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Math for Liberal Arts
MAT 110: Chapter 8 Notes
Growth: Linear vs. Exponential
David J. Gisch
February 28, 2012
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.
Growth: Linear versus Exponential
• Straightown grows by the same absolute amount each
year and Powertown grows by the same relative amount
each year.
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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.
Growth: Linear versus Exponential
Example 8.A.1: Recall simple interest versus compound
interest. Simple interest is the same amount of interest
every time as where compound interest is the same
percent of interest at each step of time. For example, let's
say we invested $500 with an interest rate of 10%.
Year
Simple
1
$1,000
$1,000
2
$1,000+100=$1,100
$1,000+100=$1,100
$1,100+110=$1,210
3
$1,100+100=$1,200
4
$1,200+100=$1,300
$1,210+121=$1,331
5
$1,300+100=$1,400
$1,331+133.10=$1,464.10
Linear: We add the
same amount, $100,
every time.
Growth: Linear versus Exponential
Compound
Exponential: We
add the same percent,
10%, every time.
Growth: Linear versus Exponential
Example 8.A.2: 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.
Example 8.A.3: You are given a choice, take $1000 each
month for the rest of your life or be given a magic penny.
The magic penny will turn into two pennies after one day.
Then double again into four pennies the next day, and so
on. Which option would you rather take?
If the bottle is half-full at 11:59, when will the bottle be
completely full?
After 30 Years!
• $1,000 option: You have $1000 12 30
• Penny: You have $0.01 2
$360,000
$10,737,418.24
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Linear or Exponential?
Linear or Exponential?
Example 8.A.4: The price of milk is increasing by 3 cents
per week.
Example 8.A.5: The price of a house is increasing by 2%
per year.
(a) Is this exponential or linear?
(a) Is this exponential or linear?
(b) If the price of milk is $3.65 today, what will it be in 5
weeks?
(b) If the price of the house is $175,000 today, what will it
be in 5 years?
Exponential Growth & Decay
• The time required for each doubling in exponential
growth is called doubling time.
• The time required for each halving in exponential decay
is called halving time.
Doubling Time and Half-Life
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Doubling Time
Doubling Time (Exponential Growth)
• After a time , an exponentially growing quantity with a
doubling time of
increases in size by a factor of
2⁄
. The new value of the growing quantity is
related to its initial value (at 0) by
2⁄
Example 8.B.1: Recall this chart from the last section. Can
you use the chart to create the formula for Powertown?
Whatever unit of time is used to measure your doubling period is the unit of
time you should use for .
For example, is a bacteria doubles every 8 hours then must be measured in
hours.
Doubling Time (Exponential Growth)
Example 8.B.2: Using your equation, what will the
population be in 30 years? Does this match the chart?
10,000
2
10,000 2
441,636
⁄
⁄
Doubling Time (Exponential Growth)
Example 8.B.3: World Population Growth: World
population doubled from 3 billion in 1960 to 6 billion in
2000. Suppose that the world population continued to
grow (from 2000 on) with a doubling time of 40 years.
What would be the population in 2050?
• Always identify your initial value and year first!
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Approximate Double Time Formula (The Rule of 70)
For a quantity growing exponentially at a rate of P% per
time period, the doubling time is approximately
70
This approximation works best for small growth rates and
breaks down for growth rates over about 15%.
Doubling Time (Exponential Growth)
Example 8.B.4: A community of rabbits begins with an
initial population of 100 and grows 7% per month.
(a) What is the approximate doubling time?
(b) By what factor does the population increase in 18 months?
(c) What is the population after 3 years?
Doubling Time (Exponential Growth)
Doubling Time (Exponential Growth)
Example 8.B.5: A community of zombies doubles every 6
hours.
Example 8.B.6: The number of DMACC students doubles in
every 16 years.
(a) What is the approximate rate (percent) of increase?
(a) What is the approximate rate (percent) of increase?
(b) By what factor does the population increase in 24 hours?
(b) If the population was 18,000 students in 2000, what will the
population be in 2030?
(c) What is the population after one week?
(c) By what factor did the population increase in in that period?
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Half-Life Time (Exponential Decay)
• After a time , an exponentially decreasing quantity with
a half-life time of
decreases in size by a factor of
⁄
. The new value of the decreasing quantity is
related to its initial value (at 0) by
⁄
1
2
Approximate Half-Life Time Formula (The Rule of 70)
For a quantity decaying exponentially at a rate of P% per
time period, the half-life time is approximately
70
This approximation works best for small decay rates and
breaks down for decay rates over about 15%.
Whatever unit of time is used to measure your half-life period is the unit of
time you should use for .
For example, if a radio isotope decays with a half-life of 5100 years, then
must be measured in years.
Half-Life Time (Exponential Decay)
Example 8.B.7: Carbon-14 is used to carbon-date decaying
remains, whether it be plant or animal. The half-life of
Carbon-14 is 5730 years.
Half-Life Time (Exponential Decay)
Example 8.B.8: You start with 10 pounds of compost. It
takes 3 months to break down to 5 pounds.
(a) Write an equation modeling the amount of compost.
(a) Write an equation modeling the amount of Carbon-14 of an object.
(b) How many pounds will their be after 1 year?
(b) Using guess and check, if a bone has 10% of its original carbon-14
left, how old is the bone?
(c) By what factor did the population decrease in in that period?
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Half-Life Time (Exponential Decay)
Example 8.B.9: Since 1900 the buying power of one dollar
has decreased 3% per year.
(a) What is the approximate half-life time?
(b) How much buying power does a 1900 dollar approximately have
today?
(c) By what factor did the 1900 dollar decrease in in that period?
Exact Formulas
Exact Formulas
• For more precise work use these exact formulas.
• For an exponentially growing quantity, the doubling
time is
log 2
log 1
• For an exponentially decreasing quantity, the doubling
time is
log 2
log 1
• In both cases is the percent (as a decimal).
Log is a mathematical function similar to a square root. I could teach you
how to calculate it by hand and what it truly means but is easier if we just
skip that and know it is a button on our calculator.
Exact Formulas
Example 8.B.10: Calculate each of the following.
Example 8.B.11: Calculate each of the following.
(a) What is the approximate half-life time of a quantity that decays by
7% per month?
(a) What is the approximate doubling time of a quantity that increases
by 4% per year?
(b) What is the exact half-life time of a quantity that decays by 7% per
month?
(b) What is the exact doubling time of a quantity that increases by 4%
per year?
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Exponential Growth/Decay
Example 8.B.12: If you had an account that had a
compound interest rate of 5% per month, how long would it
take for your money to double?
Exponential Growth/Decay
Example 8.B.13: Urban encroachment is causing the area
of forest to decline at a rate of 4.25% per year.
(a) What is the exact half-life time?
Month
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Interest Balance
1000.00
50.00 1050.00
52.50 1102.50
55.13
1157.63
57.88
1215.51
60.78 1276.28
63.81 1340.10
67.00 1407.10
70.36 1477.46
73.87
1551.33
77.57 1628.89
81.44 1710.34
85.52 1795.86
89.79 1885.65
94.28 1979.93
99.00 2078.93
log 2
log 1 .05
log 2
log 1 .0425
14.206
15.96
(a) If the forest in that local area started with 500,000 acres, how
much will be left after 20 years?
500,000 0.5
⁄
.
209,768.09
The amount has doubled between month 14 and 15.
Also, notice that we started with month 0.
Acres in Perspective
• One square mile is 640 acres.
• Roughly 209,768 acres.
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