Chapter 13.3, Problem 34E

To determine

To estimate : The rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines.

Answer to Problem 34E

The rate of population growth on 1920 and in 1980 are 16 millions/year and 78.5 millions/year respectively.

Explanation of Solution

Given information:

The table gives the world’s population in the 20^th century.

Calculation :

The rate of population growth is the slope of the secant line.

The slope of the secant is given by:

  $\begin{array}{l}\text{Slope}=\frac{\Delta P}{\Delta t}\\ \text{Slope}=\frac{P_2-P_1}{t_2-t_1}\end{array}$

where P is population.

For population growth in 1920, there is two secant lines over the intervals $\left[1910,1920\right]$ and $\left[1920,1930\right]$.

So, the secant line over the interval $\left[1910,1920\right]$ is:

  $\begin{array}{l}{\text{S}}_1=\frac{1860-1750}{1920-1910}\\ {\text{S}}_1=11\text{ millions/year}\end{array}$

Similarly, the secant line over the interval $\left[1920,1930\right]$ is:

  $\begin{array}{l}{\text{S}}_2=\frac{2070-1860}{1930-1920}\\ {\text{S}}_2=21\text{ millions/year}\end{array}$

So, the average rate at 1920 is:

  $\begin{array}{l}\text{S}=\frac{11+21}{2}\\ \text{S}=16\text{ millions/year}\end{array}$

Similarly, for population growth in 1980, there is two secant lines over the intervals $\left[1970,1980\right]$ and $\left[1980,1990\right]$.

So, the secant line over the interval $\left[1970,1980\right]$ is:

  $\begin{array}{l}{\text{S}}_1=\frac{4450-3710}{1980-1970}\\ {\text{S}}_1=74\text{ millions/year}\end{array}$

Similarly, the secant line over the interval $\left[1980,1990\right]$ is:

  $\begin{array}{l}{\text{S}}_2=\frac{5280-4450}{1990-1980}\\ {\text{S}}_2=83\text{ millions/year}\end{array}$

So, the average rate at 1980 is:

  $\begin{array}{l}\text{S}=\frac{83+74}{2}\\ \text{S}=78.5\text{ millions/year}\end{array}$

Hence,

The rate of population growth on 1920 and in 1980 are 16 millions/year and 78.5 millions/year respectively.