Section 8ASection 8AGrowth: Linear vs. Growth: Linear vs.

ExponentialExponentialPages 490-495Pages 490-495

Growth: Linear vs ExponentialGrowth: Linear vs Exponential

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Imagine two communities, Straightown and Imagine two communities, Straightown and Powertown, each with an initial population Powertown, each with an initial population of 10,000 people. Straightown grows at a of 10,000 people. Straightown grows at a constant rate of 500 people per year. constant rate of 500 people per year. Powertown grows at a constant rate of 5% Powertown grows at a constant rate of 5% per year. per year.

Compare the population growth of Compare the population growth of Straightown and Powertown.Straightown and Powertown.

Year Straightown

00 10,00010,000

11 10,50010,500

22

33

1010

1515

2020

4040

Straightown: initially 10,000 people and growing at a rate of 500 people per year

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Year Straightown

00 10,00010,000

11 10,50010,500

22 11,00011,000

33

1010

1515

2020

4040

Straightown: initially 10,000 people and growing at a rate of 500 people per year

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Year Straightown

00 10,00010,000

11 10,50010,500

22 11,00011,000

33 11,50011,500

1010

1515

2020

4040

Straightown: initially 10,000 people and growing at a rate of 500 people per year

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Straightown: initially 10,000 people and growing at a rate of 500 people per year

Year Straightown

00 10,00010,000

11 10,50010,500

22 11,00011,000

33 11,50011,500

1010 1000010000 ++ ( (1010x500) x500) ==1500015000

1515

2020

4040

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Straightown: initially 10,000 people and growing at a rate of 500 people per year

Year Straightown

00 10,00010,000

11 10,50010,500

22 11,00011,000

33 11,50011,500

1010 10000 + (10x500) 10000 + (10x500) =15000=15000

1515 10000 +10000 + ( (1515x500) x500) =17500=17500

2020

4040

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Straightown: initially 10,000 people and growing at a rate of 500 people per year

Year Straightown

00 10,00010,000

11 10,50010,500

22 11,00011,000

33 11,50011,500

1010 10000 + (10x500) 10000 + (10x500) =15000=15000

1515 10000 + (15x500) 10000 + (15x500) =17500=17500

2020 10000 +10000 + ( (2020x500) x500) ==2000020000

4040

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Straightown: initially 10,000 people and growing at a rate of 500 people per year

Year Straightown

00 10,00010,000

11 10,50010,500

22 11,00011,000

33 11,50011,500

1010 10000 + (10x500) 10000 + (10x500) =15000=15000

1515 10000 + (15x500) 10000 + (15x500) =17500=17500

2020 10000 + (20x500) 10000 + (20x500) =20000=20000

4040 10000 +10000 + ( (4040x500) x500) ==3000030000

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Powertown: initially 10,000 people and growing at a rate of 5% per year

Year Powertown

00 10,00010,000

11 10000 x (10000 x (1.051.05) = ) = 10,50010,500

22

33

1010

1515

2020

4040

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Powertown: initially 10,000 people and growing at a rate of 5% per year

Year Powertown

00 10,00010,000

11 10000 x (1.05) = 10,50010000 x (1.05) = 10,500

22 10000 x (10000 x (1.051.05))22 = = 11,02511,025

33

1010

1515

2020

4040

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Powertown: initially 10,000 people and growing at a rate of 5% per year

Year Powertown

00 10,00010,000

11 10000 x (1.05) = 10,50010000 x (1.05) = 10,500

22 10000 x (1.05)10000 x (1.05)22 = 11,025 = 11,025

33 10000 x (10000 x (1.051.05))33 = = 11,57611,576

1010

1515

2020

4040

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Powertown: initially 10,000 people and growing at a rate of 5% per year

Year Powertown

00 10,00010,000

11 10000 x (1.05) = 10,50010000 x (1.05) = 10,500

22 10000 x (1.05)10000 x (1.05)22 = 11,025 = 11,025

33 10000 x (1.05)10000 x (1.05)33 = 11,576 = 11,576

1010 10000 x (10000 x (1.051.05))1010 = = 16,28916,289

1515

2020

4040

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Powertown: initially 10,000 people and growing at a rate of 5% per year

Year Powertown

00 10,00010,000

11 10000 x (1.05) = 10,50010000 x (1.05) = 10,500

22 10000 x (1.05)10000 x (1.05)22 = 11,025 = 11,025

33 10000 x (1.05)10000 x (1.05)33 = 11,576 = 11,576

1010 10000 x (1.05)10000 x (1.05)1010 = = 16,28916,289

1515 10000 x (10000 x (1.051.05))1515 = = 20,78920,789

2020

4040

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Powertown: initially 10,000 people and growing at a rate of 5% per year

Year Powertown

00 10,00010,000

11 10000 x (1.05) = 10,50010000 x (1.05) = 10,500

22 10000 x (1.05)10000 x (1.05)22 = 11,025 = 11,025

33 10000 x (1.05)10000 x (1.05)33 = 11,576 = 11,576

1010 10000 x (1.05)10000 x (1.05)1010 = = 16,28916,289

1515 10000 x (1.05)10000 x (1.05)1515 = = 20,78920,789

2020 10000 x (10000 x (1.051.05))2020 = = 26,53326,533

4040

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Powertown: initially 10,000 people and growing at a rate of 5% per year

Year Powertown

00 10,00010,000

11 10000 x (1.05) = 10,50010000 x (1.05) = 10,500

22 10000 x (1.05)10000 x (1.05)22 = 11,025 = 11,025

33 10000 x (1.05)10000 x (1.05)33 = 11,576 = 11,576

1010 10000 x (1.05)10000 x (1.05)1010 = = 16,28916,289

1515 10000 x (1.05)10000 x (1.05)1515 = = 20,78920,789

2020 10000 x (1.05)10000 x (1.05)2020 = = 26,533 26,533

4040 10000 x (10000 x (1.051.05))4040 = = 70,40070,400

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Population Comparison Population Comparison Year

Straightown

11 10,50010,500

22 11,00011,000

33 11,50011,500

1010 15,00015,000

1515 17,50017,500

2020 20,00020,000

4040 30,00030,000

Powertown

10,50010,500

11,02511,025

11,57611,576

16,28916,289

20,78920,789

26,53326,533

70,40070,400

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Growth: Linear versus ExponentialGrowth: Linear versus Exponential8-A

Two Basic Growth PatternsTwo Basic Growth Patterns8-A

Linear Growth (Decay) occurs when a quantity increases (decreases) by the same absolute amount in each unit of time.

Example: Straightown -- 500 each year

Exponential Growth (Decay) occurs when a quantity increases (decreases) by the same relative amount—that is, by the same percentage—in each unit of time.

Example: Powertown: -- 5% each year

Linear/Exponential Growth/Decay?Linear/Exponential Growth/Decay?

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The number of students at Wilson High School has increased by 50 in each of the past four years.

• Which kind of growth is this?Linear Growth

• If the student populations was 750 four years ago, what is it today?

4 years ago: 750Now (4 years later): 750 + (4 x 50) =

950

Linear/Exponential Growth/Decay?Linear/Exponential Growth/Decay?

The price of milk has been rising with inflation at 3.5% per year.

• Which kind of growth is this?Exponential Growth

• If the price was $1.80/gallon two years ago, what is it now?

2 years ago: $1.80/gallonNow (2 years later): $1.80 × (1.035)2 = $1.93/gallon

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Tax law allows you to depreciate the value of your equipment by $200 per year.

Linear/Exponential Growth/Decay?Linear/Exponential Growth/Decay?

• Which kind of growth is this?Linear Decay

• If you purchased the equipment three years ago for $1000, what is its depreciated value now?

3 years ago: $1000Now (3 years later): $1000 – (3 x 200) = $400

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Linear/Exponential Growth/Decay?Linear/Exponential Growth/Decay?

The memory capacity of state-of-the-art computer hard drives is doubling approximately every two years.

• Which kind of growth is this?[doubling means increasing

by 100%]Exponential Growth

• If the company’s top of the line drive holds 300 gigabytes today, what will it hold in six years?

Now: 300 gigabytes2 years: 600 gigabytes4 years: 1200 gigabytes6 years: 2400 gigabytes

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Linear/Exponential Growth/Decay?Linear/Exponential Growth/Decay?

The price of DVD recorders has been falling by about 25% per year.

• Which kind of growth is this?Exponential Decay

• If the price is $200 today, what can you expect it to be in 2 years?

Now: $2002 years: 200 x (0.75)2 = $112.50

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More PracticeMore PracticeThe population of Danbury is increasing by

505 people per year. If the population is 15,000 today, what will it be in three years?

16,515

The price of computer memory is decreasing at a rate of 12% per year. If a memory chip costs $80 today, what will it cost in 2 years?

$61.95

During the worst periods of hyper inflation in Brazil, the price of food increased at a rate of 30% per month. If your food bill was $100 one month during this period, what was it two months later?

$169

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The Impact of DoublingThe Impact of Doubling

Parable 1 From Hero to Headless in 64 Easy Steps

Parable 2 The Magic Penny

Parable 3 Bacteria in a Bottle

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Parable 1Parable 1From Hero to Headless in 64 Easy From Hero to Headless in 64 Easy

StepsStepsParable 1 “If you please, king, put one

grain of wheat on the first square of my chessboard,” said the inventor. “ Then place two grains on the second square, four grains on the third square, eight grains on the fourth square and so on.” The king gladly agreed, thinking the man a fool for asking for a few grains of wheat when he could have had gold or jewels.

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Parable 1Parable 1

SquaSquare re

Grains on squareGrains on square

11 1 = 21 = 200

22 2 = 22 = 21 1

33 4 = 24 = 22 2 = 2= 2××22

44 8 = 28 = 23 3 = 2= 2××22××22

55 16 = 216 = 24 4 = = 22××22××22××22

. . . . . . . . . . . .

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Parable 1Parable 1

SquaSquare re

Grains on Grains on squaresquare

11 1 = 21 = 200

22 2 = 22 = 211

33 4 = 24 = 222

44 8 = 28 = 233

55 16 = 216 = 244

. . . . . . . . . . . .

64 226363

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Parable 1Parable 1SquaSquarere

Grains on Grains on squaresquare

Total Total Grains on Grains on chessboarchessboardd

Formula for Formula for total on total on boardboard

11 1 = 21 = 200 11 221 1 – 1– 1

22 2 = 22 = 211 1+2 = 31+2 = 3 222 2 – 1– 1

33 4 = 24 = 222 3+4 = 73+4 = 7 223 3 – 1– 1

44 8 = 28 = 233 7+8 = 157+8 = 15 224 4 – 1– 1

55 16 = 216 = 244 15 + 16 15 + 16 = 31= 31

225 5 – 1– 1

. . . . . . . . . . . . . . .. . . . . . . . . 64 226363 2264 64 - 1- 1 2264 64 - 1- 1

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Parable 1Parable 1From Hero to Headless in 64 Easy From Hero to Headless in 64 Easy

StepsStepsParable 1 “If you please, king, put one grain of wheat on the first square of my chessboard,” said the inventor. “ Then place two grains on the second square, four grains on the third square, eight grains on the fourth square and so on.” The king gladly agreed, thinking the man a fool for asking for a few grains of wheat when he could have had gold or jewels.

264 – 1 = 1.8×1019 = ≈ 18 billion, billion grains of wheat

This is more than all the grains of wheat harvested in human history.

The king never finished paying the inventor and according to legend, instead had him beheaded.

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Parable 2Parable 2The Magic PennyThe Magic Penny

Parable 2 A leprechaun promises you fantastic wealth and hands you a penny. You place the penny under your pillow and the next morning, to your surprise, you find two pennies. The following morning 4 pennies and the next morning 8 pennies. Each magic penny turns into two magic pennies.

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Parable 2Parable 2

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Day Day Amount under Amount under pillowpillow

00 $0.01$0.01

11 $0.02$0.02

22 $0.04$0.04

33 $0.08$0.08

44 $0.16$0.16

. . . . . .

Parable 2Parable 2

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Day Day Amount under Amount under pillowpillow

Amount under Amount under pillowpillow

00 $0.01$0.01 $0.01 = $0.01 = $0.01×2$0.01×200

11 $0.02$0.02 $0.02 = $0.02 = $0.01×2$0.01×211

22 $0.04$0.04 $0.04 = $0.04 = $0.01×2$0.01×222

33 $0.08$0.08 $0.08 = $0.08 = $0.01×2$0.01×233

44 $0.16$0.16 $0.16 = $0.16 = $0.01×2$0.01×244

. . . . . . t $0.01×2$0.01×2tt

Parable 2Parable 2

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TimeTime Amount under pillowAmount under pillow

1 week (7 1 week (7 days)days)

$0.01×2$0.01×277= $1.28= $1.28

2 weeks (14 2 weeks (14 days) days)

1 month (30 1 month (30 days) days)

50 days50 days

Parable 2Parable 2

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TimeTime Amount under pillowAmount under pillow

1 week (7 1 week (7 days)days)

$0.01×2$0.01×277= $1.28= $1.28

2 weeks (14 2 weeks (14 days) days)

$0.01×2$0.01×21414= = $163.84$163.84

1 month (30 1 month (30 days) days)

50 days50 days

Parable 2Parable 2

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TimeTime Amount under pillowAmount under pillow

1 week (7 1 week (7 days)days)

$0.01×2$0.01×277= $1.28= $1.28

2 weeks (14 2 weeks (14 days) days)

$0.01×2$0.01×21414= = $163.84$163.84

1 month (30 1 month (30 days) days)

$0.01×2$0.01×23030= = $10,737,418.24$10,737,418.24

50 days50 days

Parable 2Parable 2

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TimeTime Amount under pillowAmount under pillow

1 week (7 1 week (7 days)days)

$0.01×2$0.01×277= $1.28= $1.28

2 weeks (14 2 weeks (14 days) days)

$0.01×2$0.01×21414= = $163.84$163.84

1 month (30 1 month (30 days) days)

$0.01×2$0.01×23030= = $10,737,418.24$10,737,418.24

50 days50 days $0.01×2$0.01×25050= $11.3 = $11.3 trilliontrillion

Parable 2Parable 2The Magic PennyThe Magic Penny

Parable 2 A leprechaun promises you fantastic wealth and hands you a penny. You place the penny under your pillow and the next morning, to your surprise, you find two pennies. The following morning 4 pennies and the next morning 8 pennies. Each magic penny turns into two magic pennies. WOW!

The US government needs to look for a leprechaun with a magic penny.

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Parable 3Parable 3Bacteria in a BottleBacteria in a Bottle

Parable 3 Suppose you place a single bacterium in a bottle at 11:00 am. It grows and at 11:01 divides into two bacteria. These two bacteria each grow and at 11:02 divide into four bacteria, which grow and at 11:03 divide into eight bacteria, and so on.

Question0: If the bottle is full at NOON, how many bacteria are in the bottle?

Question1: When was the bottle half full?

Question2: If you (a mathematically sophisticated bacterium) warn of impending disaster at 11:56, will anyone believe you?

Question3: At 11:59, your fellow bacteria find 3 more bottles to fill. How much time have they gained for the bacteria civilization?

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Single bacteria in a bottle at 11:00 am 2 bacteria at 11:01 4 bacteria at 11:02 8 bacteria at 11:03 . . . At 12:00 (60 minutes later) the bottle is full and

contains 260 ≈ 1.15 x1018

Question0: If the bottle is full at NOON, how many bacteria are in the bottle?

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Single bacteria in a bottle at 11:00 am 2 bacteria at 11:01 4 bacteria at 11:02 8 bacteria at 11:03 . . . Bottle is full at 12:00 (60 minutes later) and so is 1/2 full at 11:59 am

Question1: When was the bottle half full?

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½ full at 11:59¼ full at 11:58⅛ full at 11:57 full at 11:56

At 11:56 the amount of unused space is 15 times the amount of used space.

116

Question2: If you (a mathematically sophisticated bacterium) warn of impending disaster at 11:56, will anyone believe you?

Your mathematically unsophisticated bacteria friends will not believe you when you warn of impending disaster at 11:56.

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There are . . .enough bacteria to fill 1 bottle at 12:00enough bacteria to fill 2 bottles at 12:01 enough bacteria to fill 4 bottles at 12:02

Question3: At 11:59, your fellow bacteria find 3 more bottles to fill. How much time have they gained for the bacteria civilization?

They have gained only 2 additional minutes for the bacteria civilization.

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By 1:00- there are 2120 bacteria.

This is enough bacteria to cover the entire surface of the Earth in a layer more than 2 meters deep!

Question4: Is this scary?

After 5 ½ hours, at this rate . . .the volume of bacteria would exceed the volume of the known universe.

Yes, this is scary!

Key Facts about Key Facts about Exponential GrowthExponential Growth

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• Exponential growth cannot continue indefinitely. After only a relatively small number of doublings, exponentially growing quantities reach impossible proportions.

• Exponential growth leads to repeated doublings. With each doubling, the amount of increase is approximately equal to the sum of all preceding doublings.

Repeated DoublingsRepeated Doublings8-A

Homework :Homework :

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