Question:
Looking below is this answer correct? If you follow the math, you reach the entire population before 8 hours have passed yet the solution says 8.86 hours....
Is the solution giving us the numbers in the set and not the actual elapsed hours???
Does anyone know why?
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?
math - Reiny, Sunday, February 24, 2008 at 9:34pm
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
Wow, you went all the way back to Feb 2008 to find this!
You are correct, the way I defined it, n is the number of terms, not time.
10:00 ---> sum of 1 term
11:00 --- > sum of 2 terms
...
19:00 --- > sum of 8 terms (3^8 - 1)/(3-1) = 3280
20:00 ----> sum of 9 terms (3^9 -1)/(3-1) = 9841
so at 19:00 hours not everybody has heard it, but at 20:00 everybody, and then some, has heard the joke.
so 20:00 hours or 8:00 pm
I think its 8 hours
The solution provided for this problem is correct, but it seems like there might be some confusion about how to interpret the answer. Let me explain.
In this problem, a visitor tells a joke to the police chief of Humorville at 10:00 a.m. According to the given information, each person who hears the joke tells three other people who have not heard it yet, and tells no one else. The question asks for the time it takes for everyone in Humorville to hear the joke.
The mathematician Reiny analyzed the situation and converted it into a geometric series with the formula: Sn = a(3^n - 1)/(3-1), where Sn is the sum of the first n terms, a is the initial term, and r is the common ratio.
In this case, a = 1 (as one person heard the joke initially), r = 3 (each person tells the joke to three others), and Sn = 8500 (the total population of Humorville).
By substituting these values into the equation, Reiny obtained the equation: 1(3^n - 1)/(3-1) = 8500.
Solving this equation, he found that n ≈ 8.86 hours, which represents the time required for everyone in Humorville to hear the joke.
So, the answer of 8.86 hours does represent the elapsed time it takes for the joke to reach the entire population of Humorville.