Answer:
a) dy/dt = ky(1-y)
b) 3:36pm
Step-by-step explanation:
a) Let the number of people who have heard the rumor = p
Let those who have not heard the rumor= q
Total population = p+q
Fraction of those that heard the rumor = p/p+q = y
Fraction of those who did not hear the rumor = q/p+q = 1-y
The rate at which the rumor spreads = dy/dt
dy/dt varies directly to y(1-y)
dy/dt = ky(1-y) where k is a constant
b) Recall that dy/dt = ky(1-y)
y(t) = y/(y+(1-y) e^-kt)
At 8 am , t= 0
y = p/ p+q
y(0) = 280/3500
y(0) = 0.08
By noon(12pm), t = 4
At this time half of the population has heard the rumor
y(4) = 0.5
Recall that y(t) = y/(y+(1-y) e^-kt)
y(t) = y0/(y0+(1-y0) e^-kt)
y(t) = 0.08/(0.08+(1-0.08) e^-kt)
y(t) = 0.08/(0.08+0 92 e^-kt)
To find k, put y(4) = 0.5 into the equation
y(4) = y/(y+(1-y) e^-4k)
0.5 = 0.08/(0.08+0.92e^-4k)
0.08 + 0.92e^-4k = 0.08/0.5
0.92e^-4k = 0.16 - 0.08
0.92e^-4k = 0.08
e^-4k = 0.08/0.92
e^-4k = 0.087
-4k = ln(0.087)
-4k = -2.422
k = -2.422/ -4
k = 0.611
y(t) = 0.08/(0.08+ 0.92 e^-0.611t)
The time by which 90% of the population would have heard the rumor is
0.9 = 0.08/(0.08+0.92e^-0.611t)
0.08 + 0.92e^-0.611t = 0.08/0.9
0.92e^-0.611t= (0.08/0.9) - 0.08
e^-0.611t = [(0.08/0.9)-0.08] / 0.92
e^-0.611t = 0.00966
-0.611t = ln(0.00966)
-0.611t = -4.640
t = -4.640/ -0.611
t = 7.6hrs
t = 7 hrs + (0.6*60)mins
t= 7 hrs + 36mins
t = 7hrs 36 mins
Therefore 8 am + 7 hrs 36 mins = 3:36pm
The time by which 90% of the rumor spreads = 3:36pm
