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| Math Central | Quandaries & Queries |
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Question from Sean, a student: Could u help me with this question. The number, N, of people who have heard a rumor spread by mass media at time, t, in days is modelled by N (t) = a / 1+ be^ -kt a. If 50 people have heard the rumour initially and 300,000 people hear the rumour eventually, find a and b b. If the rumour is initially spreading at the rate of 500 people per day, find k. Thanks. |
Sean,
Part a.
If 50 heard the rumor initially, (at time t = 0 days) then N(0) = 50. Substitute t = 0 into the expression for N(t) to get an equation for a and b.
If 300,000 people eventually hear the rumor then the limit of N(t) as t approaches infinity is 300,000. This gives a second equation in a and b.
Solve the two equations for a and b.
Part b.
The rate at which the rumor is spreading on day 0 is the derivative of N(t), evaluated when t = 0. Thus N'(0) = 500 people per day. Evaluate N'(0) and solve this equation for k. This equation can be simplified to e-kt = "a number" and hence you will need to take the natural log of both sides to solve for k.
I hope this helps,
Penny
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