   Chapter 4.5, Problem 57E

Chapter
Section
Textbook Problem

A model for the spread of a rumor is given by the equation p ( t ) = 1 1 + a e − k t where p(t) is the proportion of the population that knows the rumor at time t and a and k are positive constants.(a) When will half the population have heard the rumor?(b) When is the rate of spread of the rumor greatest?(c) Sketch the graph of p.

(a)

To determine

To find: The time at when the half of the population would have heard the rumor.

Explanation

Given:

The model for the spread of a rumor of the equation is as given below.

p(t)=11+aekt (1)

Calculation:

Calculate the time at which the half of the population would have heard the rumor.

The value of p(t) at the stage when half the population would have heard the rumor is 12 .

Substitute 12 for p(t) in the equation (1).

12=11+aekt2=1+aekt21=aekt1=aekt1a=ekt

Apply natural log to the base on both side to eliminate exponential.

ln(1a)=lnektln(a1)=ln(ekt)lna=ktln(e)

(b)

To determine

To find: The time at when the rate of spread of the rumor becomes the greatest.

(c)

To determine

To Sketch: The graph of p.

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