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Exponential Growth and Decay Investigation

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```									                          Exponential Growth and Decay Investigation

Part I - Folding Paper Activity

This is an easy activity which illustrates both exponential growth and decay.
1. Fold a sheet of paper in half as many times as possible.
2. After each fold, record three columns of information in a chart as illustrated below.
Number of Folds (n)             Number of Regions Formed on           Area of Each Region Relative to
the Paper (R)                   the Whole Sheet of Paper (A)
½
1                                     2                     (The area of each region is half
the area of the whole sheet.)

2                                     4

3

4

5

6

3. Look for a pattern in the completed chart.
If n represents the number of folds, develop a formula for finding R, the number of regions formed on the
paper.
If n represents the number of folds, develop a formula for finding A, the area of region relative to the
whole sheet of paper.
Use these formulas to extend the chart to include information on more folds than are physically possible.
4. Make two graphs of the information; one with the number of regions on the y-axis and the number of
folds on the x-axis and, the other with the area of region on the y-axis and the number of folds on the
x-axis.
a) How are the two graphs similar? different?

b) Which of the two graphs represents an increasing function? a decreasing function?

c) Which of the two graphs represents an exponential growth? an exponential decay?

d) While the graph of area versus number of folds is constantly decreasing, why will it never reach the
x-axis?
Part II - Deriving an Exponential Growth Formula

1. Prerequisite Knowledge
Growth which can be related to an exponential function is referred to as exponential growth. Since cells
increase their numbers by duplicating themselves or doubling, their growth is exponential.
The doubling period is the amount of time that it takes for cells to duplicate themselves.
The number of doubling periods is the number of times that a cell culture doubles in a given amount of
time. For instance, if a bacteria strain doubles every 2 hours, the cell culture would double 5 times in 10
hours. Thus, the number of doubling periods in 10 hours is 5.

2. Complete the following chart to solve the following problem: If a cell culture doubles every 3 minutes and
there are 1000 bacteria present originally, how many will there be after 15 minutes?

Time Elapsed, t           Number of             Calculation of the Number of           Number of Cells
Doubling Periods         Cells Present After Time, t         Present After Time, t

3 minutes                    1                          2 (1000)                        2 (1000)

6 minutes                    2                        2 ( 2(1000) )                     22 (1000)

9 minutes                    3                       2 ( 22 (1000) )                    23 (1000)

12 minutes

15 minutes

3. Consider the completed chart above in answering the following questions.
Question 1: What is the relationship between the exponent to which 2 is raised in column 4 and the
number of doubling periods (column 2)?

Question 2: What is the relationship between the time elapsed and the number of doubling periods?

Question 3: Determine an expression that can be used to find the number of doubling periods by
answering the question "If a cell culture doubles every d minutes, how many cells
would be present after t minutes?"

4. Develop an exponential growth formula for the doubling of cells.

5. If a cell culture doubles every d minutes and there were No present originally, how many will there be
after t minutes?

Time Elapsed, t           Number of                Number of Cells                 Variables Defined
Doubling Periods         Present After Time, t

t minutes                                   N=              where     N = number of cells present
after time,t
t = time elapsed
d = doubling time
No = number of cells present
originally
Part III - Deriving an Exponential Decay Formula

1. Prerequisite Knowledge
Decay which can be related to an exponential function is referred to as exponential decay. Since the
amount of radioactivity in a given element decreases or halves over time, its decay is exponential.
The half life is the amount of time that it takes for a given element to decrease to half of its original
amount. The number of half life periods is the number of times that a cell culture halves in a given
amount of time. For instance, if an element has a half life of 15 years, the element would half 4 times in
60 years. Thus, the number of half life periods in 60 years is 4.

2. Complete the following chart to solve the following problem: The half life of strontium-90 is 25 years. If
5000 mg of strontium is present originally, find the mass present after 125 years.

Time            Number of Half              Calculation of the Mass of             Mass of Strontium-90
Elapsed, t         Life Periods           Strontium-90 Present After Time, t         Present After Time, t

25 years                1                             ½ (5000)                            ½ (5000)

50 years                2                          ½ ( ½ (5000) )                        (½) 2 (5000)

75 years                3                         ½ ( ½ 2 (5000) )                       (½) 3 (5000)

100 years

125 years

3. Consider the completed chart above in answering the following questions.
Question 1: What is the relationship between the exponent to which ½ is raised in column 4 and the
number of half life periods (column 2)?

Question 2: What is the relationship between the time elapsed and the number of half life periods?

Question 3: Determine an expression that can be used to find the number of half life periods by
answering the question, "If a radioactive element halves every h years, what mass
would remain after t years?"

4. Develop an exponential decay formula for the decay of a radioactive element. If a radioactive element
halves every h years and there was a mass of Mo present originally, what mass will remain after t years?

Time Elapsed, t           Number of               Mass of Radioactive
Doubling Periods          Element Remaining                 Variables Defined
After Time, t

t years                                     M=               where   M = mass present after t years
t = time elapsed in years
h = half life in years
Mo = original mass
Part IV - Exponential Decay Applet

1. Go to the exponential decay applet located at the site
http://lectureonline.cl.msu.edu/~mmp/applist/decay/decay.htm

2. Adjust the half life slider to determine how changing the half life affects the slope of the exponential
function. Set the half life to 5, 10 and 20 seconds being sure to click on the start key and then reset key
between each setting.
What happens to the slope of the exponential function curve (in green) as the half life increases?
Provide an explanation as to why this happens.

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