per year} \end{aligned} \end{equation}$



The average rate of change was $11,204$ people per year. This means that on average, from 1650-1750, the population of the colonies increased by $11,204$ people per year.

b. Was the average annual rate of change in the population from 1650 to 1750 greater than or less than the average annual rate of change in the population from 1700 to 1750?


$\begin{equation} \begin{aligned} \text{average annual rate of change of population} =& \frac{\text{population in 1750 - population in 1700}}{1750-1700} \\ \\ =& \frac{1,170,800-250,900}{1750-1700} \\ \\ =& 18,398 \text{ people per year} \end{aligned} \end{equation}$



The population from 1650-1750 is less than the population from 1700-1750.

c. During which decade was the average annual rate of change the least? What was the average annual rate of change during that decade?

Based from the table, the least average annual rate of change was from 1610-1620, the average annual rate of change was


$\begin{equation} \begin{aligned} \frac{\text{population in 1620 - population in 1610}}{1620-1610} =& \frac{2,300-350}{1620-1610} \\ \\ =& 195 \text{ people per year} \end{aligned} \end{equation}$