The population of humorville is 8500 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 Friday, 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?
after n hours, 1+3+3^2+3^3 + ... + 3^n = (3^n - 1)/2 people have heard the joke.
So, we want n such that
(3^n - 1)/2 = 8500
3^n = 17001
n = 8.8666 hours = 8 hrs 52 min
sum of 1 + 3 + 9 + 27 ..... 3^(n-1)
geometric series
a = 1
r = 3
To determine how long it took for everyone in Humorville to hear the joke, we can calculate the number of people who hear the joke each hour until it reaches the entire population.
1) Initially, the police chief heard the joke at 10:00 a.m. So, we start with one person who knows the joke.
2) In the first hour (11:00 a.m.), that person tells three other people.
Total people who know the joke: 1 + 3 = 4
3) In the second hour (12:00 p.m.), those four people each tell three more people.
Total people who know the joke: 4 + (3x4) = 16
4) In the third hour (1:00 p.m.), these sixteen people each tell three more people.
Total people who know the joke: 16 + (3x16) = 64
5) This pattern continues, with each subsequent hour adding the number of people who heard the joke in the previous hour multiplied by three.
Using this logic, we can represent the number of people who hear the joke after 'n' hours as 3^n.
6) We need to find the smallest value of 'n' for which 3^n is greater than or equal to 8500.
To find this value, we can use logarithms:
log base 3 of 8500 = log(8500)/log(3) ≈ 7.8
This means that after approximately 7 hours and 48 minutes, more than 8500 people will have heard the joke.
Therefore, it took approximately 7 hours and 48 minutes for everyone in Humorville to hear the joke.
To determine how long it took for everyone in Humorville to hear the joke, we can break down the problem into smaller steps.
Step 1: Calculate the number of people who hear the joke at each hour.
- At 10:00 a.m., the visitor tells the police chief the joke. So, only the police chief has heard the joke.
- In the first hour (11:00 a.m.), the police chief tells the joke to 3 other people.
- In the second hour (12:00 p.m.), those 3 people tell the joke to 9 other people (3 people each).
- In the third hour (1:00 p.m.), those 9 people tell the joke to 27 other people (3 people each).
- This pattern continues, and we can see that the number of people who hear the joke at each hour forms a geometric progression with a common ratio of 3.
Step 2: Calculate the total number of people who have heard the joke at each hour.
- At 10:00 a.m., only 1 person (the police chief) has heard the joke.
- At 11:00 a.m., a total of 4 people (1 + 3) have heard the joke.
- At 12:00 p.m., a total of 13 people (1 + 3 + 9) have heard the joke.
- At 1:00 p.m., a total of 40 people (1 + 3 + 9 + 27) have heard the joke.
- This pattern continues, and we can compute the total number of people who have heard the joke at each hour using the formula: total = 1 + (3^n - 1) / 2, where n is the number of hours.
Step 3: Solve for the number of hours needed to reach the entire population.
- We want to find the number of hours (n) needed to reach the entire population of 8500 people.
- We can set up the equation: 8500 = 1 + (3^n - 1) / 2.
- Solve this equation for n to find the number of hours needed.
Using this approach, we can calculate the number of hours it took for everyone in Humorville to hear the joke.