Question

Suppose a rumor is going around a group of 210 people. Initially, only 34 members of the group have heard the rumor, but 3 days later 69 people have heard it. Using a logistic growth model, how many people are expected to have heard the rumor after 6 days total have passed since it was initially spread? (Round your answer to the nearest whole person.)

Answer 1

116 people are expected to have heard the rumor after 6 days total have passed since it was initially spread.

Explanation:

Logistic function:

The logistic function is given by:

`P(t) = (K)/(1 + Ae^(-kt))`

In which:

`A = (K - P(0))/(P(0))`

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
`K = 210`

Initially, only 34 members of the group have heard the rumor:

This means that
`P(0) = 34` and:

`A = (210 - 34)/(34) = 5.1765`

So

`P(t) = (210)/(1 + 5.1765e^(-kt))`

3 days later 69 people have heard it.

This means that
`P(3) = 69`, and we use this to find k.

`69 = (210)/(1 + 5.1765e^(-3k))`

`69 + 357.1785 e^(-3k) = 210`

`357.1785 e^(-3k) = 141`

`e^(-3k) = (141)/(357.1785)`

`\ln{e^(-3k)} = \ln{(141)/(357.1785)}`

`-3k = \ln{(141)/(357.1785)}`

`k = -\frac{\ln{(141)/(357.1785)}}{3}`

`k = 0.3098`

So

`P(t) = (210)/(1 + 5.1765e^(-0.3098t))`

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:

`P(6) = (210)/(1 + 5.1765e^(-0.3098*6)) = 116`

116 people are expected to have heard the rumor after 6 days total have passed since it was initially spread.