Answer:
3:36 PM
Step-by-step explanation:
Let fraction (y) of the people that heard the rumor , the differential equation that is satisfied by the by y is,
[tex]\frac{dy}{dt} = ky(1-y)[/tex]
Solving the differential equation,
[tex]y(t) = \frac{y_{o}}{[y_{o} + (1 + y_{o})e^{-kt}]}[/tex]
The total number of inhabitants of the town = 2000
The number of people that heard the rumor = 160
At 8 AM, let t = 0,
[tex]y(0) = \frac{160}{2000}[/tex]
= 0.08
By noon, half of the town as heard the rumor.
Then,
[tex]y(4) = \frac{1}{2}[/tex]
Therefore,
[tex]\frac{1}{2} = \frac{0.08}{[0.08 + (1 - 0.08)e^{-4k}]}[/tex]
[tex]\frac{0.08}{[0.08 + 0.92e^{-4k}]} = \frac{1}{2}[/tex]
[tex]0.08 + 0.92e^{-4k} = 0.16[/tex]
[tex]0.92e^{-4k} = 0.16 - 0.08[/tex]
[tex]0.92e^{-4k} = 0.08[/tex]
[tex]e^{-4k} = \frac {0.08}{0.92}[/tex]
[tex]k = - \frac{1}{4} In(\frac {0.08}{0.92})[/tex]
k ≈ 0.06106
Calculating time, t when y(t) = 90%
⇒ y(t) = 0.9
[tex]\frac{0.08}{[0.08 + 0.92e^{-0.06106t}]} =0.9[/tex]
[tex]0.08 + 0.92e^{-0.06106t} = \frac {0.9}{0.08}[/tex]
[tex]0.08 + 0.92e^{-0.06106t} = 0.089[/tex]
[tex]0.92e^{-0.06106t} = 0.089 - 0.08[/tex]
[tex]0.92e^{-0.06106t} = 0.0089[/tex]
[tex]t = - \frac{1}{0.06106}In (\frac{0.0089}{0.92})[/tex]
t = 7.59 hours
⇒ 7 hours 36 minutes
From 8 A.M. plus 7 hours 36 minutes = 3:36 PM
At 3:36 PM will 90% of the population have heard the rumor
