Modeling with Exponential and Logarithmic Functions

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Modeling with Exponential and Logarithmic
Functions
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Exponential Growth and Decay Models

• The mathematical model for exponential growth or
  decay is given by
• f (t) A0ekt or A A0ekt.
• If k 0, the function models the amount or size
  of a growing entity. A0 is the original amount or
  size of the growing entity at time t 0. A is
  the amount at time t, and k is a constant
  representing the growth rate.
• If k of a decaying entity. A0 is the original amount
  or size of the decaying entity at time t 0. A
  is the amount at time t, and k is a constant
  representing the decay rate.

decreasing
A0
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Example
The graph below shows the growth of the Mexico
City metropolitan area from 1970 through 2000. In
1970, the population of Mexico City was 9.4
million. By 1990, it had grown to 20.2 million.

• Find the exponential growth function that models
  the data.
• By what year will the population reach 40 million?

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Example cont.
Solution
a. We use the exponential growth model A
A0ekt in which t is the number of years since
1970. This means that 1970 corresponds to t 0.
At that time there were 9.4 million inhabitants,
so we substitute 9.4 for A0 in the growth
model. A 9.4 ekt
We are given that there were 20.2 million
inhabitants in 1990. Because 1990 is 20 years
after 1970, when t 20 the value of A is 20.2.
Substituting these numbers into the growth model
will enable us to find k, the growth rate. We
know that k 0 because the problem involves
growth.
A 9.4 ekt Use the growth model with A0 9.4.
20.2 9.4 ek20 When t 20, A 20.2.
Substitute these values.
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Example cont.
Solution
20.2/ 9.4 ek20 Isolate the exponential factor
by dividing both sides by 9.4.
ln(20.2/ 9.4) lnek20 Take the natural
logarithm on both sides.
20.2/ 9.4 20k Simplify the right side by using
ln ex x.
0.038 k Divide both sides by 20 and solve for k.
We substitute 0.038 for k in the growth model to
obtain the exponential growth function for Mexico
City. It is A 9.4 e0.038t where t is measured
in years since 1970.
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Example cont.
Solution
b. To find the year in which the population will
grow to 40 million, we substitute 40 in for A in
the model from part (a) and solve for t.
A 9.4 e0.038t This is the model from part (a).
40 9.4 e0.038t Substitute 40 for A.
40/9.4 e0.038t Divide both sides by 9.4.
ln(40/9.4) lne0.038t Take the natural
logarithm on both sides.
ln(40/9.4) 0.038t Simplify the right side by
using ln ex x.
ln(40/9.4)/0.038 t Solve for t by dividing
both sides by 0.038
Because 38 is the number of years after 1970, the
model indicates that the population of Mexico
City will reach 40 million by 2008 (1970 38).
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Text Example

• Use the fact that after 5715 years a given amount
  of carbon-14 will have decayed to half the
  original amount to find the exponential decay
  model for carbon-14.
• In 1947, earthenware jars containing what are
  known as the Dead Sea Scrolls were found by an
  Arab Bedouin herdsman. Analysis indicated that
  the scroll wrappings contained 76 of their
  original carbon-14. Estimate the age of the Dead
  Sea Scrolls.

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Text Example cont.
Solution A0/2 A0ek5715 After 5715 years, A
A0/2
1/2 ekt5715 Divide both sides of the equation
by A0.
ln(1/2) ln ek5715 Take the natural logarithm
on both sides.
ln(1/2) 5715k ln ex x.
k ln(1/2)/5715-0.000121 Solve for k.
Substituting for k in the decay model, the model
for carbon-14 is A A0e 0.000121t.
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Text Example cont.
Solution
A A0e-0.000121t This is the decay model for
carbon-14.
0.76A0 A0e-0.000121t A .76A0 since
76 of the initial amount remains.
0.76 e-0.000121t Divide both sides of the
equation by A0.
ln 0.76 ln e-0.000121t Take the natural
logarithm on both sides.
ln 0.76 -0.000121t ln ex x.
tln(0.76)/(-0.000121) Solver for t.
The Dead Sea Scrolls are approximately 2268 years
old plus the number of years between 1947 and the
current year.