Respuesta :
Using the logistic equation, it is found that:
- The model is:
[tex]P(t) = \frac{500000}{1 + 624e^{-0.4t}}[/tex]
- It will take 10 days before 100,000 people are infected.
The logistic equation for a population is given by:
[tex]P(t) = \frac{K}{1 + Ae^{-kt}}[/tex]
[tex]A = \frac{K - P(0)}{P(0)}[/tex]
In which:
- K is the carrying capacity.
- P(0) is the initial value.
- k is the growth rate, as a decimal.
In this problem:
- Population of half a million, which is the carrying capacity, hence [tex]K = 50000[/tex].
- The strain increases by 40% per week, hence [tex]k = 0.4[/tex].
- Initially, there were 800 cases, hence [tex]P(0) = 800[/tex].
Then:
[tex]A = \frac{500000 - 800}{800} = 624[/tex]
[tex]P(t) = \frac{500000}{1 + 624e^{-0.4t}}[/tex]
The time until 100,000 people are infected is t for which P(t) = 100000, hence:
[tex]P(t) = \frac{500000}{1 + 624e^{-0.4t}}[/tex]
[tex]100000 = \frac{500000}{1 + 624e^{-0.4t}}[/tex]
[tex]0.1 = \frac{1}{1 + 624e^{-0.4t}}[/tex]
[tex]0.1 + 62.4e^{-0.4t} = 1[/tex]
[tex]62.4e^{-0.4t} = 0.9[/tex]
[tex]e^{-0.4t} = \frac{0.9}{62.4}[/tex]
[tex]\ln{e^{-0.4t}} = \ln{\frac{0.9}{62.4}}[/tex]
[tex]-0.4t = \ln{\frac{0.9}{62.4}}[/tex]
[tex]t = -\frac{\ln{\frac{0.9}{62.4}}}{0.4}[/tex]
[tex]t = 10.6[/tex]
It will take 10 days before 100,000 people are infected.
A similar problem is given at https://brainly.com/question/25609786
