Question

The exponential models describe the population of the indicated country, A, in millions, t years after 2010. Which countries have a decreasing populat

Exponential models
The exponential models describe the population of the indicated country, A, in millions, t years after 2010. Which countries have a decreasing population? By what percentage is the population of these countries decreasing each year?
Country B $$\displaystyle{A}={1193.1}{e}^{{{0.006}{t}}}$$
Country C $$\displaystyle{A}={36.5}{e}^{{{0.017}{t}}}$$
Country D $$\displaystyle{A}={121.7}{e}^{{-{0.005}{t}}}$$
Country E $$\displaystyle{A}={145.3}{e}^{{-{0.004}{t}}}$$

2020-11-11

Result used:
The exponential model is, $$\displaystyle{A}{\left({t}\right)}={A}_{{0}}{e}^{{{k}{t}}}$$
1. if $$k>0$$, the population is increasing.
2. if $$k<0$$, the population is decreasing
From the given four countries, it is observed that the countries D and E only have the negative k values.
Thus, Country D and Country E have the decreasing populations.
Use the formula $$(1 +ak)$$ to convert the continuous compound growth to annual compound growth.
Obtain the decrease percent growth of D.
$$\displaystyle{1}+{a}{k}={e}^{{-{0.005}}}$$
$$\displaystyle{a}{k}={0.99501}-{1}$$
$$\displaystyle{a}{k}=-{0.00498}$$
Multiply -0.00498 with 100
$$ak=-0.498\%$$
$$\displaystyle{a}{k}\approx-{0.5}\%$$
Obtain the decrease percent growth of E.
$$\displaystyle{1}+{a}{k}={e}^{{-{0.004}}}$$
$$\displaystyle{a}{k}={0.99600}-{1}$$
$$\displaystyle{a}{k}=-{0.00399}$$
Multiply -0.00399 with 100.
$$\displaystyle{a}{k}=-{0.399}\%$$
$$\displaystyle{a}{k}\approx-{0.4}\%$$
Thus, the population of Country D is decreasing by 0.5% and the popualtion of country E is decreasing by 0.4% each year.