algebra

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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??

  • algebra -

    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

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