Transcript Linear vs. Exponential

Section 8A
Growth: Linear vs.
Exponential
Pages 516-524
8-A
Growth: Linear vs Exponential
pg516 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.
8-A
Straightown: initially 10,000 people and
growing at a rate of 500 people per year
Year
0
1
2
3
10
15
20
40
Straightown
10,000
10,500
8-A
Straightown: initially 10,000 people and
growing at a rate of 500 people per year
Year
0
1
2
3
10
15
20
40
Straightown
10,000
10,500
11,000
8-A
Straightown: initially 10,000 people and
growing at a rate of 500 people per year
Year
0
1
2
3
10
15
20
40
Straightown
10,000
10,500
11,000
11,500
8-A
Straightown: initially 10,000 people and
growing at a rate of 500 people per year
Year
0
1
2
3
10
15
20
40
Straightown
10,000
10,500
11,000
11,500
10000 + (10x500) =15000
8-A
Straightown: initially 10,000 people and
growing at a rate of 500 people per year
Year
0
1
2
3
10
15
20
40
Straightown
10,000
10,500
11,000
11,500
10000 + (10x500) =15000
10000 + (15x500) =17500
8-A
Straightown: initially 10,000 people and
growing at a rate of 500 people per year
Year
0
1
2
3
10
15
20
40
Straightown
10,000
10,500
11,000
11,500
10000 + (10x500) =15000
10000 + (15x500) =17500
10000 + (20x500) =20000
8-A
Straightown: initially 10,000 people and
growing at a rate of 500 people per year
Year
0
1
2
3
10
15
20
40
Straightown
10000
10000
10000
10000
+
+
+
+
(10x500)
(15x500)
(20x500)
(40x500)
10,000
10,500
11,000
11,500
=15000
=17500
=20000
=30000
Powertown: initially 10,000 people and
growing at a rate of 5% per year
Year
0
1
2
3
10
15
20
40
Powertown
10,000
10000 x (1.05) = 10,500
8-A
Powertown: initially 10,000 people and
growing at a rate of 5% per year
Year
0
1
2
3
10
15
20
40
Powertown
10,000
10000 x (1.05) = 10,500
10000 x (1.05)2 = 11,025
8-A
Powertown: initially 10,000 people and
growing at a rate of 5% per year
Year
0
1
2
3
10
15
20
40
Powertown
10,000
10000 x (1.05) = 10,500
10000 x (1.05)2 = 11,025
10000 x (1.05)3 = 11,576
8-A
Powertown: initially 10,000 people and
growing at a rate of 5% per year
Year
0
1
2
3
10
15
20
40
Powertown
10000 x (1.05)
10000 x (1.05)2
10000 x (1.05)3
10000 x (1.05)10
=
=
=
=
10,000
10,500
11,025
11,576
16,289
8-A
Powertown: initially 10,000 people and
growing at a rate of 5% per year
Year
0
1
2
3
10
15
20
40
Powertown
10000 x (1.05)
10000 x (1.05)2
10000 x (1.05)3
10000 x (1.05)10
10000 x (1.05)15
=
=
=
=
=
10,000
10,500
11,025
11,576
16,289
20,789
8-A
Powertown: initially 10,000 people and
growing at a rate of 5% per year
Year
0
1
2
3
10
15
20
40
Powertown
10000 x (1.05)
10000 x (1.05)2
10000 x (1.05)3
10000 x (1.05)10
10000 x (1.05)15
10000 x (1.05)20
=
=
=
=
=
=
10,000
10,500
11,025
11,576
16,289
20,789
26,533
8-A
Powertown: initially 10,000 people and
growing at a rate of 5% per year
Year
0
1
2
3
10
15
20
40
Powertown
10000 x (1.05)
10000 x (1.05)2
10000 x (1.05)3
10000 x (1.05)10
10000 x (1.05)15
10000 x (1.05)20
10000 x (1.05)40
=
=
=
=
=
=
=
10,000
10,500
11,025
11,576
16,289
20,789
26,533
70,400
8-A
Population Comparison
Year
Straightown
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
8-A
Growth: Linear versus Exponential
8-A
Two Basic Growth Patterns
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 amount—that is, by the same
percentage—in each unit of time.
Example: Powertown: -- 5% each year
8-A
ex1/517 Linear/Exponential Growth/Decay?
a) 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
8-A
ex1/517 Linear/Exponential Growth/Decay?
b) The price of milk has been rising with inflation
at 3% per year.
• Which kind of growth is this?
Exponential Growth
• If the price was $2.00 / gallon one years ago, what
is it now?
1 years ago: $2.00/gallon
Now (1 years later): $2.00 × (1.03)1 = $2.06/gallon
8-A
ex1/517 Linear/Exponential Growth/Decay?
c) 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
8-A
ex1/517 Linear/Exponential Growth/Decay?
d) 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 company’s top of the line drive holds 800
gigabytes today, what will it hold in six years?
Now: 800 gigabytes
2 years: 1600 gigabytes
4 years: 3200 gigabytes
6 years: 6400 gigabytes (or 6.4 terabytes)
8-A
ex1/517 Linear/Exponential Growth/Decay?c
e) The price of high definition TV sets has been
falling by about 25% per year.
• Which kind of growth is this?
Exponential Decay
• If the price is $2000 today, what can you expect it
to be in 2 years?
Now: $2000
2 years: 2000 x (0.75)2 = $1125
8-A
More Practice
19/522 The population of Meadowview is increasing
at a rate of 623 people per year. If the
population is 2500 today, what will it be in four
years.
21/522 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 $120
one month during this period, what was it three
months later?
23/522 The price of computer memory is
decreasing at a rate of 12% per year. If a
memory chip costs $50 today, what will it cost in
3 years?
8-A
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
8-A
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.
4-C
8-A
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
8-A
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
8-A
Parable 1
Square
Grains on
square
Total Grains
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
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 = 1.8×1019 =
≈ 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.
Parable 2
The Magic Penny
8-A
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.
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
Amount under pillow
1 week (7 days)
2 weeks (14 days)
1 month (30 days)
50 days
$0.01×27= $1.28
8-A
Parable 2
Time
Amount under pillow
1 week (7 days)
2 weeks (14 days)
1 month (30 days)
50 days
$0.01×27= $1.28
$0.01×214= $163.84
8-A
Parable 2
Time
Amount under pillow
1 week (7 days)
$0.01×27= $1.28
2 weeks (14 days)
$0.01×214= $163.84
1 month (30 days)
$0.01×230=
$10,737,418.24
50 days
8-A
Parable 2
Time
Amount under pillow
1 week (7 days)
$0.01×27= $1.28
2 weeks (14 days)
$0.01×214= $163.84
1 month (30 days)
$0.01×230=
$10,737,418.24
$0.01×250= $11.3 trillion
50 days
Parable 2
The Magic Penny
8-A
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.
Parable 3
Bacteria in a Bottle
Parable 3 Suppose you place a single bacterium in a
bottle at 11:00 am. 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.
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 11:56, will anyone believe you?
Question3: At 11:59, your fellow bacteria find 3 more bottles
to fill. How much time have they gained for the bacteria
civilization?
8-A
8-A
Question0: If the bottle is full at NOON, how many
bacteria are in the 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
. . .
At 12:00 (60 minutes later) the bottle is full and
contains 260 ≈ 1.15 x1018
8-A
Question1: When was the bottle half full?
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 (60 minutes later) and so is
1/2 full at 11:59 am
8-A
Question2: If you (a mathematically sophisticated
bacterium) warn of impending disaster at 11:56,
will anyone believe you?
½ full at 11:59
¼ full at 11:58
⅛ full at 11:57
1
full at 11:56
16
At 11:56 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 11:56.
8-A
Question3: At 11:59, 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 12:00
enough bacteria to fill 2 bottles at 12:01
enough bacteria to fill 4 bottles at 12:02
They have gained only 2 additional minutes for
the bacteria civilization.
8-A
Question4: Is this scary?
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.
Yes, this is scary!
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