Unit 8 A Growth Linear versus Exponential Copyright

Unit 8 A Growth: Linear versus Exponential Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit A, Slide 1

Growth: Linear versus Exponential § Linear Growth occurs when a quantity grows by the same absolute amount in each unit of time. § Exponential Growth occurs when a quantity grows by the same relative amount—that is, by the same percentage—in each unit of time. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit A, Slide 2

Growth: Linear versus Exponential Straightown grows by the same absolute amount each year and Powertown grows by the same relative amount each year. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit A, Slide 3

Example 1 In each of the following situations, state whether the growth (or decay) is linear or exponential, and answer the associated questions. a. The number of students at Wilson High School has increased by 50 in each of the past four years. If the student population was 750 four years ago, what is it today? b. The price of milk has been rising 3% per year. If the price of a gallon of milk was $4 a year ago, what is it now? Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit A, Slide 4

Example 1 (cont) c. 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? d. 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? e. 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? Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit A, Slide 5

Bacteria in a Bottle Suppose you place a single bacterium in a bottle at 11: 00 a. m. 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. Now, suppose the bacteria continue to double every minute, and the bottle is full at 12: 00. (the number of bacteria at this point must be 260 because they doubled every minute for 60 minutes), but the important fact is that we have a bacterial disaster on our hands: Because the bacteria have filled the bottle, the entire bacterial colony is doomed. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit A, Slide 6

Bacteria in a Bottle Question: The disaster occurred because the bottle was completely full at 12: 00. When was the bottle half-full? Answer: Because it took one hour to fill the bottle, many people guess that it was half-full after a half-hour, or at 11: 30. However, because the bacteria double in number every minute, they must also have doubled during the last minute, which means the bottle went from being half-full to full during the final minute. That is, the bottle was half-full at 11: 59, just 1 minute before the disaster Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit A, Slide 7

Example 2 How many bottles would the bacteria fill at the end of the second hour? Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit A, Slide 8

Key Facts about Exponential Growth § Exponential growth leads to repeated doublings. With each doubling, the amount of 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. Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 8, Unit A, Slide 9