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Participation Activity \#19

A rumor spreads through a population rapidly, at a rate that depends on two factors: the more people who have heard the rumor, the more rapidly it spreads, and also the more people who have not heard the rumor, the more rapidly it spreads. These two factors make the logistic model a good one to study the spread of rumors. Clearly, there is a maximum value for the number of people hearing the rumor: the entire population. A logistic model of a rumor spread is given by

[tex]\[ f(x) = \frac{1,000}{1 + 499 e^{-0.6030 x}}. \][/tex]

In this model, the rumor starts on day 0 with
[tex]\[ f(0) = \frac{1000}{1 + 499 e^{-0.6030 \cdot 0}} = \frac{1000}{1 + 499 \cdot 1} = 2 \][/tex]
people.

Using this model, estimate the number of people hearing the rumor on day 18.

Round your answer to the nearest person.

The number of people who have heard the rumor on day 18 is about ___.



Answer :

GinnyAnswer
To estimate the number of people hearing the rumor on day 18 using the given logistic model, let's follow through the steps for this calculation:

The logistic model for the number of people, [tex]\( f(x) \)[/tex], who have heard the rumor by day [tex]\( x \)[/tex] is given by:
[tex]\[ f(x)=\frac{1,000}{1 + 499 e^{-0.6030 x}} \][/tex]

Given:
- Total population [tex]\( P = 1000 \)[/tex] people.
- Initial rumor spread constant [tex]\( k = 499 \)[/tex].
- Spread rate coefficient [tex]\( r = 0.6030 \)[/tex].
- We need to find [tex]\( f(18) \)[/tex], which is the number of people who have heard the rumor on day 18.

Substitute [tex]\( x = 18 \)[/tex] into the logistic model:

[tex]\[ f(18) = \frac{1000}{1 + 499 e^{-0.6030 \cdot 18}} \][/tex]

Now let's break down the calculation step by step:

1. Calculate the exponent part:
[tex]\[ -0.6030 \cdot 18 = -10.854 \][/tex]

2. Calculate the exponential function:
[tex]\[ e^{-10.854} \][/tex]

3. Multiply by the constant 499:
[tex]\[ 499 \cdot e^{-10.854} \][/tex]

4. Add 1 to this result:
[tex]\[ 1 + 499 \cdot e^{-10.854} \][/tex]

5. Finally, divide 1000 by this result to get [tex]\( f(18) \)[/tex]:
[tex]\[ f(18) = \frac{1000}{1 + 499 e^{-10.854}} = 990.4478779069576 \][/tex]

After calculating, the number of people who have heard the rumor by day 18 is approximately:
[tex]\[ f(18) \approx 990.4478779069576 \][/tex]

To provide a final answer rounded to the nearest person:

The number of people who have heard the rumor on day 18 is about 990.

So, the number of people who have heard the rumor on day 18 is approximately 990 when rounded to the nearest person.

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