Title
Spreading Rumors
Problem Statement
A rumor starts by
someone telling the rumor to two people on May 1. Each of those two people
are responsible for telling the rumor to two others on the next day (May
2).At this time seven people know the rumor. On May 3, the four people who
heard the rumor on May 2 must each tell two more people. It is important
for 8000 people to hear the rumor by May 14. Is this outcome likely to
occur if the pattern for spreading the rumor continues as started?
On what day will approximately onehalf of the 8000 have heard the rumor?
On what day will 256 new people be told the rumor? If the rumor process
continues until May 20,how many new people will
hear the rumor on that day?
If the rumor process continues for n days, how many new people will be told
the rumor on the nth day? What will be the total number of people who know
the rumor on that day?
Problem setup
I will need to set up Excel to
help me see how many people will spread the rumor each day. I can set up a formula where each
day, the number grows exponentially, by powers of 2.
Plans to
Solve/Investigate the Problem
As I started to work on the
Excel, and looked at the results, I noticed that each answer represented an
exponent of 2.
Investigation/Exploration
of the Problem
1May

2

2May

4

3May

8

4May

16

5May

32

6May

64

7May

128

8May

256

9May

512

10May

1024

11May

2048

12May

4096

13May

8192

14May

16384

15May

32768

16May

65536

17May

131072

18May

262144

19May

524288

20May

1048576

21May

2097152

22May

4194304

23May

8388608

24May

16777216

I can see that by Day 14,
more than 8,000 people will have heard the rumor. On Day 12, more than half the 8000
people will have heard the rumor (4096). On the 8^{th} day, exactly
256 people will have heard the rumor.
If the pattern continues until May 20^{th}, 1,048,576 people will have
heard the rumor. To solve for “n”
days, I would use the formula 2^{n }(for example, on the 4^{th}
day , 2^{4} people, or 2*2*2*2 or 16
people would have heard the rumor.
Extensions of the Problem
What would happen if 2 people each told 2 people per day? How much quicker would the 8000
people be told? What if 3
people each told 1 person per day?
Author & Contact
Meg Ramsey
mramsey@rockdale.k12.ga.us
U
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