Growth: Linear versus Exponential Chapter 8 Unit A Growth: Linear versus Exponential

Essential Question What are some similarities between linear and exponential (growth/decay)? What are some differences between linear and exponential (growth/decay)?

Essential Question What are some similarities between linear and exponential (growth/decay)? What are some differences between linear and exponential (growth/decay)?

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Definitions Linear growth: quantity grows by the same ABSOLUTE amount in each unit of time Exponential Growth: quantity grows by the same RELATIVE (percentage) amount in each unit of time

Formula Linear growth: y=mx+b (where m is +) Exponential growth: 𝑦=𝑎 (𝑏) 𝑥 a= initial population b=growth factor (1+decimal from %)

Linear or Exponential? Number of students at Wilson High has increased by 50 in each of the past 4 years. If the student population was 750 four years ago, what is it today?

Linear or Exponential? The price of milk has been rising 3% per year. If the price of a gallon of milk was $4 a year ago, how much is it now?

Linear or Exponential? Tax law allows you to depreciate the value of your equipment by $200 per year. If you purchased the equipment three years ago for $1000? What is its depreciated value today?

Linear or Exponential? The memory capacity of state-of-the- art computer storage devices is doubling approximately every two years. If a company’s top-of-the-line drive holds 16 terabytes today, what will it hold in six years?

Linear or Exponential? The price of high-definition TV sets has been falling by about 25% per year. IF the price is $1000 today, what can you expect it to be in two years?

Hero to Headless in 64 Steps “If you please, king, put 1 grain of wheat on the 1st square of my board. Then place 2 grains on the 2nd square, 4 grains on the 3rd square, 8 grains on the 4th square, and so on” Create a table with #square, grains on that square, total grains, formula for total grains

Magic Penny You meet a leprechaun who promises to give you fantastic wealth, but gives you only 1 penny. After putting the penny under your pillow at night, you wake up to find 2 pennies. The following morning, you find 4 pennies, and the 4th morning you find 8 pennies. While you sleep the magic penny turns into 2 magic pennies. When will you be rich?

Bacteria in a Bottle A single bacteria is in a bottle at 11am. It grows and at 11:01 divides into 2 bacteria. These 2 bacteria each grow and at 11:02 divide into 4 bacteria, which grow and at 11:03 divide into 8 bacteria. The disaster occurs at 12:00. When was the bottle half full?

Bacteria in a Bottle A single bacteria is in a bottle at 11am. It grows and at 11:01 divides into 2 bacteria. These 2 bacteria each grow and at 11:02 divide into 4 bacteria, which grow and at 11:03 divide into 8 bacteria. At 11:56 you try to warn everyone that unless your fellow bacteria slow their growth dramatically, the end is just 4 minutes away. Will anyone believe you?

Bacteria in a Bottle It’s 11:59 and, with the bottle half-full, your fellow bacteria are finally taking your warnings seriously. The quickly start a space program, sending little bacterial spaceships out into the lab to find more bottles. They find 4 more bottles and are able to distribute the population evenly amongst the four bottles. Given that they now have 4 bottles rather than just one, how much time have they gained for the civilization?

Bacteria in a Bottle Suppose the bacteria constantly look for more bottles. Is there any hope that further discoveries will allow the colony to continue exponential growth?

Key Facts about Exponential Growth Exponential growth leads to repeated doublings. With each doubling, the amount increase is approximately equal to the SUM of all preceding doublings Exponential growth CANNOT continue indefinitely. After only a relatively small number of doublings, exponentially growing quantities reach impossible proportions

HW: Chapter 8 Unit A D2: (see notes from parables). P. 422 #21-26, 29 D3: Chapter 8 Unit A Quick Quiz, pp. 420-421 #1-10 all