Chapter 3.5, Problem 36E

a.

To determine

The exponential growth or decay model for the population of each country.

Explanation of Solution

Given information:

The exponential growth or decay model is given by formula,

  $\begin{array}{l}y=a{e}^{bt}\text{ or }y=a{e}^{-bt}\\ t=\text{time in years}\text{.}\\ t=18\text{ corresponding to the year 2018}\text{.}\end{array}$

  

The table shows the population (in millions) of five countries in the year 2018 and the projected population (in millions) for the year 2030.

Calculations:

Here the population in Japan has decreased from the year 2018 to 2030. So it is following exponential decay model.

Now,

  $\begin{array}{l}y=a{e}^{-bt}\\ t=18\text{ corresponding to the year 2018}\text{.}\\ \text{So, t=30 for the year 2030}\end{array}$

Now putting the values we get,

  $\begin{array}{l}\text{For the year 2018}\text{.}\\ y=a{e}^{-bt}\\ 126.2=a{e}^{-18b}\\ a=\frac{126.2}{e^{-18b}}\mathrm{...........}(1)\\ \text{For the year 2030}\text{.}\\ y=a{e}^{-bt}\\ 120.8=a{e}^{-30b}\\ a=\frac{120.8}{e^{-30b}}\mathrm{................}(2)\\ From\text{ (1) and (2) we get}\text{.}\\ \frac{126.2}{e^{-18b}}=\frac{120.8}{e^{-30b}}\\ \frac{126.2}{120.8}=\frac{e^{-18b}}{e^{-30b}}\\ 1.0447=e^{12b}\\ \ln 1.0447=\ln e^{12b}\\ 0.043729=12b\\ b=\frac{0.043729}{12}=0.003644\\ \text{Now substituting the value of b in equation (1) we get}\text{.}\\ a=\frac{126.2}{e^{-18b}}\\ a=\frac{126.2}{e^{-18\times 0.003644}}\\ a=134.75\end{array}$

Thus the exponential decay model for Japan is

  $y=134.75e^{-0.003644t}$

Now for Canada.

Here the population in Canada has increased from the year 2018 to 2030. So it is following exponential growth model.

Now,

  $\begin{array}{l}y=a{e}^{bt}\\ t=18\text{ corresponding to the year 2018}\text{.}\\ \text{So, t=30 for the year 2030}\end{array}$

Now putting the values we get,

  $\begin{array}{l}\text{For the year 2018}\text{.}\\ y=a{e}^{bt}\\ 35.9=a{e}^{18b}\\ a=\frac{35.9}{e^{18b}}\mathrm{...........}(1)\\ \text{For the year 2030}\text{.}\\ y=a{e}^{bt}\\ 38.6=a{e}^{30b}\\ a=\frac{38.6}{e^{30b}}\mathrm{................}(2)\\ From\text{ (1) and (2) we get}\text{.}\\ \frac{35.9}{e^{18b}}=\frac{38.6}{e^{30b}}\\ \frac{38.6}{35.9}=\frac{e^{30b}}{e^{18b}}\\ 1.0752=e^{12b}\\ \ln 1.0752=\ln e^{12b}\\ 0.0725=12b\\ b=\frac{0.0725}{12}=0.006\\ \text{Now substituting the value of b in equation (1) we get}\text{.}\\ a=\frac{35.9}{e^{18b}}\\ a=\frac{35.9}{e^{18\times 0.006}}\\ a=32.2\end{array}$

Thus the exponential growth model for Canada is

  $y=32.2e^{0.006}$

Now for Russia.

Here the population in Russia has decreased from the year 2018 to 2030. So it is following exponential decay model.

Now,

  $\begin{array}{l}y=a{e}^{-bt}\\ t=18\text{ corresponding to the year 2018}\text{.}\\ \text{So, t=30 for the year 2030}\end{array}$

Now putting the values we get,

  $\begin{array}{l}\text{For the year 2018}\text{.}\\ y=a{e}^{-bt}\\ 142.1=a{e}^{-18b}\\ a=\frac{142.1}{e^{-18b}}\mathrm{...........}(1)\\ \text{For the year 2030}\text{.}\\ y=a{e}^{-bt}\\ =a{e}^{-30b}\\ a=\frac{138.2}{e^{-30b}}\mathrm{................}(2)\\ From\text{ (1) and (2) we get}\text{.}\\ \frac{142.1}{e^{-18b}}=\frac{138.2}{e^{-30b}}\\ \frac{142.1}{138.2}=\frac{e^{-18b}}{e^{-30b}}\\ 1.0282=e^{12b}\\ \ln 1.0282=\ln e^{12b}\\ 0.027809=12b\\ b=\frac{0.027809}{12}=0.0023174\\ \text{Now substituting the value of b in equation (1) we get}\text{.}\\ a=\frac{142.1}{e^{-18b}}\\ a=\frac{142.1}{e^{-18\times 0.0023174}}\\ a=148.15\end{array}$

Thus the exponential decay model for Russia is

  $y=148.15e^{-0.0023174t}$

Now for Philippines.

Here the population in Philippines has increased from the year 2018 to 2030. So it is following exponential growth model.

Now,

  $\begin{array}{l}y=a{e}^{bt}\\ t=18\text{ corresponding to the year 2018}\text{.}\\ \text{So, t=30 for the year 2030}\end{array}$

Now putting the values we get,

  $\begin{array}{l}\text{For the year 2018}\text{.}\\ y=a{e}^{bt}\\ 105.9=a{e}^{18b}\\ a=\frac{105.9}{e^{18b}}\mathrm{...........}(1)\\ \text{For the year 2030}\text{.}\\ y=a{e}^{bt}\\ 125.7=a{e}^{30b}\\ a=\frac{125.7}{e^{30b}}\mathrm{................}(2)\\ From\text{ (1) and (2) we get}\text{.}\\ \frac{105.9}{e^{18b}}=\frac{125.7}{e^{30b}}\\ \frac{125.7}{105.9}=\frac{e^{30b}}{e^{18b}}\\ 1.186=e^{12b}\\ \ln 1.186=\ln e^{12b}\\ 0.17=12b\\ b=\frac{0.17}{12}=0.014\\ \text{Now substituting the value of b in equation (1) we get}\text{.}\\ a=\frac{105.9}{e^{18b}}\\ a=\frac{105.9}{e^{18\times 0.014}}\\ a=82.31\end{array}$

Thus the exponential decay growth for Philippines is

  $y=82.31e^{0.014}$

Now for Turkey.

Here the population in Turkey has increased from the year 2018 to 2030. So it is following exponential growth model.

Now,

  $\begin{array}{l}y=a{e}^{bt}\\ t=18\text{ corresponding to the year 2018}\text{.}\\ \text{So, t=30 for the year 2030}\end{array}$

Now putting the values we get,

  $\begin{array}{l}\text{For the year 2018}\text{.}\\ y=a{e}^{bt}\\ 81.3=a{e}^{18b}\\ a=\frac{81.3}{e^{18b}}\mathrm{...........}(1)\\ \text{For the year 2030}\text{.}\\ y=a{e}^{bt}\\ 86.7=a{e}^{30b}\\ a=\frac{86.7}{e^{30b}}\mathrm{................}(2)\\ From\text{ (1) and (2) we get}\text{.}\\ \frac{81.3}{e^{18b}}=\frac{86.7}{e^{30b}}\\ \frac{86.7}{81.3}=\frac{e^{30b}}{e^{18b}}\\ 1.029=e^{12b}\\ \ln 1.029=\ln e^{12b}\\ 0.0285=12b\\ b=\frac{0.0285}{12}=0.00237\\ \text{Now substituting the value of b in equation (1) we get}\text{.}\\ a=\frac{81.3}{e^{18b}}\\ a=\frac{81.3}{e^{18\times 0.00237}}\\ a=77.904\end{array}$

Thus the exponential growth model for Turkey is

  $y=77.904e^{0.00237}$