Answer :
Answer:
In 12 hours 52 minutes, 90% of the population will have heard the rumor.
Step-by-step explanation:
Let f(k) = c * e^{x*k } be the function that models how many people know about the rumor at time x (in hours). Here c is the initial amount of people that knew about the rumor at time x=0, and k is the exponent. We know that
f(8) = 80
f(12) = 1900/2 = 950.
We want x so that f(x) = 1800*0.9 = 1620.
First, observe that f(12)/f(8) = 950/ 80 = 11.875, and also
950/80 = c*e^{12k}/c*e^{8k}) = e^{4k}
Therefore,
e^{4k}= 11.875
4k = ln(11.875) = 2.4744
k = 2.4744/4 = 0.618608837
Now, lets find c knowing that f(8) = 80
80 = c * e^{0.618608837 * 8} = c*141.0156250
Thus, c = 80/141.0156250 = 0.5673
We want to find x such that f(x) = 1620,
f(x) = 0.5673* e^{x * 0.618608837} = 1620
e^{x*0.618608837} = 1620/0.5673 = 2855.566406
x*0.618608837 = ln(2855.566406) = 7.957025493
x = 7.897025493/0.618608837 = 12.86277372 = 12 hors, 52 minutes
In 12 hours 52 minutes, 90% of the population will have heard the rumor.