About four times a second, someone is born somewhere in the world (Waah!)
.
But people die at only about half that rate. This means that the overall
human population of planet earth is increasing by about 2 people per
second.
If the human population is increasing, that means that
the number of available parents (the "parent pool") is increasing. More
parents means more babies... which means that if four people are being
born every second now, soon that rate is going to be five per second,
six per second, ten per second, one hundred per second.... Scientists
say that the "rate" by which humans are being added to the earth is not
"linear" but rather that population is an example of an "exponentially
increasing" process. How can we characterize this type of process?
Well for starters, how do we characterize the growth of
"linear" systems?
 We can define the increase in size as a function of time in
terms of a "rate," that is, by an increase of a constant amount
per unit time.
 We can graph the population versus time (see for example the graphing
applet in the "Gas Particles in Motion" ASPIRE lesson). For a system
showing "linear" growth, the relationship between population and time
will be a straight line whose slope is given by the "rate".
In the following ASPIRE lab, we will study a system exhibiting exponential
growth and try to determine the quantities which characterize this system.
Discuss the following with your partner. Given your experience
with scientific notation (for example in the ASPIRE "Theorem Detectives" lab), what characteristics of a number written in scientific notation do you expect to be of
importance in analysing exponential growth? How might you use this to represent exponential growth graphically? (Hint: Think about the slope.)
Your mission (should you choose to accept it):
Use the lab setup on the computer, in which you will
analyze the growth rate of bacteria over time, to identify a way of
graphically interpreting the characteristics of systems exhibiting
exponential growth.
What you will see on the computer screen when you
start the lab is a petri dish which you can seed with a number of
bacteria.
 To start: Enter a number for the initial population of bacteria (choose a number between 1 and 100). Then enter a number for the growth parameter (choose a number between 1 and 5).
 To watch the bacteria colony grow, click the "Start Experiment" button.
 Click the "Record Data" button at various intervals to record the total number of bacteria in the dish as a function of time. You will have 15 hours on the watch to take 15 measurements.
 Write down the numbers in the table on the right on a sheet of paper.
 After taking 15 measurements, click the "Graph" button to see what your data looks like.
 To take another set of data, click the "Start Over" button.
 After completing several runs of the experiment, move on to the analysis questions below.
Study the data you have just collected from your trials.
 Describe the growth of the population as a function of
time, in one or more complete sentences. You might begin
with something like this: "As time elapses, the growth of
the population...".
 What are the relationships between the input parameters
(initial number of bacteria, growth parameter) and the
growth of the population with time?
 What is eventually the cause of the population growth
slowing down and stopping? Are there corresponding factors
at work in the rate of human population growth?
 Can you illustrate the growth of the population graphically?
See for example the graphing applet in the ASPIRE
"Gas
Particles in Motion" lab. Do you experience any difficulties
in plotting this data using conventional techniques? Can
y.ou think of any techniques that might make graphing such
data easier?
Here, we illustrate a different technique for graphing
data on exponential growth. Complete the tutorial below,
then graph the data collected in your bacterial growth
trials using this technique.
 Create a graph for each of your trials. Be sure to
record the initial conditions (initial number of
bacteria, growth parameter) along with your graphs.
 What relationship can you see between the starting
parameters and the shape of these curves? To what parameters
in a "linear" growth curve would these parameters correspond?

All of the curves "plateau" at about the
same point, regardless of the initial population and growth parameter.
What causes this plateauing? What factors might cause the plateau
to lie in a different place? Are there similar factors at work in
human populations.
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