Documents
Resources
Learning Center
Upload
Plans & pricing Sign in
Sign Out

Exponential Growth and Decay Investigation

VIEWS: 248 PAGES: 4

									                          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.
5. Answer the following questions
   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.

								
To top