Respuesta :
Answer:
Differential equation
[tex]\frac{dy}{dt} =ky(1-y)[/tex]
Solution
[tex]y=\frac{1}{1+4e^{-0.327t}}[/tex]
Value of constant k=0.327 days^(-1)
The rumor reaches 80% at 8.48 days.
Step-by-step explanation:
We know
y(t): proportion of people that heard the rumor
y'(t)=ky(1-y), rate of spread of the rumor
Differential equation
[tex]\frac{dy}{dt} =ky(1-y)[/tex]
Solving the differential equation
[tex]\frac{dy}{y(1-y)}=k\cdot dt \\\\\int \frac{dx}{y(1-y)} =k \int dt \\\\-ln(1-\frac{1}{y} )+C_0=kt\\\\1-\frac{1}{y} =Ce^{-kt}\\\\\frac{1}{y} =1-Ce^{-kt}\\\\y=\frac{1}{1-Ce^{-kt}}[/tex]
Initial conditions:
[tex]y(0)=0.2\\y(3)=0.4\\\\y(0)=0.2=\frac{1}{1-Ce^0}\\\\1-C=1/0.2\\\\C=1-1/0.2= -4\\\\\\y(3)=0.4=\frac{1}{1+4e^{-3k}} \\\\1+4e^{-3k}=1/0.4\\\\e^{-3k}=(2.5-1)/4=0.375\\\\k=ln(0.375)/(-3)=0.327\\\\\\y=\frac{1}{1+4e^{-0.327t}}[/tex]
Value of constant k=0.327 days^(-1)
At what time the rumor reaches 80%?
[tex]y(t)=0.8=\frac{1}{1+4e^{-0.327t}} \\\\1+4e^{-0.327t}=1/0.8=1.25\\\\e^{-0.327t}=(1.25-1)/4=0.0625\\\\t=ln(0.0625)/(-0.327)=8.48[/tex]
The rumor reaches 80% at 8.48 days.
