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
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asked 2020-11-10
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 \(\displaystyle{A}={145.3}{e}^{{-{0.004}{t}}}\)

Answers (1)

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.

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