Answer :
Answer:
116 people are expected to have heard the rumor after 6 days total have passed since it was initially spread.
Step-by-step explanation:
Logistic function:
The logistic function is given by:
[tex]P(t) = \frac{K}{1 + Ae^{-kt}}[/tex]
In which:
[tex]A = \frac{K - P(0)}{P(0)}[/tex]
Considering that K is the carrying capacity, k is the growth/decay rate and P(0) is the initial population.
Suppose a rumor is going around a group of 210 people.
This means that [tex]K = 210[/tex]
Initially, only 34 members of the group have heard the rumor:
This means that [tex]P(0) = 34[/tex] and:
[tex]A = \frac{210 - 34}{34} = 5.1765[/tex]
So
[tex]P(t) = \frac{210}{1 + 5.1765e^{-kt}}[/tex]
3 days later 69 people have heard it.
This means that [tex]P(3) = 69[/tex], and we use this to find k.
[tex]69 = \frac{210}{1 + 5.1765e^{-3k}}[/tex]
[tex]69 + 357.1785 e^{-3k} = 210[/tex]
[tex]357.1785 e^{-3k} = 141[/tex]
[tex]e^{-3k} = \frac{141}{357.1785}[/tex]
[tex]\ln{e^{-3k}} = \ln{\frac{141}{357.1785}}[/tex]
[tex]-3k = \ln{\frac{141}{357.1785}}[/tex]
[tex]k = -\frac{\ln{\frac{141}{357.1785}}}{3}[/tex]
[tex]k = 0.3098[/tex]
So
[tex]P(t) = \frac{210}{1 + 5.1765e^{-0.3098t}}[/tex]
How many people are expected to have heard the rumor after 6 days total have passed since it was initially spread?
This is P(6), so:
[tex]P(6) = \frac{210}{1 + 5.1765e^{-0.3098*6}} = 116[/tex]
116 people are expected to have heard the rumor after 6 days total have passed since it was initially spread.