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Section 8A Growth: Linear vs. Exponential Pages 490-495

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Growth: Linear vs Exponential 8-A Imagine two communities, Straightown and Powertown, each with an initial population of 10,000 people. Straightown grows at a constant rate of 500 people per year. Powertown grows at a constant rate of 5% per year. Compare the population growth of Straightown and Powertown.

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YearStraightown 010,000 110,500 2 3 10 15 20 40 Straightown: initially 10,000 people and growing at a rate of 500 people per year 8-A

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YearStraightown 010,000 110,500 211,000 3 10 15 20 40 Straightown: initially 10,000 people and growing at a rate of 500 people per year 8-A

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YearStraightown 010,000 110,500 211,000 311,500 10 15 20 40 Straightown: initially 10,000 people and growing at a rate of 500 people per year 8-A

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Straightown: initially 10,000 people and growing at a rate of 500 people per year YearStraightown 010,000 110,500 211,000 311,500 10 10000 + (10x500) =15000 15 20 40 8-A

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Straightown: initially 10,000 people and growing at a rate of 500 people per year YearStraightown 010,000 110,500 211,000 311,500 10 10000 + (10x500) =15000 15 10000 + (15x500) =17500 20 40 8-A

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Straightown: initially 10,000 people and growing at a rate of 500 people per year YearStraightown 010,000 110,500 211,000 311,500 10 10000 + (10x500) =15000 15 10000 + (15x500) =17500 20 10000 + (20x500) =20000 40 8-A

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Straightown: initially 10,000 people and growing at a rate of 500 people per year YearStraightown 010,000 110,500 211,000 311,500 10 10000 + (10x500) =15000 15 10000 + (15x500) =17500 20 10000 + (20x500) =20000 40 10000 + (40x500) =30000 8-A

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Powertown: initially 10,000 people and growing at a rate of 5% per year YearPowertown 010,000 1 10000 x (1.05) = 10,500 2 3 10 15 20 40 8-A

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Powertown: initially 10,000 people and growing at a rate of 5% per year YearPowertown 010,000 1 10000 x (1.05) = 10,500 2 10000 x (1.05) 2 = 11,025 3 10 15 20 40 8-A

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Powertown: initially 10,000 people and growing at a rate of 5% per year YearPowertown 010,000 1 10000 x (1.05) = 10,500 2 10000 x (1.05) 2 = 11,025 3 10000 x (1.05) 3 = 11,576 10 15 20 40 8-A

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Powertown: initially 10,000 people and growing at a rate of 5% per year YearPowertown 010,000 1 10000 x (1.05) = 10,500 2 10000 x (1.05) 2 = 11,025 3 10000 x (1.05) 3 = 11,576 10 10000 x (1.05) 10 = 16,289 15 20 40 8-A

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Powertown: initially 10,000 people and growing at a rate of 5% per year YearPowertown 010,000 1 10000 x (1.05) = 10,500 2 10000 x (1.05) 2 = 11,025 3 10000 x (1.05) 3 = 11,576 10 10000 x (1.05) 10 = 16,289 15 10000 x (1.05) 15 = 20,789 20 40 8-A

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Powertown: initially 10,000 people and growing at a rate of 5% per year YearPowertown 010,000 1 10000 x (1.05) = 10,500 2 10000 x (1.05) 2 = 11,025 3 10000 x (1.05) 3 = 11,576 10 10000 x (1.05) 10 = 16,289 15 10000 x (1.05) 15 = 20,789 20 10000 x (1.05) 20 = 26,533 40 8-A

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Powertown: initially 10,000 people and growing at a rate of 5% per year YearPowertown 010,000 1 10000 x (1.05) = 10,500 2 10000 x (1.05) 2 = 11,025 3 10000 x (1.05) 3 = 11,576 10 10000 x (1.05) 10 = 16,289 15 10000 x (1.05) 15 = 20,789 20 10000 x (1.05) 20 = 26,533 40 10000 x (1.05) 40 = 70,400 8-A

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Population Comparison YearStraightown 110,500 211,000 311,500 1015,000 1517,500 2020,000 4030,000 Powertown10,500 11,025 11,576 16,289 20,789 26,533 70,400 8-A

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Growth: Linear versus Exponential 8-A

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Two Basic Growth Patterns 8-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

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Linear/Exponential Growth/Decay? 8-A 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: 750 Now (4 years later): 750 + (4 x 50) = 950

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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/gallon Now (2 years later): $1.80 × (1.035) 2 = $1.93/gallon 8-A

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Tax law allows you to depreciate the value of your equipment by $200 per year. 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: $1000 Now (3 years later): $1000 – (3 x 200) = $400 8-A

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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 gigabytes 2 years: 600 gigabytes 4 years: 1200 gigabytes 6 years: 2400 gigabytes 8-A

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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: $200 2 years: 200 x (0.75) 2 = $112.50 8-A

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More Practice The 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 8-A

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The Impact of Doubling Parable 1 From Hero to Headless in 64 Easy Steps Parable 2 The Magic Penny Parable 3 Bacteria in a Bottle 8-A

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Parable 1 From Hero to Headless in 64 Easy Steps Parable 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. 8-A

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Parable 1 Square Grains on square 1 1 = 2 0 2 2 = 2 1 3 4 = 2 2 = 2×2 4 8 = 2 3 = 2×2×2 5 16 = 2 4 = 2×2×2×2... 8-A

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Parable 1 Square Grains on square 1 1 = 2 0 2 2 = 2 1 3 4 = 2 2 4 8 = 2 3 5 16 = 2 4... 64 2 63 8-A

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Parable 1 Square Grains on square Total Grains on chessboard Formula for total on board 1 1 = 2 0 1 2 1 – 1 2 2 = 2 1 1+2 = 3 2 2 – 1 3 4 = 2 2 3+4 = 7 2 3 – 1 4 8 = 2 3 7+8 = 15 2 4 – 1 5 16 = 2 4 15 + 16 = 31 2 5 – 1... 64 2 63 2 64 - 1 8-A

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Parable 1 From Hero to Headless in 64 Easy Steps Parable 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. 2 64 – 1 = 1.8×10 19 = ≈ 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. 8-A

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Parable 2 The 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. 8-A

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Parable 2 8-A Day Amount under pillow 0$0.01 1$0.02 2$0.04 3$0.08 4$0.16...

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Parable 2 8-A Day Amount under pillow 0$0.01 $0.01 = $0.01×2 0 1$0.02 $0.02 = $0.01×2 1 2$0.04 $0.04 = $0.01×2 2 3$0.08 $0.08 = $0.01×2 3 4$0.16 $0.16 = $0.01×2 4... t $0.01×2 t

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Parable 2 8-A Time Amount under pillow 1 week (7 days) $0.01×2 7 = $1.28 2 weeks (14 days) 1 month (30 days) 50 days

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Parable 2 8-A Time Amount under pillow 1 week (7 days) $0.01×2 7 = $1.28 2 weeks (14 days) $0.01×2 14 = $163.84 1 month (30 days) 50 days

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Parable 2 8-A Time Amount under pillow 1 week (7 days) $0.01×2 7 = $1.28 2 weeks (14 days) $0.01×2 14 = $163.84 1 month (30 days) $0.01×2 30 = $10,737,418.24 50 days

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Parable 2 8-A Time Amount under pillow 1 week (7 days) $0.01×2 7 = $1.28 2 weeks (14 days) $0.01×2 14 = $163.84 1 month (30 days) $0.01×2 30 = $10,737,418.24 50 days $0.01×2 50 = $11.3 trillion

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Parable 2 The 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. 8-A

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Parable 3 Bacteria 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? 8-A

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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 2 60 ≈ 1.15 x10 18 Question0: If the bottle is full at NOON, how many bacteria are in the bottle?

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8-A 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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8-A ½ full at 11:59 ¼ full at 11:58 ⅛ full at 11:57 full at 11:56 At 11:56 the amount of unused space is 15 times the amount of used space. 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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8-A There are... enough bacteria to fill 1 bottle at 12:00 enough 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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8-A By 1:00- there are 2 120 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!

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Key Facts about Exponential Growth 8-A 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.

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Repeated Doublings 8-A

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Homework : Page 496 # 8, 10, 12, 14, 18, 26

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