Chapter 3.9, Problem 3E

To determine

 Predict the percentage of the population that will be $65$ and over in the years $2040\&2050$ .

Answer to Problem 3E

  $25\%\&30\%$

Explanation of Solution

Given information:

The graph indicates how Australia’s population is aging by showing the past and projected percentage of the population aged $65$ and over. Use a linear approximation to predict the percentage of the population that will be $65$ and over in the years $2040\&2050$ . Do you think your predictions are too high or too low? Why?

  

Calculation:

  

From the graph ot is observed that,

In $1870$ the curve is at $1$ and slowly increases

In $1900$ the curve is at $4$ and slowly increases

In $1950$ the curve is at $8$ and slowly increases

In $2000$ the curve is at $11$ and slowly increases

In $2030$ the curve is at $20$ as it rises rapidly.

With this information we can say, if ypu use linear equation to predict the chamge in population then your estimate will be too low.

In this first four the number is increasing slowly.

In the last one it is increasing rapidly.

Because the numbers are increasing at a varying rate you cannot predict with a linear graph.

Think of the graph $x^2$ .

It increases slowly at first then faster.

If you used a linear equation on that it would predict numbers lower than the actual equation.

If $A(t)$ represents the percentage of the population aged $65$ and over.

From the graph we have $A\left(2030\right)=20,A\left(2020\right)=15$ .

Estimate $A^{\text{'}}\left(2030\right)$ .

  $\begin{array}{l}{A}^{\text{'}}\left(2030\right)\approx \frac{A\left(2030\right)-A\left(2020\right)}{2030-2020}\\ =\frac{20-15}{10}\\ =0.5\end{array}$

With this estimate the linear approximation for the population in the year $2040$ .

  $\begin{array}{l}A\left(2040\right)\approx A\left(2030\right)+A^{\text{'}}\left(2030\right).\left(2040-2030\right)\\ =20+\mathrm{0.5.10}\\ =25\end{array}$

Estimate $A^{\text{'}}\left(2040\right)$

  $\begin{array}{l}{A}^{\text{'}}\left(2040\right)\approx \frac{A\left(2040\right)-A\left(2030\right)}{2040-2030}\\ =\frac{25-20}{10}\\ =0.5\\ \end{array}$

With this estimate the linear approximation for the population in the year $2050$ .

  $\begin{array}{l}A\left(2050\right)\approx A\left(2040\right)+A^{\text{'}}\left(2040\right).\left(2050-2040\right)\\ =25+\mathrm{0.5.10}\\ =30\end{array}$

Hence the linear approximation gives the value $30$ .

Hence, the predicted percentages of the population aged $65$ and above in this years $2040\&2050$ are $25\%\&30\%$ .