Respuesta :
Answer:
The number of plants in the yard after 3 months = 60
Explanation:The number of blackberry plants that started growing in the yard, p₀ = 30
The blackberry plants will spread by 100% a month
p(1) = 30 + 1(30)
p(1) = 60
The yard can only sustain 60 plants, m = 60
The logistic growth model is given as:
[tex]p=\frac{m}{1+(\frac{m}{p_0}-1)e^{-rt}}[/tex]Substitute p₀ = 30, p = 60, t = 1, and m = 60 to solve for the growth rate, r.
[tex]\begin{gathered} 60=\frac{60}{1+(\frac{60}{30}-1)e^{-r(1)}} \\ 60=\frac{60}{1+(2-1)e^{-r}} \\ 60=\frac{60}{1+e^{-r}} \\ 60(1+e^{-r})=60 \\ 1+e^{-r}=\frac{60}{60} \\ e^{-r}=1-1 \\ e^{-r}=0 \\ \ln (e^{-r})=\ln 0 \\ -r=-\infty \\ r=\infty \end{gathered}[/tex]To estimate the number of plants after 3 months:
substitute t = 3, and r = ∞ into the logistic model
[tex]\begin{gathered} p=\frac{m}{1+(\frac{m}{p_0}-1)e^{-rt}} \\ p(3)=\frac{60}{1+(\frac{60}{30}-1)e^{-\infty(3)}} \\ p(3)=\frac{60}{1+e^{-\infty}} \\ p(3)=\frac{60}{1+0} \\ p(3)=\frac{60}{1} \\ p(3)=60 \end{gathered}[/tex]The number of plants in the yard after 3 months = 60
