9.4 exponential growth and decay...
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Section 9.4 Exponential Growth and Decay Functions 561
9.4 Exponential Growth and Decay Functions
OBJECTIVES
1 Model Exponential Growth.
2 Model Exponential Decay.
Now that we can graph exponential functions, let’s learn about exponential growth and exponential decay.
A quantity that grows or decays by the same percent at regular time periods is said to have exponential growth or exponential decay. There are many real-life examples of exponential growth and decay, such as population, bacteria, viruses, and radioactive substances, just to name a few.
Recall the graphs of exponential functions.
f(x) � bx,for b � 1
(0, 1)(1, b)
For b � 1
Exponential Functionsf(x) � bx
Increasing(from left to right)
Exponential Growth
Decreasing(from left to right)
Exponential Decay
For 0 � b � 1
f(x) � bx,for 0 � b � 1
(0, 1) (1, b)
y y
xx
OBJECTIVE
1 Modeling Exponential Growth We begin with exponential growth, as described below.
y = C11 + r2x
Exponential Growth initial
amountnumber of time intervals
11 + r2 is growth factor r is growth rate (often a percent)
d
Qr
Q
EXAMPLE 1 In 1995, let’s suppose a town named Jackson had a population of
15,500 and was consistently increasing by 10% per year. If this yearly increase contin-ues, predict the city’s population in 2015. (Round to the nearest whole.)
Solution: Let’s begin to understand by calculating the city’s population each year:
Time Interval x = 1 x = 2 3 4 5 and so on …
Year 1996 1997 1998 1999 2000
Population 17,050 18,755 20,631 22,694 24,963
c c 15,500 + 0.10115,5002 17,050 + 0.10117,0502 This is an example of exponential growth, so let’s use our formula with
C = 15,500; r = 0.10; x = 2015 - 1995 = 20
Then,
y = C11 + r2x
= 15,50011 + 0.10220
= 15,50011.1220
� 104,276(Continued on next page)
562 CHAPTER 9 Exponential and Logarithmic Functions
In 2015, we predict the population of Jackson to be 104,276.
1020304050607080
100
120
150140
90
110
130
0 2 4 6 8 10 12 14 16 18 20Year (since 1995)
Popu
lati
on (
in th
ousa
nds)
0 2422
(20, 104.276)
(0, 15.5)
PRACTICE
1 In 2000, the town of Jackson (from Example 1) had a population of 25,000 and started consistently increasing by 12% per year. If this yearly increase continues, predict the city’s population in 2015. Round to the nearest whole.
Note: The exponential growth formula, y = C11 + r2x, should remind you of the compound interest formula from the previous section, A = P11 + r
n2nt. In fact, if the number of compoundings per year, n, is 1, the interest formula becomes A = P11 + r2t, which is the exponential growth formula written with different variables.
OBJECTIVE
2 Modeling Exponential Decay Now let’s study exponential decay.
y = C11 - r2x
Exponential Decay
initial amount
number of time intervals
11 - r2 is decay factor r is decay rate (often a percent)
d Qr
Q
EXAMPLE 2 A large golf country club holds a singles tournament each year.
At the start of the tournament for a particular year, there are 512 players. After each round, half the players are eliminated. How many players remain after 6 rounds?
Solution: This is an example of exponential decay.
Let’s begin to understand by calculating the number of players after a few rounds.
Round (same as interval) 1 2 3 4 and so on …
Players (at end of round) 256 128 64 32
512 - 0.5015122 256 - 0.5012562 c c
Here, C = 512; r = 12
or 50, = 0.50; x = 6
Thus,
y = 51211 - 0.5026
= 51210.5026
= 8
Section 9.4 Exponential Growth and Decay Functions 563
After 6 rounds, there are 8 players remaining.
50
150100
200250300350400450500550600650
0 1 2 3 54 6 7
(0, 512)
(6, 8)
8Rounds
Num
ber
of p
laye
rs
0
PRACTICE
2 A tournament with 800 persons is played so that after each round, the num-ber of players decreases by 30%. Find the number of players after round 9. Round your answer to the nearest whole.
The half-life of a substance is the amount of time it takes for half of the substance to decay.
original amount decay rate
EXAMPLE 3 A form of DDT pesticide (banned in 1972) has a half-life of
approximately 15 years. If a storage unit had 400 pounds of DDT, find how much DDT is remaining after 72 years. Round to the nearest tenth of a pound.
Solution: Here, we need to be careful because each time interval is 15 years, the half-life.
Time Interval 1 2 3 4 5 and so on …
Years Passed 15 2 # 15 = 30 45 60 75
Pounds of DDT 200 100 50 25 12.5
From the table, we see that after 72 years, between 4 and 5 intervals, there should be between 12.5 and 25 pounds of DDT remaining.
Let’s calculate x, the number of time intervals.
x = 72 1years2
15 1half@life2 = 4.8
Now, using our exponential decay formula and the definition of half-life, for each time
interval x, the decay rate r is 12
or 50% or 0.50.
y = 40011 - 0.5024.8:
y = 40010.5024.8
y � 14.4
PRACTICE
In 72 years, 14.4 pounds of DDT remain.
3 Use the information from Example 3 and calculate how much of a 500-gram sample of DDT will remain after 51 years. Round to the nearest tenth of a gram.
time intervals for 72 years
564 CHAPTER 9 Exponential and Logarithmic Functions
Vocabulary, Readiness & Video Check
Martin-Gay Interactive Videos
See Video 9.4
OBJECTIVE
1
OBJECTIVE
2
OBJECTIVE
2
Watch the section lecture video and answer the following questions.
1. Example 1 reviews exponential growth. Explain how you find the growth rate and the correct number of time intervals.
2. Explain how you know that Example 2 has to do with exponential decay, and not exponential growth.
3. For Example 3, which has to do with half-life, explain how to calcu-late the number of time intervals. Also, what is the decay rate for half-life and why?
9.4 Exercise Set
Practice using the exponential growth formula by completing the table below. Round final amounts to the nearest whole. See Example 1.
Original Amount
Growth Rate per
Year
Number
of Years, x
Final Amount after x Years of Growth
305 5% 8
402 7% 5
2000 11% 41
1000 47% 19
17 29% 28
29 61% 12
1.
2.
3.
4.
5.
6.
Practice using the exponential decay formula by completing the table below. Round final amounts to the nearest whole. See Example 2.
Original Amount
Decay Rate per
Year
Number
of Years, x
Final Amount after x Years
of Decay
305 5% 8
402 7% 5
10,000 12% 15
15,000 16% 11
207,000 32% 25
325,000 29% 31
7.
8.
9.
10.
11.
12.
MIXED PRACTICE
Solve. Unless noted otherwise, round answers to the nearest whole. See Examples 1 and 2.
13. Suppose a city with population 500,000 has been growing at a rate of 3% per year. If this rate continues, find the popula-tion of this city in 12 years.
14. Suppose a city with population 320,000 has been growing at a rate of 4% per year. If this rate continues, find the popula-tion of this city in 20 years.
15. The number of employees for a certain company has been decreasing each year by 5%. If the company currently has 640 employees and this rate continues, find the number of employees in 10 years.
16. The number of students attending summer school at a local community college has been decreasing each year by 7%. If 984 students currently attend summer school and this rate continues, find the number of students attending summer school in 5 years.
17. National Park Service personnel are trying to increase the size of the bison population of Theodore Roosevelt National Park. If 260 bison currently live in the park, and if the population’s rate of growth is 2.5% annually, find how many bison there should be in 10 years.
18. The size of the rat population of a wharf area grows at a rate of 8% monthly. If there are 200 rats in January, find how many rats should be expected by next January.
19. A rare isotope of a nuclear material is very unstable, decaying at a rate of 15% each second. Find how much isotope remains 10 seconds after 5 grams of the isotope is created.
20. An accidental spill of 75 grams of radioactive material in a local stream has led to the presence of radioactive debris decaying at a rate of 4% each day. Find how much debris still remains after 14 days.
Section 9.5 Logarithmic Functions 565
Practice using the exponential decay formula with half-lives by completing the table below. The first row has been completed for you. See Example 3.
Original Amount
Half-Life (in years)
Number of Years
Time Intervals, x = a YearsHalf@Life
bRounded to Tenths if Needed
Final Amount after x Time Intervals (rounded to
tenths)Is Your Final Amount
Reasonable?
60 8 10108
= 1.25 25.2 yes
a. 40 7 14
b. 40 7 11
a. 200 12 36
b. 200 12 40
21 152 500
35 119 500
21.
22.
23.
24.
Solve. Round answers to the nearest tenth. CONCEPT EXTENSIONS
31. An item is on sale for 40% off its original price. If it is then marked down an additional 60%, does this mean the item is free? Discuss why or why not.
32. Uranium U-232 has a half-life of 72 years. What eventually happens to a 10 gram sample? Does it ever completely decay and disappear? Discuss why or why not.
25. A form of nickel has a half-life of 96 years. How much of a 30-gram sample is left after 250 years?
26. A form of uranium has a half-life of 72 years. How much of a 100-gram sample is left after 500 years?
REVIEW AND PREVIEWBy inspection, find the value for x that makes each statement true. See Sections 5.1 and 9.3.
27. 2x = 8 28. 3x = 9 29. 5x =15
30. 4x = 1
OBJECTIVE
9.5 Logarithmic Functions
OBJECTIVES
1 Write Exponential Equations with Logarithmic Notation and Write Logarithmic Equations with Exponential Notation.
2 Solve Logarithmic Equations by Using Exponential Notation.
3 Identify and Graph Logarithmic Functions.
1 Using Logarithmic Notation Since the exponential function f 1x2 = 2x is a one-to-one function, it has an inverse.
We can create a table of values for f -1 by switching the coordinates in the accom-panying table of values for f 1x2 = 2x.
x y � f1x2-3
18
-214
-112
0 1
1 2
2 4
3 8
x y � f �11x218
-3
14
-2
12
-1
1 0
2 1
4 2
8 3
f �1(x)
f(x) � 2x
y � x
1�1�2�3�4�5
12345
2 3 4 5�1�2�3�4�5 x
y