A small town has 1900 inhabitants. At 8 AM, 80 people have heard a rumor. By noon half the town has heard it. At what time will 90% of the population have heard the rumor? (Do not round k in your calculation. Round the final answer to one decimal place.) 7.59 Incorrect: Your answer is incorrect. hours after the begin

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.