1
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential; Task 3.3.4
TASK 3.3.4: EXPONENTIAL DECAY — NO BEANS ABOUT IT
A genie has paid you a visit and left a container of magic colored beans with
instructions. You are to locate the magic bean for your group. You will be given boxes
in which to place the beans. You will remove one third of the beans and place these in
box 1. Then, you will remove one third of the beans in box 1 and place these beans in
the second box. In the third box you will place one third of the beans from box 2. You
will continue this process until have only one bean. This is your magic bean. This bean
can be used to create more magic beans. When the genie left you with your instructions
the genie said only whole beans can be removed from the box.
1.
Suppose the container has 2700 beans. Complete the table. Remember you can only
remove whole beans, so if you get a decimal answer, use only the whole number
portion and do not round.
Table I
Box
Process
Number of Beans
1
1
2700*( )1
3
900
2
1
2700*( )2
3
300
3
1
2700*( )3
3
100
4
1
2700*( )4
3
33
5
1
2700*( )5
3
11
6
1
2700*( )6
3
3
7
…
n
1
2700*( )7
3
…
1
…
1
2700*( )n
3
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
2
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential; Task 3.3.4
2. Find a viewing window for the problem situation.
Sketch your graph:
3.
Identify your window:
Xmin:
1
Xmax:
8
Xscl:
1
Ymin:
0
Ymax:
900
Yscl:
1
Justify your window choices.
Sample answer: The variable x stands for the number of boxes, so one to eight
boxes appear reasonable. The variable y stands for the number of beans removed
and placed in the next box, so 0 through 900 will show the number beans for
boxes 1 through 7.
4.
What if the genie gave you 1 billion magic beans? Write a function for how many
beans will be in box n.
1
b = 1,000,000,000*( )n
3
5.
Instead of one billion beans, assume the genie gave you an original container with
16,200 beans. Using the same rule, in which box would you have 200 beans?
Box 4
1
16,200*( ) 1 = 5400
3
Box 1
1
16,200*( ) 2 = 1800
3
Box 2
1
16,200*( ) 3 = 600
3
1
16,200*( ) 4 = 200
3
Box 3
Box 4
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
3
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential; Task 3.3.4
6. a. Suppose you have a box with 5 beans and you know it is box number 10. How
2
many beans were in the original container if in the decay process you lose
3
2
each time? 295,245. To find the answer, note that losing means that the
3
1
box will contain of the original. Therefore you can reverse the process by
3
multiplying each time by 3..
5
15
45
135
405
1215
3645
10,935
32,805
98,415
295,245
Box 10
Box 9
Box 8
Box 7
Box 6
Box 5
Box 4
Box 3
Box 2
Box 1
Original Container
b. Write a function rule to find the number of beans in the original container
given the beans in the nth container.
b = 5*3 n
7. In these activities you have been working with exponential growth and
exponential decay. List some similarities between the two functions and list some
major differences.
Similarities:
In both functions, the operation is multiplication, ratios are constant, both written
the same way with a starting point, equations written the same, and have 1 as a
common point.
Differences:
Differences are that in exponential growth the multiplier is greater than 1 and in
exponential decay the multiplier is less than one. In exponential decay the
function is approaching a limit of zero. In exponential growth the function is
approaching positive infinity.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
4
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential; Task 3.3.4
TASK 3.3.4: EXPONENTIAL DECAY — NO BEANS ABOUT IT
A genie has paid you a visit and left a container of magic colored beans with
instructions. You are to locate the magic bean for your group. You will be given boxes
to place the beans. You will remove one third of the beans and place these in box 1.
Then, you will remove one third of the beans in box 1 and place these beans in the
second box. In the third box you will place one third of the beans from box 2. You will
continue this process until have only one bean. This is your magic bean. This bean can
be used to create more magic beans. When the genie left you with your instructions the
genie said only whole beans can be removed from the box.
1.
Suppose the container has 2700 beans. Complete the table. Remember you can only
remove whole beans, so if you get a decimal answer, use only the whole number
portion and do not round.
Table I
Box
Process
Number of Beans
…
…
1
2
3
4
5
6
7
…
n
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
5
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 3. Exponential; Task 3.3.4
2. Find a viewing window for the problem situation.
Sketch your graph:
Identify your window:
Xmin:
Xmax:
Xscl:
Ymin:
Ymax:
Yscl:
3.
Justify your window choices.
4. What if the genie gave you 1 billion magic beans? Write a function for how many
beans will be in box n.
5. Instead of one billion beans, assume the genie gave you an original container with
16,200 beans. Using the same rule, in which box would you have 200 beans?
6. a.
Suppose you have a box with 5 beans and you know it is box number 10. How
2
many beans were in the original container if in the decay process you lose
3
each time?
b. Write a function rule to find the number of beans in the original container
given the beans in the nth container.
7. In these activities you have been working with exponential growth an exponential
decay. List some similarities between the two functions and list some major
differences.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.