Respuesta :
Answer:
[tex]P_o = \frac{143000}{e^{-20*0.01303024661}}=110193.69[/tex]
And we can round this to the nearest up integer and we got 110194.
Step-by-step explanation:
The natural growth and decay model is given by:
[tex]\frac{dP}{dt}=kP[/tex] (1)
Where P represent the population and t the time in years since 1970.
If we integrate both sides from equation (1) we got:
[tex] \int \frac{dP}{P} =\int kdt [/tex]
[tex]ln|P| =kt +c[/tex]
And if we apply exponentials on both sides we got:
[tex]P= e^{kt} e^k [/tex]
And we can assume [tex]e^k = P_o[/tex]
And we have this model:
[tex]P(t) = P_o e^{kt}[/tex]
And for this case we want to find [tex]P_o[/tex]
By 1990 we have t=20 years since 1970 and we have this equation:
[tex]143000 = P_o e^{20k}[/tex]
And we can solve for [tex]P_o[/tex] like this:
[tex]P_o = \frac{143000}{e^{20k}}[/tex] (1)
By 2019 we have 49 years since 1970 the equation is given by:
[tex]98000 = P_o e^{49k}[/tex] (2)
And replacing [tex]P_o[/tex] from equation (1) we got:
[tex]98000=\frac{143000}{e^{20k}} e^{49k} =143000 e^{29k}[/tex]
We can divide both sides by 143000 we got:
[tex]\frac{98000}{143000} =0.685 = e^{29k}[/tex]
And if we apply ln on both sides we got:
[tex]ln(0.685) = 29k[/tex]
And then k =-0.01303024661[/tex]
And replacing into equation (1) we got:
[tex]P_o = \frac{143000}{e^{-20*0.01303024661}}=110193.69[/tex]
And we can round this to the nearest up integer and we got 110194.
