# Introduction to Exponential Growth and Decay by jonathanscott

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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

a0         True or False      Why?

b0         True or False      Why?

b1         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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