Team Members
I received this data set from Elizabeth Jones.
Materials
This lab requires a large group of people and a stopwatch. (Our data only used 20 people because that is our whole class. A larger group would provide more data points and a clearer picture of the function's behavior.)
Procedure
"It" is a contagious disease. The lab simulates the spread of the disease through a population. The experiment begins with one infected person at time zero. That person infects someone else by looking him in the eye and saying "Hi! My name is _____ and now you've got it." Newly infected people also begin to spread the disease in the same manner while the originally infected person continues spreading the disease as well. At regular intervals (we used 10 seconds), the simulation was interrupted to count the number of infected people. The process continues until the entire group is infected.
Data Set
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Scatter Plot
Analysis of Data
Let x equal the time elapsed (in seconds). Let y = f(x) equal the number of people infected after x seconds.
Two factors suggest that our data represents an exponential function.
1) The shape of the graph indicates an exponential function because of a y-intercept at 1, sharply increasing function values, and sharply increasing rate of increase in function values.
2) The spread of a disease is a type of natural, physical growth
of an infected population that exemplifies the behavior of a base e exponential
function, according to our previous work with exponential and logarithmic
function.
This regression curve seems to be a good fit visually and according to the correlation value.
Conclusion
Above I supported my hypothesis that our function is exponential with my intuition (or more likely memory from precalculus) that disease spreading or growth models exponential growth.
Why is this true? I was able to convince myself of the exponential relationship
with an informal comparison to the linear Pass the Secret Function. If
one person can pass along a secret in m seconds, then increasing the number
of people by 1 increases the time to pass along the secret by m seconds.
The key factor is only one person is passing along the secret at any given
time. In constrast Now You’ve Got It increases the number of people actually
spreading the disease along with the number of infected people. Suppose
one person can infect c people during a 10 second interval. Then increasing
the elapsed time by one 10 second interval increases the number of people
by c for each person previously infected.
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Although this explanation allows use to introduce an exponent, it is still too simplistic because we are considering discreet increases in the infected population. It fails to take into account that as soon as someone catches the disease, she can begin to spread it even before the next ten second interval begins. So increasing the elapsed time by one 10 second interval increases the number of people by c for each person previously infected AND by some number less than c for each of the people being infected and beginning to spread the disease during that additional 10 second interval. This process of continuous (rather than discreet ) growth of the spreaders as well as the infected people introduces the base e.
Because of the rapid increase of the exponential function we can predict that even a small extension of the domain would produce large increase in range values. In particular after only 40 seconds, our regression equation shows that approximately 216 people would be infected!