Download Doubling Time in Exponential Growth

GSI Overview Student Notes:
What Influences our perceptions and Inferences?
What is wrong with this map? Explain.
How many Prongs do you see? Explain
What do you see?
What Influences our perceptions and Inferences?
Inference Activity
Read the following story and then indicate your response to each of the statements printed below the story.
A business man had just turned off the lights in the store when a man appeared and demanded money. The owner
opened the cash register. The contents of the cash register were scooped up and the man sped away. A member of the
police force was notified promptly.
Indicate your response to each of the following statements, by checking "True" if you believe it to be true, "False" if you
believe it to be false, and "???" if you cannot determine whether it is true or false (True False ???)
1.______
2.______
3.______
4.______
5.______
6.______
7.______
8.______
A man appeared after the owner had turned off the store lights.
A man received cash.
The robber was a man.
The man who opened the cash register was the owner.
The store owner scooped up the contents of the cash register.
Someone opened a cash register.
The cash register was empty when the owner opened it.
After the man scooped up the contents of the cash register, he ran away.
What is the role of Science?
Can science data be misused? Explain
What is Exponential Growth?
Complete Activity – Doubling Time
As the human population grows what might be the impact on
(1) resources use and waste
(2) poverty
(3) loss of biodiversity
(4) Global Climate Change
Doubling Time in Exponential Growth
PURPOSE


Investigate the mathematical concept of exponential growth, applying doubling time as a calculation method
Explore the impacts of exponential growth in biological and other processes
INTRODUCTION
Growing populations of organisms do not follow linear rates of change. One reason populations grow very rapidly is that they have
higher birth rates than death rates. Each cycle of reproduction has more offspring than the previous generation. At any point there
are more maturing producers than ever before and the increase in the base population accelerates. Mathematically, such growth is
called exponential. It is the same type of rate as describes compounding interest in a bank account. While the rate is fixed and may
be a small percentage, it is continually applied to a growing base, so that the total expands by a greater and greater amount per unit
of time.
The time in which a population or money amount doubles is a good benchmark by which to grasp and foresee the impact of
exponential growth over time. For even the smallest rate of steady growth leads eventually to doubling and redoubling. While
exponential growth in one's investments is welcome, when applied to populations, especially human populations, it can have grave
implications. Many people do not have a good grasp of exponential rates. The following two exercises will illustrate the powerful
effects of exponential growth when it is modeled as a process of doubling, or repeatedly multiplying by two.
Materials
• calculator
• encyclopedias or other sources of global resource data
Problem A
A math major is home for a vacation break and takes a job for thirty days. In negotiating for a salary, she tells her employer that
instead of a wage of $20/hr, she would accept one that pays one penny the first day, then doubles to two cents the next day, four
cents the third day, and so on for the month, The employer thinks that this is a good deal for him and agrees.
Show your work, including intermediate calculations.
1. Is this deal a good one for the boss? If so, under what conditions?
2. How is this a good deal for the math major?
3. When does the student break even—that is, on what day has she made as much as she would have earning $20
per hour?
4.
What is the total differential in the two payment methods over the 30-day period?
5.
Define exponential growth. Explain why it is so powerful.
6.
Describe an example of exponential growth in another field, such as science.
7.
Explain what external factors might put limits on this type of mathematical increase.
Doubling Time
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Problem B
Under ideal conditions some common bacteria can divide and double their numbers in less than one-half hour. Suppose one spring
day at 6 A.M., a few such bacteria fall into a can of strawberry syrup in a broken garbage bag behind a snack bar. These conditions—
warmth, moisture, and lots of food—are perfect for growth, and the population doubles every 20 minutes. But by 6 P.M. the
bacteria are overcrowded and dry and their food is gone,
As you will discover in your calculations, this story about bacteria dramatizes the uncertain state of our natural resources, even in
times of perceptible abundance.
Show Your Work. Explain any assumptions you make
1.
At what time did the can of syrup become half full?
2.
At one point during the day some forward-thinking bacteria get the idea that they are facing a crisis. Their
numbers are growing exponentially and they are using up their space and food at an ever-increasing rate. At what
time do you think that idea would come? Explain.
3.
Why would awareness of the crisis not occur before 5 P.M.? How much food remains at that time? (Imagine
hearing the bacteria politicians saying: "Don't worry, we still have 3/4 of our resources. We have more food than
we have ever eaten since we got here.")
4.
In spite of the rhetoric, a few bacteria search for more food and space. They find three more syrup cans. How
much of a time reprieve are the bacteria given by this find? When will the new cans be depleted?
5.
Suppose the global human population growth rate is about 1.3% annually. How long does it take for the human
population to double?
6.
Given your response to Question 5 and your research into Earth's natural resources, how far along are we in terms
of Earth's carrying capacity for humans? Briefly describe the kinds of factors to consider.
7.
Extra Credit - Thomas Malthus wrote An Essay on the Principle of Population in 1798. It had a deep effect on
history, influencing Darwin's ideas of evolution. Research this work.
a. What did Thomas Malthus have to say on the relationship between population growth
and our ability to grow food?
b. Mathematically, Malthus's thesis had a valid basis. Outline three reasons why Malthus'
predictions of 1798 have not come true.
c. Could his predictions come true in the future? Briefly explain.
8. To many of us, Earth does not seem crowded. There are vast, undeveloped areas even in the United
States. Explain what "part of the can" is left for us, compared to the bacteria.
At this time humans do not have the option of finding "other cans." Earth is all we have. The writer and
inventor Buckminster Fuller called it Spaceship Earth, saying we are part of a six billion-member crew
flying through an isolated region of space. If we get into trouble, there is no one to help us.
9.
Describe three actions you can take as an individual to help us avoid the fate of the
bacteria in the first can.
Complete and Explain the Following Diagram:
Living on Interest v. Living on Principal
The Story of Stuff
Introduction:
“Before”
Answer
Question
“After”
Answer
Of the 100 largest economies on the planet,
______% are governments.
Of the 100 largest economies on the planet,
______% are companies.
Extraction:
In the past 30 years, _____% of the world’s natural resources have been
consumed.
The US has less than _____% of the original forests left.
In the US, _____% of the waterways are now undrinkable
The US has 5% of the world’s population, but uses _____% of the world’s
resources, produces _____% of the world’s waste.
If everyone consumed at US rates, we would need _____ planets to support
all of us.
_____% of global fisheries are fished at or beyond capacity
_____% of the planet’s original forests are gone.
In the Amazon forest alone, we are losing _____ trees every minute
Production:
_____ synthetic chemicals are used in manufacturing today.
_____ of these chemicals have been tested for synergistic health impacts
BFR’s are chemicals used to make things flame resistant. They are also very
toxic to the _____.
The food with the highest level of many toxic contaminants is _____.
The people who get the most exposure to toxic chemicals are the _____
Globally, _____ people per day are moving into cities.
US industry releases _____ pounds of toxic chemicals per year.
Distribution:
T/F
T/F
Externalized cost of production includes: loss of natural resources, making of
pollution, low wages and lack of health benefits for workers.
We don’t always pay the true cost to produce and transport goods.
T/F
T/F
Consumption:
T/F
T/F
What percent of total material goods are still in use 6 months after they are
sold?
Today we consume ____ as much as Americans did 50 years ago.
Some companies design stuff to intentionally break quickly
Many of us throw away perfectly good things due to advertising.
We see an average of _____ advertisements per day
Today we have more stuff than ever before, but national _____ is declining.
Name the top two ways Americans spend their leisure time.
T/F
T/F
The average house size has doubled since _____
Disposal:
Each person in the US throws away approximately _____ pounds of garbage
every day.
That’s twice the amount of garbage we threw away _____ years ago.
Name two options for getting rid of waste.
The #1 source of Dioxin, the most toxic man made substance known to
science, is…
For every 1 garbage can full of waste removed from our house, _____ cans
were made “upstream.”
From Another Way… 10 Little and Big Things You Can Do, list five things you personally can do to promote sustainability: