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TiliK225 [7]
2 years ago
8

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.)
Mathematics
1 answer:
Veseljchak [2.6K]2 years ago
7 0

Answer:

106 people.

Step-by-step explanation:

Logistic equation:

The logistic equation is given by:

P(t) = \frac{K}{1+Ae^{-kt}}

In which

A = \frac{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 = \frac{191 - 38}{38} = 4.03

Then

P(t) = \frac{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) = \frac{191}{1+4.03e^{-kt}}

68 = \frac{191}{1+4.03e^{-3k}}

68 + 274.04e^{-3k} = 191

e^{-3k} = \frac{191-68}{274.04}

e^{-3k} = 0.4484

\ln{e^{-3k}} = \ln{0.4484}

-3k = \ln{0.4484}

k = -\frac{\ln{0.4484}}{3}

k = 0.2674

Then

P(t) = \frac{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) = \frac{191}{1+4.03e^{-0.2674*6}} = 105.52

Rounding to the nearest whole number, 106 people.

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