Lesson

2 Exponential Decay

LESSON OVERVIEW

Unit 6

This lesson engages students in analysis of situations where some quantity of interest decreases by some constant factor as each unit of time passes. Since the factor change for a function that decreases is a fraction between 0 and 1, exponential decay is likely to be a bit more challenging for students than the previous lesson on exponential growth. The experiment at the beginning of the lesson is designed to illustrate visually one of the common settings for exponential decay—removal or diffusion of some pollutant. You could derive a mathematical model for the pollution remaining after n days, or the change from one day to the next, but since there is an element of randomness involved in this process, it is probably better to begin looking for an algebraic rule with the other, more well-behaved, examples that follow. If you actually do this experiment you will get better results using large numbers of beans of two colors, removing two small cups at a time. Students do not actually have to count the beans which are removed. It is enough for students to understand that, if there are two cups of brown beans (the pollution) mixed with eight cups of white, then the two cups removed will subtract some but not all of the original brown beans. As you add two more small cups of white beans to compensate for the runoff, students can see that the proportion of brown beans looks less, so fewer will be removed with the next runoff. (If you actually count the brown beans removed each time, be sure to count the initial number first. In this way, you can track the pollution remaining in the container. These data are good to come back to after students know how to use exponential regression on the calculator.) To begin developing the arithmetic and algebraic rules that match exponential decay, we have chosen a simple example—the height of a bouncing ball after each rebound. The idea is that if a ball always rebounds to some fraction r (0
r
1) of its drop height, then the height of each succeeding bounce will be related to the prior bounce height by the equation NEXT  r  NOW. Repeated application of this relation leads to the exponential pattern that on bounce number n the ball will rebound to r n of its original height. Students might need some help to find this pattern, and especially to find a way to express it with algebraic rules. The example of drug decay illustrates a significant setting for exponential decay. Although we provide students with the rule in this case, it is important for them to see why the shape of that rule implies decay by repeated multiplication with the factor 0.95. By the end of this lesson, students should realize that there are decay analogs for the repeated multiplication models of Lesson 1 and that the rules take the same general form (with rate factors between zero and one). They also should have a sense of the shape of graphs that arise from rules of the form y  a(b x) when 0
b
1.

Lesson Objectives ■ To determine and explore the exponential decay model, y  a(b x), where

0
b
1, through tables, graphs, and algebraic rules ■ To compare exponential decay models with exponential growth models ■ To compare exponential models of the form y  a(b x), where 0
b
1, to linear models T439

UNIT 6

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EXPONENTIAL MODELS

LAUNCH

Master 166

full-class discussion

MASTER

Transparency Master

166 Think About This Situation The graphs below show two possible outcomes of the pollution and cleanup simulation. Pollutant Remaining

20 16 12 8 4 0 0

3

6

9

12

15

Days Since Spill

What pattern of change is shown by each graph? Which graph shows the pattern of change that you would expect for this situation? Test your idea by running the experiment several times and plotting the (time, pollutant remaining) data. What sort of equation relating pollution P and time t would you expect to match your plot of data? Test your idea using a graphing calculator or computer.

Use with page 440.

UNIT 6 • EXPONENTIAL MODELS

Unit 6

See Teaching Master 166. a The straight line shows a constant decrease in the pollutant remaining as the time increases. The curve shows large decreases initially, but the rate of decrease slows as time increases. b The curve is the pattern that would be expected. Repeat the experiment until the class sees that the curve is the correct graph. c The models for these graphs are y  4x  20 and y  20(0.8) x. Students may not have a good equation to model the cleanup situation, but you should leave the response open for now. You may wish to come back to it later in the unit. Students should realize that the graph should not be linear.

EXPLORE small-group investigation

INVESTIGATION

More Bounce to the Ounce

Many students are uncomfortable with fractions. As you circulate you may need to help them get started. 27
23
 18, 18

23
 . . . . Some will want to use their calculators to enter 27
0.66. Discourage this by pointing out that the factor 0.66 is rounded off. They could enter on their calculators 27
2  3  18, or 18
2  3  . . . , but recording answers in fraction form makes it a lot easier to see the familiar powers of two in some numerators and powers of three in the denominator. Of course, starting with 15 or some other whole number will result in the same pattern, but there may not be any powers of 2 readily apparent in the numerator. For some students, it may be appropriate to ask if we always will see the powers of 3 in the denominator.

LESSON 2

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E X P O N E N T I A L D E C AY

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EXPLORE continued 1. Bounce Number Rebound Height

0 27

1

2

18

12

3

4

5

6

7

8

9

10

8

16

3

32

9

64

27

128

81

256

243

512

729

1,024

2,187

WINDOW Xmin ⫽–1 Xmax ⫽11 Xscl ⫽2 Ymin ⫽–2 Ymax ⫽30 Yscl ⫽10 Xres ⫽1

Unit 6

a. It decreases by one-third on each bounce. The scatterplot is decreasing, but by a smaller amount each time. Students may not see the
13
. Adding a row “Change in Rebound Height” to the table may help. b. NEXT 
23
(NOW), starting at 27. x

c. y  27(
23
) d. The table values will be smaller; the plot will have the same shape but will not be as high, and the coefficient in the equation will be 15 instead of 27. 2. a–c. Responses will vary but should be close to a 0.67 rebound factor. NOTE: You may wish to use the Texas Instruments CBL or a similar piece of technology to measure the height and graph the relationship. Give the students plenty of time to practice how to measure the height with the technology. 3. A sample response using a tennis ball follows. a. Bounce Number

0

1

2

3

Rebound Height

50

30

18

10

b. NEXT  NOW  0.6, starting at 50. y  50(0.6) x

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UNIT 6

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EXPONENTIAL MODELS

Master 167

SHARE AND SUMMARIZE full-class discussion

MASTER

Transparency Master

167 Checkpoint

See Teaching Master 167. a ■ The rebound heights are always less by some constant factor. ■ The change is shown by a gradual decline in the height of the data points. b ■ The equations relating NOW and NEXT all have the same form: NEXT  b · NOW, starting at a. They are different because a and b may be different. This differences are probably due to the different materials from which the balls are made. ■ All of the equations have the form y  a(b x ), where a is equal to the original height of the ball, and the exponent (x) represents the number of bounces the ball has taken. The equations have different coefficients (a) because of the different initial heights. They also have different rates of rebounding (b) due to the type of ball. c The form is the same: NEXT  b · NOW starting at a, or y  a(b x). The patterns in this investigation are different only because b is a rational number less than 1 rather than an integer. When b  1, the curve is decreasing. The examples from Lesson 1 were all increasing curves.

Look back at the data from your two experiments. ■ How do the rebound heights change from one bounce to the next in

each case? ■ How is the pattern of change in rebound height shown by the shape

of the data plots in each case? List the equations relating NOW and NEXT and the rules (y = …) you found for predicting the rebound heights of each ball on successive bounces. ■ What do the equations relating NOW and NEXT bounce heights have

in common in each case? How, if at all, are those equations different and what might be causing the differences? ■ What do the rules beginning “y = …” have in common in each case?

How, if at all, are those equations different and what might be causing the differences? What do the tables, graphs, and equations in these examples have in common with those of the exponential growth examples in the beginning of this unit? How, if at all, are they different?

Be prepared to share and compare your data, models, and ideas with the rest of the class. Use with page 442.

UNIT 6 • EXPONENTIAL MODELS

Unit 6

When you are listening to students share their answers for the Checkpoint, you may want to remind them that the rumor grew much faster when the tree grew by a factor of 3 at each stage than when the factor was 2. Ask which will make the bounce heights decline faster, a bounce factor of
23
or a bounce factor of
25
(as in the upcoming “On Your Own”). Students will have to think about the results of multiplying by
23
or
25
: Which gives the smaller result? How can this be seen on the graph?

Different groups might have used different balls and dropped the balls from different initial heights. However, the patterns of (bounce number, rebound height) data should have some similar features.

APPLY individual task

n Your Own a. Bounce Number Rebound Height

0 25

1 10

2

3

4

5

4

8

5

16

25

32

125

WINDOW Xmin ⫽–1 Xmax ⫽6 Xscl ⫽1 Ymin ⫽–1 Ymax ⫽30 Yscl ⫽5 Xres ⫽1

b. NEXT 
25
(NOW ), starting at 25. y  25 (
25
)

x

LESSON 2

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E X P O N E N T I A L D E C AY

T442

APPLY continued c.

MORE ASSIGNMENT

pp. 448–454

Unit 6

Students can now begin Modeling Task 1 or Organizing Task 1 from the MORE assignment following Investigation 3.

■

The rebound height of the softball is consistently less than what would be expected from a new softball. Furthermore, the ratio of the rebound height from one drop to the next is not consistent. So the data may indicate that this softball is not top quality. ■ Responses may vary, depending on what height ratio the student chooses. A sample table of rebound heights for the ball dropped from 20 feet is provided here. For this table, the ratio from one bounce to the next is the same as the ratio for the corresponding bounces in the original table. Bounce Number

1

2

3

4

5

Rebound Height

7.6

3

1.2

0.4

0.1

d. y  20(
25
) x or NEXT  (
25
) (NOW ), starting at 20.

EXPLORE small-group investigation

INVESTIGATION

Sierpinski Carpets

It is apparently not as obvious to students as it is to teachers that if we cut out 1 square in 9 there will be
89
of the original carpet area left. Students may find the large hole in the center distracting. Even those who understand that
91
is removed may have trouble connecting this to
89
remaining. Perhaps it is the question “
89
of what?” that needs to be stated clearly. Expect to see more counting than seems necessary, until students convince themselves of the pattern. If students spend too much time coloring and counting, you might want to do this as a large group, though this still will leave some students unconvinced. 1.

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UNIT 6

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EXPONENTIAL MODELS

EXPLORE continued

Master 168 MASTER

2. a. b.



Checkpoint

8

9 8

9

This is of c.

Transparency Master

168

8

9 64

72

8

, 9

which is

64

81

Summarize the ways in which the table, graph, and equations for the Sierpinski carpet pattern are similar to, and different from, those for the following patterns:

of the original square meter of carpet.

the bouncing ball patterns of Investigation 1; the calling tree, king’s chessboard, and bacteria growth patterns of Lesson 1.

512 8



576 9

Be prepared to share your summaries of similarities and differences with the entire class.

512
of the original. This is
89
of
89
of
89
, which is
729 4,096 d. After 4 cutouts:
89

89

89

89


6,561

After 5 cutouts: 3.

5

(
89
)

32,768

59,049

NEXT  NOW

(
89
), starting at 1 a. After 10: (
89
)  0.31 m 2 b. After 20: (
89
)  0.09 m 2 After 30: (
89
)  0.03 m 2 y  (
89
)

Use with page 444.

UNIT 6 • EXPONENTIAL MODELS

Unit 6

10 20 30

4.

x

a.

X

Y1

X

Y1

0 1 2 3 4 5 6

1 .88889 .79012 .70233 .6243 .55493 .49327

4 5 6 7 8 9 10

.6243 .55493 .49327 .43846 .38974 .34644 .30795

X⫽0

+ + +

+

+

+

+

+

X⫽4

+ + +

b. 6 cutouts are needed in order to get more hole than carpet. 8 6 262,144



≈ 0.49 531,441 9

()

(

)

NOTE: Some groups may need help understanding that there is more hole than carpet remaining when y is less than 50%.

SHARE AND SUMMARIZE full-class discussion

See Teaching Master 168. a The patterns are the same. Slight differences are due to a larger value for the base in the Sierpinski problem and a smaller value for the coefficient. b The Sierpinski pattern represents an exponential decrease, whereas the problems in Lesson 1 represented an exponential increase. In one case, the base is less than one, and in the other it is greater than one. All of the graphs show a changing rate of change. Be sure students understand that exponential decrease occurs when the base is less than one, and exponential growth or increase occurs when the base is greater than one.

See additional Teaching Notes on page T481F.

LESSON 2

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E X P O N E N T I A L D E C AY

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APPLY individual task

n Your Own a. The cutout number is in L1 and the area of the remaining carpet is in L 2. L1

L2

0 1 2 3 4 5 6

9 8 7. 1 1 1 1 6.321 5.6187 4.9944 4.4394

L3

L2(1)⫽9

L1

L2

5 6 7 8 9 10

4.9944 4.4394 3.9462 3.5077 3.118 2.7715

L3

L2(12)⫽

b.

Unit 6

y

x

c. NEXT  NOW 
89
, starting at 9 x d. y  9 
89
e. Six cuts are needed. ASSIGNMENT pp. 448–454 ■ In a table of values, you can look for a y value that is less than 4.5. ■ In the plot of the data, you could use the trace function to determine when the value Students can now begin of y is less than 4.5. Modeling Task 2 or Reflecting Task 4 from the MORE assign- f. The data show the same patterns but these answers are all larger because the second carpet was larger. In the NOW-NEXT equation, we start at 9 instead of 1; the coefficient in ment following Investigation 3. the other equation is 9 instead of 1. The answer to Part e is still 6 cuts.

()

MORE

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UNIT 6

•

EXPONENTIAL MODELS

()

EXPLORE small-group investigation

Medicine and Mathematics

INVESTIGATION

Unit 6

Students often mix up how much insulin is lost every minute (5%) with how much is left (95%). As you circulate, be sure they are writing carefully what multiplying by 0.95 does. You may wish to ask questions such as the ones below. What variables are we tracking in the graph? (Amount of insulin remaining against time) What does the equation say about these variables? (If you get vague answers for this one, such as “It is decreasing by 95%,” you may be able to stir up some reaction if you deliberately misinterpret this. “The number of diabetics is decreasing by 95%? The amount of blood is decreasing by 95%? The amount of insulin is decreasing by 95%?” All of these are, of course, incorrect. The amount of insulin in the blood is decreasing by 5%, and the amount remaining is 95% of what it was, for every unit of time.) You may see a similar problem occurring in contexts which develop the concept of a gain of 5%, resulting in a next value of 105% of whatever the current value is. The implied 100% “start” may be the missing idea. 1. About 14 minutes 2. Approximate values of the data from the graph are in the table below. x

0

y

10

3

6

9

8.75

7.5

6.5

12 5.75

15 4.9

18 4.2

21 3.75

24

27

3

etc.

The equation fits the data fairly well but generally gives values a little less than those on the graph. 10 is the initial amount of insulin taken and 0.95 is the fraction of the original insulin left after 1 minute. 3. NEXT  NOW(0.95), starting at 10.

LESSON 2

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E X P O N E N T I A L D E C AY

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EXPLORE continued

Master 169 MASTER

Transparency Master

169 Checkpoint

In this unit, you have seen that patterns of exponential change can be modeled by equations of the form y ⫽ a(b x). What equation relates NOW and NEXT y values of this model? What does the value of a tell about the situation being modeled? About the tables and graphs of (x, y) values? What does the value of b tell about the situation being modeled? About the tables and graphs of (x, y) values? How is the information provided by values of a and b in exponential equations like y = a(b x) similar to, and different from, that provided by a and b in linear equations like y = a + bx?

Be prepared to compare your responses with those from other groups.

UNIT 6 • EXPONENTIAL MODELS

Unit 6

Use with page 447.

4. The calculations tell you both the amount left after a particular time, and that the amount left is continuously decaying. a. The amount of insulin left after a minute and a half is 10(0.95) 1.5, or about 9.3 units. b. The amount left after four and a half minutes is 10(0.95) 4.5, or about 7.9 units. c. The amount left after eighteen and three-quarters minutes is 10(0.95)18.75, or about 3.8 units. 5. a. Time in Minutes

0

1.5

4.5

7.5

10.5

13.5

16.5

19.5

Insulin in Blood

10

9.3

7.9

6.8

5.8

5.0

4.3

3.7

b. These values are close for the first few minutes but then are slightly lower than the values on the graph. c. After approximately 13.5 minutes, the insulin is half gone.

SHARE AND SUMMARIZE full-class discussion

See Teaching Master 169. a NEXT  NOW  b, starting at a.

CONSTRUCTING A MATH TOOLKIT: Students should add a summary of the Checkpoint responses for Parts a–c to their Toolkits (Teaching Master 200).

NOTE: Responses will vary, but students should be able to prove that it would have been difficult, if not impossible, for the toddler to drink enough (256 tsp) antifreeze Sunday morning for two teaspoons of ethylene glycol to be present in the youngster on Monday morning. T447

UNIT 6

•

b The value of a tells us the beginning value of y. In the table, it is the value of y when x equals zero, and it is the value of the y-intercept of the graph. c The value of b tells us how the values of y are changing. Each successive term can be obtained by multiplying the preceding term by b. Assuming the x values change by one, it is the constant ratio between two consecutive y values in the table. If b is greater than one, the graph and table values will be increasing; if b is less than one and greater than zero, the graph and table values will be decreasing. d In both linear and exponential equations, a provides the value of y when x is zero. In both types of equations the value of b will tell us if the values are increasing or decreasing. For linear equations, if b is greater than zero the y values will be increasing, but for exponential models the value of b must be greater than one for the values to be increasing. Also, linear models change by a constant amount (b), whereas exponential models change by a constant factor (b). JOURNAL ENTRY: Imagine that you are a medical examiner assigned to the following case. A toddler died mysteriously at 12:05 on a Monday afternoon. An autopsy revealed that the youngster’s system contained 2 teaspoons of ethylene glycol, the primary ingredient in antifreeze. When questioned by the police, the parents were certain that the child had not been left alone, except for a short time early the previous morning when the parents were folding the laundry, around 7 or 8 o’clock. After some mathematical calculations, you conclude that it was not likely that the child had drunk antifreeze while unattended. (The metabolic half-life of ethylene glycol is approximately 4 hours.) Consequently, you continue the investigation. Searching through a medical database, you find references to a rare disease that causes the body to produce a chemical similar to ethylene glycol. Using further tests, you conclude that this disease was indeed the cause of death. Write a report explaining how you came to the conclusion that it was unlikely that the child had drunk antifreeze. Include in your explanation any calculations you may have made.

EXPONENTIAL MODELS

APPLY individual task

n Your Own a. Time Amount left (mg)

12:00

1:00

2:00

3:00

4:00

5:00

300

180

108

64.8

38.88

23.33

b.

WINDOW Xmin ⫽–1 Xmax ⫽6 Xscl ⫽1 Ymin ⫽0 Ymax ⫽330 Yscl ⫽50 Xres ⫽1

X

Y1

X

Y1

X

Y1

0 .25 .5 .75 1 1.25 1.5

300 264.03 232.38 204.52 180 158.42 139.43

1.75 2 2.25 2.5 2.75 3 3.25

122.71 108 95.052 83.656 73.627 64.8 57.031

3.5 3.75 4 4.25 4.5 4.75 5

50.194 44.176 38.88 34.219 30.116 26.506 23.328

X⫽0

e.

X⫽1.75

Unit 6

c. d.

The plot shows that penicillin decays at a decreasing rate. y  300(0.6) x

X⫽3.5

The half-life of penicillin is between 1 hour 15 minutes and 1 and a half hours. The amount of penicillin left in the blood will be less than 10 mg after about 6.7 hours. 1

X=6.7021277

Y=9.7782833

WINDOW Xmin ⫽0 Xmax ⫽10 Xscl ⫽1 Ymin ⫽0 Ymax ⫽50 Yscl ⫽30 Xres ⫽1

LESSON 2

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E X P O N E N T I A L D E C AY

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