Question

The population of Humorville is 9800 people. In one hour, each person who hears a joke tells three other people who have not head it, and tells no one else. Last Friday, a visitor from out of town told the mayor a new joke at 10:00 am. How long did it take for everyone in Humorville to head the joke?

Answer 1

At 7 hours from the visit telling the joke, everyone in town would have heard of it

Explanation:

Exponential Grow

This is a good example of exponential growth, where the speed at which a rumor is spread out depends on the actual number of persons who already know it.

The visitor told the mayor a new joke at 10:00 am.

Total people who heard the joke=1

This person tells the joke to 3 people in one hour, so at 11:00 am, 1+3=4 persons heard the joke

Total people who heard the joke=4

Those persons take one hour to tell the joke to every 3 persons each. Thus at 11:00

4 + 4*3 = 16 persons heard the joke. This succession grows very quickly. At 12:00

16 + 16*3 = 64 persons heard the joke

We can note the number of persons hearing the joke is an even power of 2, that is

`2^0=1`

`2^2=4`

`2^4=16`

`2^6=64`

We can predict the result for each hour since the exponent is double the number of hours passed since the joke started to spread. The number of persons who have heard the joke after t hours is

`N=2^(2t)`

We can iterate until we find the value of t so that

`2^(2t)<9800`

Let's better find the limit value of t

`2^(2t)=9800`

Taking logarithms

`log(2^(2t))=log9800`

`2tlog(2)=log(9800)`

Thus

`\displaystyle t=(log(9800))/(2log 2)`

`t=6.6`

So at 7 hours from the visit telling the joke (between 16:00 and 17:00), everyone in town would know it

Note that

`2^(12)=4096`

`2^(14)=16384`