Adapted from Connected Math 2: Growing, Growing, Growing Introduction to Exponential Growth and Decay Part 1: Exploring Exponential Growth Many single-celled organisms reproduce by dividing into two identical cells. Suppose an amoeba splits into two amoebas every half hour. a. An experiment starts with one amoeba. Complete the table to show the number of amoebas at the end of each hour over a 6 hour period. Amount of Number of Time (hr) Amoebas 0 1 1 2 3 4 5 6 b. How does the number of amoebas change from one hour to the next? c. Write an equation for the number of amoebas a after t hours. d. After how many hours will the number of amoebas reach one million? e. Suppose a new experiment starts with 2 amoebas. Make a table to show the number of amoebas at the end of each hour over a 6 hour period. Amount of Number of Time (hr) Amoebas 0 2 1 2 3 4 5 6 f. What do you notice about your two tables (what is alike and what is different)? Adapted from Connected Math 2: Growing, Growing, Growing g. Write an equation for the number of amoebas a after t hours. h. What do you notice about your two equations (what is alike and what is different)? i. Sketch a graph of each amoeba experiment on the same coordinate plane. j. An exponential function can be written in the form y = ab x, where x is the independent variable and y is the dependent variable. What do you think they b and the a represent? k. Explore the following statements that refer to exponential functions to determine if they are true or false and justify your answers. a0 True or False Why? b0 True or False Why? b1 True or False Why? Adapted from Connected Math 2: Growing, Growing, Growing Part 2: Exploring Exponential Decay After an animal receives a preventive flea medicine, the medicine breaks down in the animal’s bloodstream. With each hour, there is less medicine in the blood. The table and graph show the amount of medicine in a dog’s bloodstream each hour for 6 hours after receiving a 400-milligram dose. Breakdown of Medicine Breakdown of Medicine 450 Time Since Active 400 Dose (hr) Medicine in Blood (mg) A ct ive M e d icin e in B lo o d ( m g ) 350 0 400 300 250 1 100 Active Medicine 200 in Blood (mg) 2 25 150 3 6.25 100 4 1.5625 50 5 0.3907 0 0 1 2 3 4 5 6 6 0.0977 Time Since Dose (hr) Study the pattern of change in the graph and table. a. How does the amount of active medicine in the dog’s blood change from one hour to the next? b. Write an equation to model the relationship between the number of hours h since the dose is given and the milligrams of active medicine m. c. How is the graph for this problem similar to the graphs you created in part 1? How is it different? d. How is your equation similar and different to the ones you created in part 1? Adapted from Connected Math 2: Growing, Growing, Growing Part 3: Cooling Down Throughout the year we have explored the idea of thermal equilibrium. One of the ways we demonstrated this is by graphing the effect of leaving a very hot liquid exposed to a much cooler environment (room temperature). Below is a sample set of data from one of those experiments. Use what you have learned about exponential growth and decay to answer the questions that follow. Cooling Water Time Water Room Temp. Cooling Water (min) Temp. Temp. Diff. (C) (C) (C) 70 0 89 27 62 T e m p e r a tu r e D if fe r e n ce (C °) 60 5 71 27 44 50 10 59 27 32 40 15 52 27 25 20 47 27 20 30 25 43 27 16 20 30 40 27 13 10 35 37 27 10 0 40 35 27 8 1 2 3 4 5 6 7 8 9 10 11 12 13 45 33 27 6 Time (min) 50 32 27 5 55 31 27 4 60 30 27 3 a. Estimate the decay factor for the relationship between temperature difference and time in this experiment. b. Find an equation for the (time, temperature difference) data. Your equation should allow you to predict the difference at the end of any 5 minute interval. c. What do you think the graph of the (time, temperature difference) data would look like if you had continued the experiment for several more hours? d. What factors might affect the rate at which a cup of hot liquid cools?
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