Section 8A Growth: Linear vs. Exponential

1
Section 8AGrowth Linear vs. Exponential

• Pages 490-495

2
Growth Linear versus Exponential
8-A

• 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.

3
Population Comparison
Year Straighttown
1 10,500
2
3
10
15
20
40
4
Population Comparison
Year Straighttown
1 10,500
2 11,000
3
10
15
20
40
5
Population Comparison
Year Straighttown
1 10,500
2 11,000
3 11,500
10
15
20
40
6
Population Comparison
Year Straighttown
1 10,500
2 11,000
3 11,500
10 10,000(50010) 15,000
15
20
40
7
Population Comparison
Year Straighttown
1 10,500
2 11,000
3 11,500
10 10,000(50010) 15,000
15 10,000(50015) 17,500
20
40
8
Population Comparison
Year Straighttown
1 10,500
2 11,000
3 11,500
10 10,000(50010) 15,000
15 10,000(50015) 17,500
20 10,000(50020) 20,000
40
9
Population Comparison
Year Straighttown
1 10,500
2 11,000
3 11,500
10 10,000(50010) 15,000
15 10,000(50015) 17,500
20 10,000(50020) 20,000
40 10,000(50040) 30,000
10
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






11
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
10,500 1.05 11,025





12
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
10,500 1.05 11,025
11,025 1.05 11,576




13
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
10,500 1.05 11,025
11,025 1.05 11,576
10,000 (1.05)10 16,289



14
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
10,500 1.05 11,025
11,025 1.05 11,576
10,000 (1.05)10 16,289
10,000 (1.05)15 20,789


15
Population Comparison
Year Straightown
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
10,500 1.05 11,025
11,025 1.05 11,576
10,000 (1.05)10 16,289
10,000 (1.05)15 20,789
10,000 (1.05)20 26,533

16
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
10,500 1.05 11,025
11,025 1.05 11,576
10,000 (1.05)10 16,289
10,000 (1.05)15 20,789
10,000 (1.05)20 26,533
10,000 (1.05)40 70,400
17
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
10,500
11,025
11,576
16,289
20,789
26,533
70,400
18
Growth Linear versus Exponential
8-A
19
Growth Linear versus Exponential
8-A
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 amountthat is,
by the same percentagein each unit of
time. Example Powertown -- 5 each year
20
Linear or Exponential Growth?
8-A

• 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 1.8631.035 1.93 / gallon

21
Linear or Exponential Decay?
8-A

• 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

22
Linear or Exponential Growth?
8-A
The memory capacity of state-of-the art computer
hard drives is doubling approximately every two
years. If a companys 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
23
Parable 1 Chess Board
8-A
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 . . .
24
Parable 1
4-C
Square Grains on square
1 1 20
2 2 21
3 4 22 22
4 8 23 222
5 16 24 2222
. . . . . .
25
Parable 1
4-C
Square Grains on square
1 1 20
2 2 21
3 4 22
4 8 23
5 16 24
. . . . . .
64 263
26
Parable 1
4-C
Square Grains on square Total Grains on chessboard
1 1 20 1
2 2 21 12 3
3 4 22
4 8 23
5 16 24
. . . . . .
64 263
27
Parable 1
4-C
Square Grains on square Total Grains on chessboard
1 1 20 1
2 2 21 12 3
3 4 22 34 7
4 8 23
5 16 24
. . . . . . . . .
64 263
28
Parable 1
4-C
Square Grains on square Total Grains on chessboard
1 1 20 1
2 2 21 12 3
3 4 22 34 7
4 8 23 78 15
5 16 24
. . . . . . . . .
64 263
29
Parable 1
4-C
Square Grains on square Total Grains on chessboard
1 1 20 1
2 2 21 12 3
3 4 22 34 7
4 8 23 78 15
5 16 24 15 16 31
. . . . . . . . .
64 263
30
Parable 1
4-C
Square Grains on square Total Grains on chessboard
1 1 20 1
2 2 21 12 3
3 4 22 34 7
4 8 23 78 15
5 16 24 15 16 31
. . . . . . . . .
64 263 264 - 1
31
Parable 1
4-C
Square Grains on square Total Grains on chessboard Formula for total on board
1 1 20 1 21 1
2 2 21 12 3 22 1
3 4 22 34 7 23 1
4 8 23 78 15 24 1
5 16 24 15 16 31 25 1
. . . . . . . . . . . .
64 263 264 - 1 264 - 1
32
Parable 1 Chess Board
8-A
264 1 1.81019 18 billion, billion
grains of wheat This is more than all the grains
of wheat harvested in human history.
33
Parable 2 Magic Penny
8-A
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???
34
Parable 2
8-A
Day Amount under pillow
0 0.01
1 0.02
2 0.04
3 0.08
4 0.16
. . .
35
Parable 2
8-A
Day Amount under pillow Amount under pillow
0 0.01 0.01 0.0120
1 0.02 0.02 0.0121
2 0.04 0.04 0.0122
3 0.08 0.08 0.0123
4 0.16 0.16 0.0124
. . .
t 0.012t
36
Parable 2
8-A
Time Amount under pillow
1 week (7 days) 0.0127 1.28
2 weeks (14 days)
1 month (30 days)
50 days
37
Parable 2
8-A
Time Amount under pillow
1 week (7 days) 0.0127 1.28
2 weeks (14 days) 0.01214 163.84
1 month (30 days)
50 days
38
Parable 2
8-A
Time Amount under pillow
1 week (7 days) 0.0127 1.28
2 weeks (14 days) 0.01214 163.84
1 month (30 days) 0.01230 10,737,418.24
50 days
39
Parable 2
8-A
Time Amount under pillow
1 week (7 days) 0.0127 1.28
2 weeks (14 days) 0.01214 163.84
1 month (30 days) 0.01230 10,737,418.24
50 days 0.01250 11.3 trillion
40
Parable 3 Bacteria in a Bottle
8-A
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 (an hour later). Q1 How many bacteria
are in the bottle then? 260 1.15 x1018
41
Parable 3 Bacteria in a Bottle
8-A
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 (an hour later). Q2 When was the bottle
half full? At 1159!
42
Parable 3 Bacteria in a Bottle
8-A
Q3 If you warn the bacteria at 1156 of the
impending disaster - will anyone believe you?
½full at 1159 ¼ full at 1158 ? full at 1157
full at 1156 the amount of unused space
is 15 times the amount of used space The bottle
fills VERY rapidly in the last 3 minutes.
43
Parable 3 Bacteria in a Bottle
8-A
Q4 At 1159 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
1200 enough bacteria to fill 2 bottles at 1201
enough bacteria to fill 4 bottles at 1202
Theyve gained only 2 additional minutes!
44
Parable 3 Bacteria in a Bottle
8-A
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.
45
Key Facts about Exponential Growth
8-A
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.
46
Repeated Doublings
8-A
47
Key Facts about Exponential Growth
8-A
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.
48
Parable 1
8-A
Square Grains on square Total Grains Formula for total
1 1 20 1 21-1
2 2 21 12 3 22-1
3 4 22 34 7 23-1
4 8 23 78 15 24-1
5 16 24 15 16 31 25-1
. . . . . . . . . . . .
64 263 264-1
49

• Homework for Friday
• Page 496
• 8, 10, 12, 14, 18, 26