13. Consider the spreading of a rumor through a company of 1000 employees, allwo

ID: 2987715 • Letter: 1

Question

13. Consider the spreading of a rumor through a company of 1000 employees, allworking in

the same building.We assume that the spreading of a rumor is similar to the spreading of

a contagious disease (see Example 3, Section 1.2) in that the number of people hearing

the rumor each day is proportional to the product of the number who have heard the

rumor previously and the number who have not heard the rumor. This is given by

rn+1 = rn + 1000krn(1000 ? rn)

where k is a parameter that depends on how fast the rumor spreads and n is the number

of days. Assume k = 0.001 and further assume that four people initially have heard

rumor. How soon will all 1000 employees have heard the rumor?

My question is on this problem #15 by using problem 13

15.Again consider the spreading of a rumor (see Problem 13 in this section), but now

assume a company with 2000 employees. The rumor concerns the number of mandatory

terminations that this company must absorb. Based on the model presented in

Problem 13, build a model for the company with the following rumor growth rates to

determine the number who have heard the rumor after 1 week.

a. k = 0.25

b. k = 0.025

Step 1: Identify the Problem. Predict the spread of a rumor in a controlled environment.

Step 2: Make Simplifying Assumptions.

Variables:

r(n) = number of employees who have heard the rumor at time n

n = number of days

k = parameter that depends on how fast the rumor spreads

Assumptions:

The spread of a rumor is similar to the spread of disease, in that the number of people

hearing the rumor each day is proportional to the product of the number hearing the

rumor and the number who have not heard the rumor.

k = 0:001 and r(0) = 4.

Step 3: Construct the Model. Based on the above assumptions, our model is given by

r(n + 1) = r(n) + 1000kr(n) kr(n)

, or

r(n + 1) = r(n) + kr(n)(1000 r(n)):

Step 4: Solve and Interpret the Model. This discrete dynamical system can be solved

analytically using Maple. Notice that one of the equilibrium solutions is 1000 (i.e., everyone).

Thus, we need to determine how long before we attain the equilibrium solution.

It appears that after 12 days, all 1,000 employees have heard the rumor

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