Section 9C Exponential Modeling (pages 585

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Section 9CExponential Modeling(pages 585 601)
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Exponential Growth (Decay) occurs when a quantity
increases (decreases) by the same relative
amountthat is, by the same percentagein each
unit of time. (Powertown -- 5 each year,
investments)
A Exponential Function grows (or decays) by the
same relative amount per unit of time.
Independent variable tDependent variable
Qdecimal growth rate rInitial Value
Q0 (dependent) initial value x (1
r)independent or Q Q0 x (1 r)t
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Comments/587

• Q Q0 x (1 r)t
• Units for r and for t must be the same.
• If Q0 is the initial value at time t0, then t
  must be measured in units since t0.
• In this formula, r is the decimal form of the
  percentage growth/decay rate
• If r gt 0, then quantity grows exponentially. If
  r lt 0, then quantity decays exponentially.

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Ex1/587 The 2000 census found a US population of
about 281 million. a) Write a function for the US
population that assumes a exponential
growth at 0.7 per year. b) Use the function
to predict the US population in 2100.
initial value (in 2000) 281 milliongrowth rate
.007 per yearindependent variable years since
2000dependent variable population
Q 281million x (1.007)t
year 2100 is 100 years since 2000, so t 100
Q 281million x (1.007)100 564 million
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Ex2/588 Chinas one-child policy was originally
implemented with the goal of reducing Chinas
population to 700 million by 2050. Chinas 2000
population was about 1.2 billion. Suppose
Chinas population declines at a rate of 0.5 per
year. a) write a function for the exponential
decay of the populationb) will this rate of
decline be sufficient to meet the original goal?
initial value (in 2000) 1.2 billionrate -.005
per yearindependent variable years since
2000dependent variable population
Q 1.2billion x (1-.005)t
year 2050 is 50 years since 2000, so t 50
Q 1.2billion x (.995)50 .934billion
With this model, the predicted population in 2050
is 934,000,000 and so the goal of 700,000,000
will not be met.
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What do graphs look like?
43/599 Your starting salary at a new job is 2000
per month and you get annual raises of 5 per
year.a) create an exponential function.b)
create a table showing Q values for the first 15
units of time.c) make a graph of the exponential
function.
Q 24000 x (1.05)t
curvy!
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Using the graph
43/599 Your starting salary at a new job is 2000
per month and you get annual raises of 5 per
year.Using the graph determinea) your salary
after 12 years.b) when your salary will be
30000.
When t 12, Q _______.
Q 30,000 when t _____
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Can we solve exactly using the function?
Use Properties of Logarithms
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Can we solve exactly using the function?
Q 24000 x (1.05)t
30000 24000 x (1.05)t
1.25 (1.05)t
The salary will be 30,000 in 4.6 years
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What do graphs look like?
39/598 A privately owned forest that had 1
million acres of old growth is being clear cut at
a rate of 7 per year.a) create an exponential
function.b) create a table showing Q values for
the first 15 units of time.c) make a graph of
the exponential function.
Q 1million x (1-.07)t 1mill.x (.93)t
curvy!
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When will the acreage be reduced by half?
Q 1000000 x (0.93)t
500000 1000000 x (0.93)t
The acreage will be reduced by half in 9.55 years.
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Other forms for Exponential Functions
Independent variable tDependent variable
Qdoubling time TdInitial Value Q0
Independent variable tDependent variable
Qhalf-life THInitial Value Q0
Units for t and Td (and TH) must be the same.
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51/599 The drug Valium is eliminated from the
bloodstream exponentially with a half-life of 36
hours. Suppose that a patient receives an initial
dose of 20 milligrams of Valium at midnight.a)
function How much Valium is in the patients
blood at noon the next day?b) graph Estimate
when the Valium concentration will reach 10 of
its initial level.
initial value (at midnight) 20
milligramshalf-life 36 hoursindependent
variable hours since midnightdependent
variable milligrams of Valium
noon the next day is 12 hours past midnight
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51/599 The drug Valium is eliminated from the
bloodstream exponentially with a half-life of 36
hours. Suppose that a patient receives an initial
dose of 20 milligrams of Valium at midnight.a)
function How much Valium is in the patients
blood at noon the next day?b) graph Estimate
when the Valium concentration will reach 10 of
its initial level.
10 of its initial value is 10 of 20 mg or
.10x20 2 mg
Q is 2 mg when t is about _________
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Can we solve exactly using the function?
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53/599 Uranium-238 has a half-life of 4.5 billion
years.You find a rock containing a mixture of
uranium-238 and lead. You determine that 85 of
the original uranium-238 remains the other 15
decayed into lead. How old is the rock?
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Homework Pages 598-599 38,40,42,52,54a
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