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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.)

User Norlin
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3 votes

Answer:

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.

User Amosmos
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