 Jaya Legge

2021-02-06

The exponential models the population of the indicated country, A, in millions, t years after 20006. Which country has the greatest growth rate? By what percentage is the population of that country increasing each year?
Country 1: $A=26.3{e}^{0.029t}$
Country 2: $A=127.7{e}^{0.007t}$
Country 3: $A=148.5{e}^{-0.0091t}$
Country 4: $A=1094.2{e}^{0.016t}$ Caren

Given:
The exponential model that describe the population of different countries:
Country 1: $A=26.3{e}^{0.029t}$
Country 2: $A=127.7{e}^{0.007t}$
Country 3: $A=148.5{e}^{-0.0091t}$
Country 4: $A=1094.2{e}^{0.016t}$
Population growth is equal to differentiation of exponential model with respect to ‘t’,
So now differentiating the given function one by one:
$\frac{d}{dt}\left({A}_{1}\right)=\frac{d}{dt}\left(26.3{e}^{0.029t}\right)$
$\frac{d{A}_{1}}{dt}=\left(26.3×0.029{e}^{0.029t}\right)$
$\frac{d{A}_{1}}{dt}=\left(0.7627{e}^{0.029t}\right)$
$\frac{d}{dt}\left({A}_{2}\right)=\frac{d}{dt}\left(127.7{e}^{0.007t}\right)$
$\frac{d{A}_{1}}{dt}=\left(127.7×0.007{e}^{0.007t}\right)$
$\frac{d{A}_{1}}{dt}=\left(0.8939{e}^{0.007t}\right)$
$\frac{d}{dt}\left({A}_{3}\right)=\frac{d}{dt}\left(148.5{e}^{-0.009t}\right)$
$\frac{d{A}_{3}}{dt}=\frac{d}{dt}\left(148.5×\left(-0.009\right){e}^{-0.009t}\right)$
$\frac{d{A}_{3}}{dt}=-\left(1.3365{e}^{-0.009t}\right)$
$\frac{d}{dt}\left({A}_{4}\right)=\frac{d}{dt}\left(1094.2{e}^{0.016t}\right)$
$\frac{d{A}_{4}}{dt}=\frac{d}{dt}\left(1094.2×0.016{e}^{0.016t}\right)$
$\frac{d{A}_{4}}{dt}=\left(17.5072{e}^{0.016t}\right)$
To check which one is larger, by comparing the quantities of growth rate it can be concluded that:
Country 4: ${A}_{4}=1094.2{e}^{0.016t}$
Has the greates growth rate.
To find the percentage change in population:
Country 4: ${A}_{4}=1092.2{e}^{0.016t}$
Now put $t=0$,
${A}_{4}=1094.2{e}^{0.016×0}$
${A}_{4}=1094.2{e}^{0}$
${A}_{4}=1094.2$
Now put $t=1,$
${A}_{4}=1094.2{e}^{0.016×1}$
${A}_{4}=1094.2{e}^{0.016}$
${A}_{4}=1111.8$
Now,
Percentage change $=\frac{1111.8-1094.2}{1094.2}×100$
Percentage change $=1.60$
Country 4: $A=1094.2{e}^{0.016t}$
Has the greates growth rate.
Percentage change in population each year $=1.60$

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