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 Friday, a visitor from out of town told the police chief a new joke at 10:00a.m. How long did it take for everyone in Humorville to hear it?

after one hour, 3

after two hours, 9+3
after three hours, 27+9+3

so this is a geometric series,

3,9,27, ...3^t

The sum of a geometric series is

St=(1-3^t)/(1-3)=(1-3^t)/-2

or
8500*(-2)=1-3^t
-17000=-3^t
log17000=tlog3
t=8.86, but t is an integer, so t=9 hours.

check my work.

To determine how long it took for everyone in Humorville to hear the joke, we can follow these steps:

Step 1: Count the number of people who hear the joke in each hour and keep track of the total number of people who have heard it.

At 10:00 a.m., the police chief hears the joke, so the total number of people who have heard it is 1.

At 11:00 a.m., each of the 1 person who heard the joke in the previous hour tells 3 other people, resulting in 1 * 3 = 3 additional people who heard the joke. Therefore, the total number of people who have heard it by 11:00 a.m. is 1 + 3 = 4.

At 12:00 p.m., each of the 4 people who heard the joke in the previous two hours tells 3 other people, resulting in 4 * 3 = 12 additional people who heard the joke. Therefore, the total number of people who have heard it by 12:00 p.m. is 4 + 12 = 16.

We can continue this pattern for each subsequent hour.

Step 2: Determine the time it takes for the total number of people who heard the joke to reach the population of Humorville (8,500).

By observing the pattern, we see that in each hour, the number of people who hear the joke triples. So, it takes log base 3 of the population to reach the total number of people who heard the joke.

Using the logarithmic formula: log base 3 of 8,500 = log(8,500) / log(3).

By calculating this, we find that log base 3 of 8,500 is approximately 5.95.

Since we cannot have a partial hour, we round up to the nearest whole hour, which means it took 6 hours for everyone in Humorville to hear the joke.

Therefore, the answer is: It took 6 hours for everyone in Humorville to hear the joke.