Question

Suppose a rumor is going around a group of 191 people. Initially, only 38 members of the group have heard the rumor, but 3 days later 68 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

106 people.

Explanation:

Logistic equation:

The logistic equation is given by:

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

In which

`A = (K - P_0)/(P_0)`

K is the carrying capacity, k is the growth/decay rate, t is the time and P_0 is the initial value.

Suppose a rumor is going around a group of 191 people. Initially, only 38 members of the group have heard the rumor.

This means that
`K = 191, P_0 = 38`, so:

`A = (191 - 38)/(38) = 4.03`

Then

`P(t) = (191)/(1+4.03e^(-kt))`

3 days later 68 people have heard it.

This means that
`P(3) = 68`. We use this to find k.

`P(t) = (191)/(1+4.03e^(-kt))`

`68 = (191)/(1+4.03e^(-3k))`

`68 + 274.04e^(-3k) = 191`

`e^(-3k) = (191-68)/(274.04)`

`e^(-3k) = 0.4484`

`\ln{e^(-3k)} = ln(0.4484)`

`-3k = ln(0.4484)`

`k = -(ln(0.4484))/(3)`

`k = 0.2674`

Then

`P(t) = (191)/(1+4.03e^(-0.2674t))`

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) = (191)/(1+4.03e^(-0.2674*6)) = 105.52`

Rounding to the nearest whole number, 106 people.