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 peo

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.)

in progress 0
Latifah 1 week 2021-07-22T09:47:41+00:00 1 Answers 7 views 0

Answers ( )

    0
    2021-07-22T09:48:58+00:00

    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.

Leave an answer

Browse

Giải phương trình 1 ẩn: x + 2 - 2(x + 1) = -x . Hỏi x = ? ( )