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| PAGE: 638 | SET: Exercises | PROBLEM: 9 |
(a)
Let the number of people who have heard the rumor be 
, and those havenot be 
. Fraction of the people who have heard the rumor 
.
Fraction of the people who have not heard the rumor 
.
.
Fraction of the people who have not heard the rumor is 
.
The rate of spread is proportional to the product of fraction of the people who have heard the rumor and fraction of the people who have not heard the rumor.
, where 
is a proportional constant.
(b)
The logistic differential equation is 
, where 
is a carrying capacity.
Compare 
with logistic differential equation.
Here, 
and 
.
The solution logistic model equation is 
, where 
.
Substitute 
and 
in .
.
Substitute 
, and 
in .
.
The solution of differential equation is 
.
(c)
The town has total people 
.
At 8 AM the number of people heard the rumor is 
.
Consider 
for initial time 8 AM.
Fraction of the people who have heard the rumor is 
.
By noon the half of the people heard the rumor.
Here 
.
.
Substitute the initial value 
in .
.
Substitute 
in 
.
.
Substitute 
in .
.
At what time will 
of the population have heard the rumor.
.
.
When , the time is 8 AM.
, the time is
.
of the rumor has spread by about 
.
(a) , where is a proportional constant.
(b) The solution of differential equation is .
(c) of the rumor has spread by about 
.
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