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 herd it, and tells no one else. Last friday, a visitor from out of town told the police cheif a new joke at 10:00a.m. How lomg did it take for the joke to travel through Humorville and for everyone to hear it??
now 1 person knows the joke -- 1
after 1 hour: 3^1 new people will know the joke
after 2 hours 3^2 new people will know the joke
after n hours 3^n new people will know the joke
so 1 + 3 + 3^2 + 3^3 + ... 3^n = 8500
geometric series, a = 1 , r = 3 , n = ?
S(n) = a(r^n - 1)/(r-1)
8500 = 1(3^n - 1)/2
17000 = 3^n - 1
3^n = 17001
log(3^n) = log17001
nlog3 = log 17001
n = log 17001/ log 3 = 8.86 or appr 9 hours
To determine how long it took for the joke to travel through Humorville and for everyone to hear it, we can break down the problem into several steps:
1. Calculate the number of people who hear the joke at each hour: Since each person who hears the joke tells three others who have not heard it, the number of people who hear the joke doubles every hour. Starting with the police chief who heard the joke initially, we can calculate as follows:
- 10:00 a.m. - 1 person (police chief)
- 11:00 a.m. - 2 people
- 12:00 p.m. - 4 people
- 1:00 p.m. - 8 people
- 2:00 p.m. - 16 people
- 3:00 p.m. - 32 people
- 4:00 p.m. - 64 people
2. Continue calculating the number of people who hear the joke until it exceeds the population of Humorville (8500 people): Starting at 4:00 p.m., we can continue calculating the number of people who hear the joke in the next few hours:
- 5:00 p.m. - 128 people
- 6:00 p.m. - 256 people
- 7:00 p.m. - 512 people
- 8:00 p.m. - 1024 people
3. Determine the total time it takes for the joke to reach everyone: By 8:00 p.m., the number of people who have heard the joke exceeds the population of Humorville, so everyone in the town has heard it by that time. Therefore, it took a total of 10 hours for the joke to travel through Humorville and for everyone to hear it (from 10:00 a.m. to 8:00 p.m.).
Keep in mind that this calculation assumes an ideal scenario where everyone accurately tells the joke to three new people every hour and no other factors such as people being unavailable or not passing on the joke disrupt the process.