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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.)

User Darrel Lee
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1 Answer

1 vote

Answer:

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.

User Nisarg Thakkar
by
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