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Sagot :
Because populations grow exponentially, the equation for modelling the growth is:
[tex]P(t)=P_0\cdot e^{a\cdot t}\text{.}[/tex]Where:
• P(t) is the population after t years,
,• P_0 is the initial population,
,• a is the growth factor.
To find the growth factor, we use the consider:
• t = 19 years,
,• P(t) = 2 P_0 (we know that after t = 19 years the population will be doubled).
Replacing these data in the equation above and solving for a, we get:
[tex]\begin{gathered} 2P_0=P_0\cdot e^{a\cdot19\text{ years}}, \\ \frac{2P_0}{P_0}=e^{a\cdot19\text{ years}}, \\ 2=e^{a\cdot19\text{ years}}, \\ \ln 2=a\cdot19\text{ years}\cdot\ln e \\ \ln 2=a\cdot19\text{ years}, \\ a=\frac{\ln2}{19\text{ years}}\cong\frac{0.03648}{\text{year}}\text{.} \end{gathered}[/tex]The annual percent growth rate r is:
[tex]r=a\cdot100\%=\frac{0.03648}{\text{year}}.100\%=\frac{3.648\%}{\text{year}}\text{.}[/tex]Answer
• a = 0.03648/year
,• r = 3.648%/year
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