Transcript 8A PP with notes

Exponential Astonishment
Copyright © 2011 Pearson Education, Inc.
Unit 8A
Growth: Linear versus
Exponential
Copyright © 2011 Pearson Education, Inc.
Slide 8-3
Activity: Towers of Hanoi






The game is called Towers of Hanoi
The game begins with al the disks stacked on one
peg in order of decreasing size.
The object of the game is to move the entire stack
of disks to a different peg; following 2 rules
1. Only 1 disk can be moved at a time
2. A larger disk can never be placed on top of a
smaller disk.
Play the game with seven disks, looking for the
most efficient strategy for moving disks.
Copyright © 2011 Pearson Education, Inc.
Slide 8-4
Strategy
CN (Game 1-6)






1. Look at the series of goals. How many total moves will it take to
reach the goal of having 7 disks on another peg?
2. Find a general formula for the second and third columns after n
steps.
3.Use the formula for the total moves to predict the number of moves
required to complete the game with 10 disks.
4. The game is related to a legend…. How many total moves are
required to move the entire set of 64 disks?
5. The legend holds that upon completion….. Based on your answer
to #4, how many years will it take to move the entire stack of 64 disks.
(moving 1 disk/sec and the universe is about 14 billion yrs. Old.
6. Briefly comment on what this game illustrates about the nature of
exponential growth.
Copyright © 2011 Pearson Education, Inc.
Slide 8-5
8-A
Growth: Linear versus Exponential
 Linear Growth occurs when a quantity grows
by some fixed absolute amount in each unit
of time.
 Exponential Growth occurs when a quantity
grows by the same fixed relative amount—
that is, by the same percentage—in each unit
of time.
Copyright © 2011 Pearson Education, Inc.
Slide 8-6
8-A
Growth: Linear versus Exponential
Straightown grows by the same absolute amount each
year and Powertown grows by the same relative amount
each year.
Copyright © 2011 Pearson Education, Inc.
Slide 8-7
8-A
Linear or Exponential? 1-5
CN (1)

In each of the following situations state whether
the growth (or decay) is linear or exponential, and
answer the associated questions.

The number of students at Wilson High School
has increased by 50 in each of the past four
years.
1. If the student population was 750 four years
ago, what is it today?

Copyright © 2011 Pearson Education, Inc.
Slide 8-8
8-A
Linear or Exponential?
CN (2)

The price of mile has been rising 3% per year.

2. If the price of a gallon of milk was $4 a year
ago, what is it now?
Copyright © 2011 Pearson Education, Inc.
Slide 8-9
8-A
Linear or Exponential?
CN (3)

Tax law allows you to depreciate the value of your
equipment by $200 per year.

3. If you purchased the equipment three years
ago for $1000, what is its depreciated value
today?
Copyright © 2011 Pearson Education, Inc.
Slide 8-10
Linear or Exponential?
CN (4)


8-A
The memory capacity of state-of-the-art computer
hard drives is doubling approximately every two
years.
4. If a company’s top-of-the-line drive holds 22.5
terabytes today, what will it hold in six years?
Copyright © 2011 Pearson Education, Inc.
Slide 8-11
8-A
Linear or Exponential?
CN (5)

The price of high-definition TV sets has been
falling by about 25% per year.

5. If the price is $1000 today, what can you
expect it to be in two years?
Copyright © 2011 Pearson Education, Inc.
Slide 8-12
8-A
The Impact of Doublings
Parable 1:

From Hero to Headless in 64 easy steps.

Legend has it that when chess was invented in
ancient times, a king was so enchanted that he
said to the inventor, “Name your reward.”

Grains of wheat for 64 spaces on a chessboard.

The king never finished paying the inventor and
according to legend, had him beheaded.
Copyright © 2011 Pearson Education, Inc.
Slide 8-13
8-A
Parable 2:

The Magic Penny

One lucky day you meet a leprechaun who
promises to give you fantastic wealth, but hands
you a penny before disappearing.
You place the penny under your pillow and the
next morning you find two pennies..


By the end of 52 days you have enough to pay off
the national debt of the US.
Copyright © 2011 Pearson Education, Inc.
Slide 8-14
8-A
Parable 3:
Example
Bacteria in a Bottle: Suppose you put a single
bacterium in a bottle at 11:00 a.m. It grows and at 11:01,
it divides into two bacteria. These two bacteria grow and
at 11:02 divide into four bacteria, which grow and at
11:03 divide into eight bacteria, and so on. Thus, the
bacteria doubles every minute.
If the bottle is half-full at 11:59, when will the bottle be
completely full?
Since the bacteria doubles every minute, the bottle will
be full after one more minute. That is, the bottle will be
completely full at 12:00 p.m.
Copyright © 2011 Pearson Education, Inc.
Slide 8-15
8-A
Doubling Lessons



The three parables reveal at least two key lessons
about the repeated doublings that arise with
exponential growth.
1. You’ll notice that the number of grains on each
square is nearly equal to the total number of
grains on all previous squares combined.
2. All three parables show quantities growing to
impossible proportions.
Copyright © 2011 Pearson Education, Inc.
Slide 8-16
8-A
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 © 2011 Pearson Education, Inc.
Slide 8-17
Homework
CN (12)

8-A
12. Please answer the quick quiz multiple choice
questions 1-10 on p. 478.
Copyright © 2011 Pearson Education, Inc.
Slide 8-18
Homework
8A




8-A
CN 1-12 (Game 1-6, Notes 1-5, Quick Quiz)
P. 478: 1-8
1 web
 31. Computing Power
 32. Web Growth
1 world
 33. Linear Growth
 34. Exponential Growth
Copyright © 2011 Pearson Education, Inc.
Slide 8-19