the population of humorville is 8,500 people. in this town, jokes travel fast. in one hour, each person who hears a joke tells three other people who have not heard it, and tells no one else. last frieday, a visitor from out of town told the police chief a new joke at 10:00 a.m. how long did it take for everyone in humorville to hear the joke?
at 10:00 one person heard it
at 11:00 1 + 3 heard it
at 12:00 1 + 3 + 9 hear it
..
.
looks like a geometric series with
a=1
r = 3, and Sn = 8500
1(3^n - 1)/(3-1) = 8500
3^n - 1 = 17000
3^n = 17001
n = log 17001/lg3
n = 8.86 hours or appr 9 hours
so at 19:00 hrs, or in the US way of stating time, at 7:00 pm
poopie
To find out how long it took for everyone in Humorville to hear the joke, we need to understand the process of how the joke spreads in the town.
Let's break down the information given:
1. The joke starts with the police chief, who is the first person to hear it.
2. Each person who hears the joke tells three other people who have not heard it.
3. There are 8,500 people in Humorville.
Based on these details, we can see that the joke spreads exponentially. The police chief tells 3 people, and each of those 3 people tell 3 other people, and so on.
To determine how many "rounds" it takes for the joke to reach everyone, we can use the concept of powers of 3. Let's start with the number of people who heard the joke after the first round:
1st round: 1 (police chief)
2nd round: 3
3rd round: 3^2 = 9
4th round: 3^3 = 27
5th round: 3^4 = 81
...
nth round: 3^(n-1)
We want to find the minimum value for "n" where the number of people who have heard the joke exceeds the total population of Humorville (8,500).
Using a simple equation, we can solve for "n":
3^(n-1) ≥ 8,500
Taking the logarithm of both sides with base 3, we get:
(n-1) ≥ log3(8,500)
To calculate this, we can use a calculator or an online tool that computes logarithms with different bases. Evaluating the logarithm, we find:
(n-1) ≥ 8.4613
Rounding up, we get:
n ≥ 9
Since "n" represents the number of rounds, we need to add one more round (the initial round) to get the total time required for everyone to hear the joke:
Total time = n + 1 = 9 + 1 = 10 rounds
Therefore, it would take 10 rounds or 10 hours for everyone in Humorville to hear the joke.