A small town has 1000 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 your final answer to one decimal place.)
Question
A small town has 1000 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 your final answer to one decimal place.)
Solution
This problem can be solved using the exponential growth formula, which is:
N = N0 * e^(kt)
where: N is the final amount of the quantity, N0 is the initial amount of the quantity, k is the growth rate, t is the time elapsed.
In this case, the quantity is the number of people who have heard the rumor.
At 8 AM, 80 people have heard the rumor. By noon (4 hours later), half the town (500 people) have heard it. We can use these two points to solve for k, the growth rate.
First, we plug in the values into the formula:
500 = 80 * e^(4k)
Next, we solve for k:
Divide both sides by 80:
6.25 = e^(4k)
Take the natural logarithm of both sides:
ln(6.25) = 4k
Solve for k:
k = ln(6.25) / 4 ≈ 0.4621
Now that we have k, we can find out when 90% of the town (900 people) will have heard the rumor. We plug in the values into the formula and solve for t:
900 = 80 * e^(0.4621t)
Solve for t:
t = ln(900/80) / 0.4621 ≈ 6.3 hours
Since the time is measured from 8 AM, 90% of the town will have heard the rumor by approximately 2:18 PM.
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