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**Section 8A Growth: Linear vs. Exponential**

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**Growth: Linear vs Exponential**

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. Make sure students understand that the principles above also apply to linear and exponential decay as well.

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8-A Straightown: initially 10,000 people and growing at a rate of 500 people per year Year Straightown 10,000 1 10,500 2 3 10 15 20 40

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8-A Straightown: initially 10,000 people and growing at a rate of 500 people per year Year Straightown 10,000 1 10,500 2 11,000 3 10 15 20 40

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8-A Straightown: initially 10,000 people and growing at a rate of 500 people per year Year Straightown 10,000 1 10,500 2 11,000 3 11,500 10 15 20 40

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8-A Straightown: initially 10,000 people and growing at a rate of 500 people per year Year Straightown 10,000 1 10,500 2 11,000 3 11,500 10 (10x500) =15000 15 20 40

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8-A Straightown: initially 10,000 people and growing at a rate of 500 people per year Year Straightown 10,000 1 10,500 2 11,000 3 11,500 10 (10x500) =15000 15 (15x500) =17500 20 40

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8-A Straightown: initially 10,000 people and growing at a rate of 500 people per year Year Straightown 10,000 1 10,500 2 11,000 3 11,500 10 (10x500) =15000 15 (15x500) =17500 20 (20x500) =20000 40

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8-A Straightown: initially 10,000 people and growing at a rate of 500 people per year Year Straightown 10,000 1 10,500 2 11,000 3 11,500 10 (10x500) =15000 15 (15x500) =17500 20 (20x500) =20000 40 (40x500) =30000

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

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

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

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

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

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

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

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**Population Comparison**

Year Straightown 1 10,500 2 11,000 3 11,500 10 15,000 15 17,500 20 20,000 40 30,000 Powertown 10,500 11,025 11,576 16,289 20,789 26,533 70,400

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**Growth: Linear versus Exponential**

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**Two Basic Growth Patterns**

Linear Growth (Decay) occurs when a quantity increases (decreases) by the same absolute amount in each unit of time. Example: Straightown 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 Make sure students understand that the principles above also apply to linear and exponential decay as well.

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**Linear/Exponential Growth/Decay?**

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): (4 x 50) = 950 Make sure students understand that the principles above also apply to linear and exponential decay as well.

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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

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**Linear/Exponential Growth/Decay?**

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

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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

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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

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8-A 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 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 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

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8-A 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

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**From Hero to Headless in 64 Easy Steps**

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.

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8-A Parable 1 Square Grains on square 1 1 = 20 2 2 = 21 3 4 = 22 = 2×2 4 8 = 23 = 2×2×2 5 16 = 24 = 2×2×2×2 . . .

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8-A Parable 1 Square Grains on square 1 1 = 20 2 2 = 21 3 4 = 22 4 8 = 23 5 16 = 24 . . . 64 263

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**Formula for total on board**

Parable 1 8-A Square Grains on square Total Grains on chessboard Formula for total on board 1 1 = 20 21 – 1 2 2 = 21 1+2 = 3 22 – 1 3 4 = 22 3+4 = 7 23 – 1 4 8 = 23 7+8 = 15 24 – 1 5 16 = 24 = 31 25 – 1 . . . 64 263

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**From Hero to Headless in 64 Easy Steps**

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. 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 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.

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8-A Parable 2 Day Amount under pillow $0.01 1 $0.02 2 $0.04 3 $0.08 4 $0.16 . . . Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

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8-A Parable 2 Day Amount under pillow $0.01 $0.01 = $0.01×20 1 $0.02 $0.02 = $0.01×21 2 $0.04 $0.04 = $0.01×22 3 $0.08 $0.08 = $0.01×23 4 $0.16 $0.16 = $0.01×24 . . . t $0.01×2t Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

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**Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)**

Time Amount under pillow 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days) 1 month (30 days) 50 days Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

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**Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)**

Time Amount under pillow 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days) $0.01×214= $163.84 1 month (30 days) 50 days Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

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**Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)**

Time Amount under pillow 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days) $0.01×214= $163.84 1 month (30 days) $0.01×230= $10,737,418.24 50 days Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

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**Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)**

Time Amount under pillow 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days) $0.01×214= $163.84 1 month (30 days) $0.01×230= $10,737,418.24 50 days $0.01×250= $11.3 trillion Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

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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.

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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?

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**Single bacteria in a bottle at 11:00 am 2 bacteria at 11:01 **

Question0: If the bottle is full at NOON, how many bacteria are in the bottle? 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 ≈ 1.15 x1018 Make sure students understand that the principles above also apply to linear and exponential decay as well.

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**Question1: When was the bottle half full?**

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 Make sure students understand that the principles above also apply to linear and exponential decay as well.

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8-A Question2: If you (a mathematically sophisticated bacterium) warn of impending disaster at 11:56, will anyone believe you? 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. Make sure students understand that the principles above also apply to linear and exponential decay as well. Your mathematically unsophisticated bacteria friends will not believe you when you warn of impending disaster at 11:56.

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**enough bacteria to fill 1 bottle at 12:00 **

Question3: At 11:59, your fellow bacteria find 3 more bottles to fill. How much time have they gained for the bacteria civilization? 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 Make sure students understand that the principles above also apply to linear and exponential decay as well. They have gained only 2 additional minutes for the bacteria civilization.

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**Question4: Is this scary?**

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! After 5 ½ hours, at this rate . . . the volume of bacteria would exceed the volume of the known universe. Make sure students understand that the principles above also apply to linear and exponential decay as well. Yes, this is scary!

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**Key Facts about Exponential Growth**

• 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. Discussing each of the parables from the text are vital. Students should not miss this content and all the ramifications to the quantitative world around us.

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8-A Repeated Doublings

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

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