   Chapter 3.4, Problem 84E

Chapter
Section
Textbook Problem

Under certain circumstance a rumor spreads according to the equation p ( t ) = 1 1 + a e − k t where p(t) is the proportion of the population that has heard the rumor at time t and a and k are positive constants. [In Section 9.4 we will see that this is a reasonable equation for p(t).](a) Find limt→∞ p(t).(b) Find the rate of spread of the rumor.(c) Graph p for the case a= 10, k = 0.5 with 1 measured in hours. Use the graph to estimate how long it will take for 80% of the population to hear the rumor.

(a)

To determine

To find: The limit of the function limtp(t).

Explanation

Calculation:

Obtain limit of the function.

That is, compute limtp(t).

limtp(t)=limt(11+aekt)=limt(11+aekt<

(b)

To determine

To find: The rate of spread of the rumor.

(c)

To determine

To sketch: The graph p(t) and estimate the time such that 80% of the population to hear the rumor.

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