VIEWS: 248 PAGES: 4 POSTED ON: 2/15/2011
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.