Transcript Section 3B Putting Numbers in Perspective

Section 8A
Growth: Linear vs.
Exponential
Pages 490-495
8-A
Growth: Linear versus Exponential
Two Communities:
Straighttown, Powertown
Initial Population in each town = 10,000
Straighttown grows by 5% of 10,000 or 500
people per year.
Powertown grows by 5% per year.
Population Comparison
Year
1
2
3
10
15
20
40
Straighttown
10,500
Population Comparison
Year
Straighttown
1
10,500
2
11,000
3
10
15
20
40
Population Comparison
Year
Straighttown
1
10,500
2
11,000
3
11,500
10
15
20
40
Population Comparison
Year
Straighttown
1
10,500
2
11,000
3
11,500
10
10,000+(500×10) = 15,000
15
20
40
Population Comparison
Year
Straighttown
1
10,500
2
11,000
3
11,500
10
10,000+(500×10) = 15,000
15
10,000+(500×15) = 17,500
20
40
Population Comparison
Year
Straighttown
1
10,500
2
11,000
3
11,500
10
10,000+(500×10) = 15,000
15
10,000+(500×15) = 17,500
20
10,000+(500×20) = 20,000
40
Population Comparison
Year
Straighttown
1
10,500
2
11,000
3
11,500
10
10,000+(500×10) = 15,000
15
10,000+(500×15) = 17,500
20
10,000+(500×20) = 20,000
40
10,000+(500×40) =30,000
Population Comparison
Year
Straighttown
1
10,500
2
11,000
3
11,500
10
15,000
15
17,500
20
20,000
40
30,000
Powertown
10,000 × 1.05 = 10,500
Population Comparison
Year
Straighttown
Powertown
1
10,500
10,000 × 1.05 = 10,500
2
11,000
10,500 × 1.05 = 11,025
3
11,500
10
15,000
15
17,500
20
20,000
40
30,000
Population Comparison
Year Straighttown
Powertown
1
10,500
10,000 × 1.05 = 10,500
2
11,000
10,500 × 1.05 = 11,025
3
11,500
11,025 × 1.05 = 11,576
10
15,000
15
17,500
20
20,000
40
30,000
Population Comparison
Year
Straighttown
Powertown
1
10,500
10,000 × 1.05 = 10,500
2
11,000
10,500 × 1.05 = 11,025
3
11,500
11,025 × 1.05 = 11,576
10
15,000
10,000 × (1.05)10 = 16,289
15
17,500
20
20,000
40
30,000
Population Comparison
Year
Straighttown
Powertown
1
10,500
10,000 × 1.05 = 10,500
2
11,000
10,500 × 1.05 = 11,025
3
11,500
11,025 × 1.05 = 11,576
10
15,000
10,000 × (1.05)10 = 16,289
15
17,500
10,000 × (1.05)15 = 20,789
20
20,000
40
30,000
Population Comparison
Year Straightown
Powertown
1
10,500
10,000 × 1.05 = 10,500
2
11,000
10,500 × 1.05 = 11,025
3
11,500
11,025 × 1.05 = 11,576
10
15,000
10,000 × (1.05)10 = 16,289
15
17,500
10,000 × (1.05)15 = 20,789
20
20,000
10,000 × (1.05)20 = 26,533
40
30,000
Population Comparison
Year
Straighttown
Powertown
1
10,500
10,000 × 1.05 = 10,500
2
11,000
10,500 × 1.05 = 11,025
3
11,500
11,025 × 1.05 = 11,576
10
15,000
10,000 × (1.05)10 = 16,289
15
17,500
10,000 × (1.05)15 = 20,789
20
20,000
10,000 × (1.05)20 = 26,533
40
30,000
10,000 × (1.05)40 =70,400
Population Comparison
Year
Straighttown
Powertown
1
10,500
10,500
2
11,000
11,025
3
11,500
11,576
10
15,000
16,289
15
17,500
20,789
20
20,000
26,533
40
30,000
70,400
8-A
Growth: Linear versus Exponential
8-A
Growth: Linear versus Exponential
Linear Growth occurs when a quantity grows
by some fixed absolute amount in each unit
of time.
Example: Straighttown -- 500 each year
Exponential Growth occurs when a quantity
grows by the same fixed relative amount—
that is, by the same percentage—in each unit
of time.
Example: Powertown: -- 5% each year
8-A
Linear or Exponential Growth?
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 2 years ago,
what is it now?
1 year ago: $1.80 ×1.035 = $1.863 / gallon
Now: $1.863×1.035 = $1.93 / gallon
8-A
Linear or Exponential Decay?
Tax law allows you to depreciate the value
of your equipment by $200 per year.
• Which kind of decay is this?
Linear Decay
• If you purchased the equipment 3 years
ago for $1000, what is the depreciated
value today?
$1000 – ($3 years × $200/year) = $400
8-A
Linear or Exponential Growth?
The memory capacity of state-of-the art computer
hard drives is doubling approximately every
two years.
If a company’s top-of-the-line drive holds 300
gigabytes today, what will it hold in 6 years?
300  600  1200  2400 gigabytes
today
2 years
4 years
6 years
Which type of growth is this?
Exponential Growth
8-A
Parable 1: Chess Board
1 grain of wheat on first square
2 grains on the second square
4 grains on the third square
8 grains on the fourth square
...
4-C
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
...
...
4-C
Parable 1
Square
Grains on
square
1
1 = 20
2
2 = 21
3
4 = 22
4
8 = 23
5
16 = 24
...
...
64
263
4-C
Parable 1
Square
Grains on
square
Total Grains
on
chessboard
1
1 = 20
1
2
2 = 21
1+2 = 3
3
4 = 22
4
8 = 23
5
16 = 24
...
...
64
263
4-C
Parable 1
Square
Grains on Total Grains
square
on
chessboard
1
1 = 20
1
2
2 = 21
1+2 = 3
3
4 = 22
3+4 = 7
4
8 = 23
5
16 = 24
...
...
64
263
...
4-C
Parable 1
Square
Grains on
square
Total Grains
on
chessboard
1
1 = 20
1
2
2 = 21
1+2 = 3
3
4 = 22
3+4 = 7
4
8 = 23
7+8 = 15
5
16 = 24
...
...
64
263
...
4-C
Parable 1
Square
Grains on
square
Total Grains
on chessboard
1
1 = 20
1
2
2 = 21
1+2 = 3
3
4 = 22
3+4 = 7
4
8 = 23
7+8 = 15
5
16 = 24
15 + 16 = 31
...
...
...
64
263
4-C
Parable 1
Square
Grains on Total Grains
square
on
chessboard
1
1 = 20
1
2
2 = 21
1+2 = 3
3
4 = 22
3+4 = 7
4
8 = 23
7+8 = 15
5
16 = 24 15 + 16 = 31
...
...
...
64
263
264 - 1
4-C
Parable 1
Square
Grains on Total Grains
square
on
chessboard
Formula for
total on
board
1
1 = 20
1
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
16 = 24 15 + 16 = 31
25 – 1
5
...
...
...
...
64
263
264 - 1
264 - 1
8-A
Parable 1: Chess Board
264 – 1 = 1.8×1019 =
≈ 18 billion, billion grains of wheat
This is more than all the grains of wheat
harvested in human history.
8-A
Parable 2: Magic Penny
1 penny under your pillow
2 pennies the next morning
4 pennies the next morning
8 pennies the next morning
...
Will you ever get “fantastically” wealthy?
With pennies???
8-A
Parable 2
Day
Amount under pillow
0
$0.01
1
$0.02
2
$0.04
3
$0.08
4
$0.16
...
8-A
Parable 2
Day
Amount under
pillow
Amount under pillow
0
$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
8-A
Parable 2
Time
1 week (7 days)
2 weeks (14
days)
1 month (30
days)
50 days
Amount under pillow
$0.01×27= $1.28
8-A
Parable 2
Time
1 week (7 days)
2 weeks (14
days)
1 month (30
days)
50 days
Amount under pillow
$0.01×27= $1.28
$0.01×214= $163.84
8-A
Parable 2
Time
1 week (7 days)
2 weeks (14
days)
1 month (30
days)
50 days
Amount under pillow
$0.01×27= $1.28
$0.01×214= $163.84
$0.01×230=
$10,737,418.24
8-A
Parable 2
Time
1 week (7 days)
2 weeks (14
days)
1 month (30
days)
50 days
Amount under pillow
$0.01×27= $1.28
$0.01×214= $163.84
$0.01×230=
$10,737,418.24
$0.01×250= $11.3
trillion
8-A
Parable 3: Bacteria in a 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
...
Bottle is full at 12:00 (an hour later).
Q1: How many bacteria are in the bottle
then?
260
≈1.15 x1018
8-A
Parable 3: Bacteria in a 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
...
Bottle is full at 12:00 (an hour later).
Q2: When was the bottle half full?
At 11:59!
Parable 3: Bacteria in a Bottle
Q3: If you warn the bacteria at 11:56 of the
impending disaster - will anyone believe
you?
½full at 11:59
¼ full at 11:58
⅛ full at 11:57
1
16 full at 11:56 –
the amount of unused space is 15
times the amount of used space
The bottle fills VERY rapidly in the last 3
minutes.
8-A
Parable 3: Bacteria in a Bottle
Q4: At 11:59 the bacteria spread to 3 more
bottles. Now that they have 4 total bottles
to fill, how much time have they gained?
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
They’ve gained only 2 additional
minutes!
8-A
Parable 3: Bacteria in a Bottle
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.
8-A
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.
8-A
8-A
Repeated Doublings
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.
8-A
8-A
Parable 1
Square
Grains on
square
Total Grains
Formula for
total
1
1 = 20
1
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
15 + 16 =
31
...
25-1
...
...
64
263
...
264-1
Homework for Friday:
Page 496
# 8, 10, 12, 14, 18, 26