1. EXPONENTIAL RELATIONS
MBF3C
Lesson #4: Modelling Exponential Growth and Decay

2. Item Value
The value of an item increases by 2% each year. The graph shows the value of this item over time.
Suzanne says that the relation could be exponential, but it could also be quadratic. Do you agree? Explain.
Referring to the graph, after how many years was the value of the item $200?

3. Item Value
c) The item's value can be modelled by the relation C = 100(1.02)t, where C represents the value of the item in dollars and t represents the time in years. Use the relation. After how many years was the value of the item $200?
C = 100(1.02)t
C = 100(1.02)50
=269.16
C = 100(1.02)t
C = 100(1.02)48
=258.70
C = 100(1.02)t
C = 100(1.02)40
=220.80
C = 100(1.02)t
C = 100(1.02) =200

4. Item Value
d) Determine how long it took for the value of the item to reach $300. Use both the graph and the relation. How do the answers compare?
USING THE EQUATION:
C = 100(1.02)t
C = 100(1.02)56
=303.17
C = 100(1.02)55
=297.17
USING THE GRAPH, WE CAN SEE THAT WHEN THE VALUE OF THE ITEM REACHES $300, approximately 56 YEARS HAVE PASSED

5. Exponential Growth and Decay Word Problems
The general formula that can be used for all exponential growth and decay problems is:
where A represents the initial value
b represents the growth or decay rate
x represents the length of time
y=A(b)x

6. Exponential Growth: Two types of problems
eg. 1
Jason decides to grow bacteria for a living. The bacteria doubles in size every 3 minutes. If there are 1000 bacteria initially present, how many will there be in 15 minutes? The question involves doubling period.
The self-interest decision making is encouraged because we do a very bad job at "training" and educating people to look at the data. We do an even worse job at presenting the raw data for objective analysis. Instead, we are a nation and community of SPIN doctors. This causes people to argue from a position of belief rather than a position of knowledge.
Now, of course, the problem is made worse by the perception that we are all afraid of math and that "formulas" don't apply to real life.
"Math, formulas and other things that don't apply to real life"
(--anonymous comment from student evaluation as the part they liked least about the course)
So we live in a society that is afraid of and doesn't understand numbers.
This again is a recipe for disaster as it means the public can be sold most anything
Clearly, Exponential growth, in general, is not understood by the lay public. If exponential use of a resource is not accounted for in planning - disaster can happen.
The difference between linear growth and exponential growth is astonishing. Exponential growth means that some quantity grows by a fixed percentage rate from one year to the next.
In this animation, one can clearly see that no matter what the growth rate is, exponential growth stars out being in a period of slow growth and then quickly changes over to rapid growth with a characteristic doubling time.

7. Doubling Time and Half-Life
Exponential growth leads to repeated doubling. The time required for the quantity to double is called the doubling time.
Exponential decay leads to repeated halving. The time required for the quantity to diminish by ½ is called the half-life.

8. Doubling Time
Exponential growth leads to repeated doubling. The time required for the quantity to double is called the doubling time.
Examples
Doubling time for bacteria was 1 minute.
Doubles each week

9. Kendra won a contest. She will be paid a sum each week for 26 weeks
Kendra won a contest. She will be paid a sum each week for 26 weeks. The first week she will be paid is 1 cent. The amount doubles each week.
How much will she be paid in the eighth week?
How much will she be paid in the 20th week?
How much will she be paid in the last week (week 26)?

10. WHAT IS THE DOUBLING PERIOD?
Kendra won a contest. She will be paid a sum each week for 26 weeks. The first week she will be paid is 1 cent. The amount doubles each week.
How much will she be paid in the eighth week?
How much will she be paid in the 20th week?
How much will she be paid in the last week (week 26)?
WHAT IS THE DOUBLING PERIOD?

11. After 1 week the payout has doubled once. 0.01×21 = $0.02
After 2 weeks the payout has doubled twice.
0.01×22 = $0.04
After 3 weeks the payout has doubled three times.
0.01×23 = $0.08
After 4 weeks the payout has doubled four times.
0.01×24 = $0.16
After 20 weeks the payout has doubled twenty times.
0.01×220 = $
After 26 weeks the payout has doubled twenty-six times.
0.01×226 = $671,088.64

12. After 1 week the payout has doubled once. 0.01×21 = $0.02
After 2 weeks the payout has doubled twice.
0.01×22 = $0.04
After 3 weeks the payout has doubled three times.
0.01×23 = $0.08
After 4 weeks the payout has doubled four times.
0.01×24 = $0.16
After 20 weeks the payout has doubled twenty times.
0.01×220 = $
After 26 weeks the payout has doubled twenty-six times.
0.01×226 = $671,088.64
Exponent
1/1 = 1
2/1 = 2
3/1 = 3
4/1 = 4
20/1 = 20
26/1 = 26

13. M=Mo (𝟐) P=Po (2) DOUBLING TIME FORMULA y=A(2) P=I(𝑏)
𝑡 𝑑
New value
y=A(2)
Time
Doubling period
M=Mo (𝟐)
Rate of increase
Amount before increase
When the base of an exponential relation is 2, the relation is describing doubling time.
M final quantity
Mo initial quantity
t represents time
h represents the doubling time
𝟐 represents doubling
P=Po (2)
𝑡 𝒅
𝑡 ℎ
P=I(𝑏)
Terms indicating doubling time Doubles, Doubling time, Growing by a factor of 2

14. If we know doubling time, can we determine new values?
Consider an initial population of 10,000 with a doubling time of 10 years.
Years
Population
Exponent
After 10 years the population has doubled once.
10,000×21 = 20000
1/10 = 1
After 20 years the population has doubled twice.
10,000×22 = 40000
20/10 = 2
After 30 years the population has doubled three times.
10,000×23 = 80000
30/10 = 3
After t years the population has doubled t / 10 times.
10,000×2
t /10

15. Doubling time of population is approximately 14 years.

16. Exponential Growth: Two types of problems
eg. 2
Mrs. James is accumulating money in her Swiss bank account from her oil stocks investment. The stocks give her an interest rate of 30% per year. She's looking to retire in 22 years from now. If she invests dollars today how much money will she make when she retires? This question involves growth rate.

17. Terms indicating growth
Exponential Growth FORMULA
Exponential Growth: is growth when the quantity increases by the same percent in each unit of time.
Time
Amount after increase
Rate of increase
Amount before increase t
Terms indicating growth
Increases
Appreciates
Grows
Climbs
Matures
Rises
Soar
triples, doubles, quadruples, etc.
More???
P=I(1+b)
This growth function is used to find growth in population, interest, or any quantity that grows exponentially.

18. You deposit $500 in a bank account that pays 6% interest compounded yearly. Find the value of the account after 1 year, 2 years, 3 years, 20 years.
Example:
Use
C = r = t = 1,2,3,20
1 year
2 years
Step 1: Determine if it’s an example of exponential growth or decay and write the formula.
Step 2: Create an equation from the problem.
Step 3: Substitute values with information given (make sure the rate % is converted to a decimal)
Step 4: Simplify and solve (if doing by hand, make sure you use BEDMAS!)
3 years
20 years
y = 500( )
y = $
y = $
The longer you invest the more interest you earn!

19. Exponential Decay: Two types of problems
eg. 3
When Earnest Rutherford discovered radioactive decay of radium, he noticed it has a half life of 100 days. This means whatever mass of radium he has initially it will be half as much in 100 days. He starts with 320 mg of radium and stores it for 40 days. How much will be left after that time? This question involves half-life.

20. Terms indicating half-time half-life, half-time
HALF TIME TIME FORMULA
𝑡 ℎ
y=A( 1 2 )
New value
Time
When the base of an exponential relation is 1/2, the relation is describing half-time.
Half-life
Rate of decrease
Amount before increase
M=Mo ( 𝟏 𝟐 )
𝑡 𝒉
𝑡 ℎ
P=Po ( 𝟏 𝟐 )
𝑡 𝒉
P=I(𝑏)
M final quantity
Mo initial quantity
t represents time
h represents the half-life
𝟏 𝟐 represents half-life
The half life is the time it takes for a quantity to decay to half of its original size
Terms indicating half-time half-life, half-time

21. The world population doubles every 51. 8 years
The world population doubles every 51.8 years. At the beginning of the twentieth century, the world population was 1.6 billion. What was the population at the end of the twentieth century?
Time (in years)
Population
1.6 billion

22. Fran Data Date Sept. 6 1,159,000 Sept. 7 804,000 Sept. 8 515,000
Customers without Power
Sept. 6
1,159,000
Sept. 7
804,000
Sept. 8
515,000
Sept. 9
340,500
Sept. 10
195,200
Sept. 11
136,300
Sept. 12
77,000
Sept. 13
37,600

23. Exponential Decay: Two types of problems
eg. 4
Carly buys a new car for $20,000. It is a nice one, but the value depreciates at a rate of 6% per year. If she wants to sell her car in 8 years from now how much will it be worth? This question involves depreciation.
Terms indicating decay Loses, falls, depreciates, drops. declines, decays, regresses

24. EXPONENTIAL DECAY FORMULA
Exponential Decay: is decrease of a quantity by the same percent over a period of time.
Things that have exponential decay: medication, radioactive waste, bodies, car value(depreciation).
Time
*Remember, sometimes n is used to number of years.
Rate of decrease
Amount after decrease
Initial amount
before decrease
*Remember, sometimes r is used to denote this.
*Remember, sometimes P, M or C
is used to denote this.
*Remember, sometimes Io r Po is used to denote this.
Example:
A popular medication loses 23% of its effectiveness each hour. Find how much of the 200 mg of each pill is working after 5 hours.
Step 1: Determine if it’s an example of exponential growth or decay and write the formula.
Step 2: Create an equation from the problem.
Step 3: Substitute values with information given (make sure the rate % is converted to a decimal)
Step 4: Simplify and solve (if doing by hand, make sure you use BEDMAS!)
y = 200(.2706)
y = 54.13mg

25. Step 2: Create an equation from the problem.
The average new car loses 17% of its value each year. Find the value of a $20,000 car after 3 years.
Example:
Depreciation
Step 1: Determine if it’s an example of exponential growth or decay and write the formula.
Step 2: Create an equation from the problem.
Step 3: Substitute values with information given (make sure the rate % is converted to a decimal)
Step 4: Simplify and solve (if doing by hand, make sure you use BEDMAS!)
y = 20000(.5717)
y =$11,435.74

26. PROBLEMS RE: POPULATION DECAY OR GROWTH

27. Lesson 4 Success Criteria Up To now:
I can explain how the equation of an exponential function relates to exponential growth and decay
I can identify formulas for:
Doubling time
Exponential growth
Half-life
Exponential decay
I can use the equation that represents exponential growth and decay to solve problems.
I can convert a growth or decay rate from a percentage to a decimal.
I can tell the difference between an equation that is an example of exponential growth and exponential decay.

28. ONE MORE FOR THIS LESSON:
I can write an equation for an exponential growth or decay application

29. Example 1:
The number of caterpillars in a nest increases by 7.6% every 2 months. There are 750 caterpillars now. Write an equation for the caterpillar population P after m months.
Time
Amount after increase
Rate of increase
Amount before increase
P=750
(1+7.6) m
P=750(8.6) m

30. Example 2:
A bee hive population triples every 4 weeks. Initially there are 300 bees. Write an equation for the population P of bees after w weeks.
P=300(3)w/4

31. Example 3:
A population of 1500 mice decreases by 20% every day. Write an equation for the population of mice P after d days.
P=1500(0.80)d

32. Example 4:
The mass of a plant goes up 18% every 3 months. The current mass of the plant is 500 grams. Write an equation for the mass of the plant M after m months.
M=500(1.18)m/3

33. Example 5:
The population of Cambridge goes up 2% every year. In 2010 the population was 130 thousand. Write an equation that gives the population Cambridge in thousands, T, after y years since 2010.
T=130(1.02)y

34. Example 6: Which of these functions describe exponential decay? Explain

35. Example 7: In 1990, a sum of $100 is invested at a rate of 6% per year for 15 years.
(a) What is the growth rate (i.e. the value of "b")
(b) What is the initial amount?
(c) Write an equation that models the growth of the investment, and use it to determine the value of the investment after 15 years?

36. Example 8: In each case, write an equation that models the situation described. Explain what each part of the equation represents
a. The percent colour remaining if your blue jeans lose 1% of their original colour every time they are washed
b. The population if a town had 2500 residents in 1990 and grew at a rate of 0.5% each year after that forty years.
c. The population of a colony if a single bacterium doubles every day, what is the population P after t days.

37. Example 9
The Petersons purchased their house for $ in Since then, the value of their house has increased 5 % per year. State the value of the house in the year

38. A car purchased for $24,500 will depreciate at a
rate of 18% per year for the first 6 years. Write an
equation and graph over the first 6 years.
$30,000
$20,000
$10,000

39. $100 growth at 10% for 5 years. $100 depreciate at 10% for 5 years.
Compare Growth and Decay Models
Exponential Growth
Exponential Decay
$100 growth at 10%
for 5 years.
$100 depreciate at 10%
for 5 years.
160
140
120
100 80 60 40 20
160
140
120
100 80 60 40 20

40. Classify each as exponential growth or
exponential decay. 1) 4)
Exponential
Growth
Exponential
Decay 2) 5)
Exponential
Growth
Exponential
Decay 3) 6)
Exponential
Growth
Exponential
Growth

41. Find the value of a downtown office building that
cost 12 million dollars to build 20 years ago and
depreciated at 9% per year.