Section 8A Growth: Linear vs. Exponential

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Section 8AGrowth Linear vs. Exponential

• Pages 490-495

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Growth Linear vs Exponential
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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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Straightown initially 10,000 people and growing
at a rate of 500 people per year
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Straightown initially 10,000 people and growing
at a rate of 500 people per year
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Straightown initially 10,000 people and growing
at a rate of 500 people per year
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Straightown initially 10,000 people and growing
at a rate of 500 people per year
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Straightown initially 10,000 people and growing
at a rate of 500 people per year
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Straightown initially 10,000 people and growing
at a rate of 500 people per year
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Straightown initially 10,000 people and growing
at a rate of 500 people per year
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Powertown initially 10,000 people and growing at
a rate of 5 per year
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Powertown initially 10,000 people and growing at
a rate of 5 per year
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Powertown initially 10,000 people and growing at
a rate of 5 per year
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Powertown initially 10,000 people and growing at
a rate of 5 per year
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Powertown initially 10,000 people and growing at
a rate of 5 per year
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Powertown initially 10,000 people and growing at
a rate of 5 per year
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Powertown initially 10,000 people and growing at
a rate of 5 per year
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Population Comparison
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Growth Linear versus Exponential
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Two Basic Growth Patterns
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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 amountthat is, by the same
percentagein each unit of time. Example
Powertown -- 5 each year
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Linear/Exponential Growth/Decay?
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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

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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 companys 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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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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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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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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Parable 1
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Parable 1
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Parable 1
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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.
264 1 1.81019 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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Parable 2
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Parable 2
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Parable 2
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Parable 2
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Parable 2
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Parable 2
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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 1100 am. It grows and at 1101
divides into two bacteria. These two bacteria
each grow and at 1102 divide into four bacteria,
which grow and at 1103 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 1156,
will anyone believe you?
Question3 At 1159, your fellow bacteria find 3
more bottles to fill. How much time have they
gained for the bacteria civilization?
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Question0 If the bottle is full at NOON, how
many bacteria are in the bottle?
Single bacteria in a bottle at 1100 am 2
bacteria at 1101 4 bacteria at 1102 8
bacteria at 1103 . . . At 1200 (60 minutes
later) the bottle is full and contains 260
1.15 x1018
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Question1 When was the bottle half full?
Single bacteria in a bottle at 1100 am 2
bacteria at 1101 4 bacteria at 1102 8
bacteria at 1103 . . . Bottle is full at
1200 (60 minutes later) and so is 1/2 full at
1159 am
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Question2 If you (a mathematically sophisticated
bacterium) warn of impending disaster at 1156,
will anyone believe you?
½ full at 1159 ¼ full at 1158 ? full at
1157 full at 1156 At 1156 the amount of
unused space is 15 times the amount of used space.
Your mathematically unsophisticated bacteria
friends will not believe you when you warn of
impending disaster at 1156.
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Question3 At 1159, 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 1200 enough bacteria to fill 2 bottles at
1201 enough bacteria to fill 4 bottles at
1202
They have gained only 2 additional minutes for
the bacteria civilization.
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Question4 Is this scary?
By 100- 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.
Yes, this is scary!
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Key Facts about Exponential Growth
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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
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• Homework
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