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Solve applied problems involving exponential growth and decay. Solve applied problems involving compound interest.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
5.6 Applications and Models: Growth and Decay; and Compound Interest
Population Growth
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
The function
P(t) = P0 ekt, k > 0
can model many kinds of population growths.
In this function:
P0 = population at time 0,
P(t) = population after time t,
t = amount of time,
k = exponential growth rate.
The growth rate unit must be the same as the time unit.
Example
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
In 2009, the population of Mexico was about 111.2 million, and the exponential growth rate was 1.13% per year.a) Find the exponential growth function.
b) Graph the exponential growth function.
c) Estimate the population in 2014.
d) After how long will the population be double what it was in 2009?
Interest Compound Continuously
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
The function
P(t) = P0ekt can be used to calculate interest that is compounded continuously.
In this function:
P0 = amount of money invested, P(t) = balance of the account after t years, t = years, k = interest rate compounded continuously.
Example
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Suppose that $2000 is invested at interest rate k, compounded continuously, and grows to $2504.65 after 5 years. a. What is the interest rate?
b. Find the exponential growth function.
c. What will the balance be after 10 years?
d. After how long will the $2000 have doubled?
Growth Rate and Doubling Time
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
The growth rate k and doubling time T are related by
kT = ln 2 or or
Note that the relationship between k and T does not depend on P0 .
k ln2
TT
ln2
k
Example
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
The population of Kenya is now doubling every 25.8 years. What is the exponential growth rate?
Models of Limited Growth
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
In previous examples, we have modeled population growth. However, in some populations, there can be factors that prevent a population from exceeding some limiting value.
One model of such growth is
which is called a logistic function. This function increases toward a limiting value a as t approaches infinity. Thus, y = a is the horizontal asymptote of the graph.
P(t) a
1be kt
Models of Limited Growth - Graph
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
P(t) a
1be kt
Exponential Decay
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Decay, or decline, of a population is represented by the function
P(t) = P0ekt, k > 0.
In this function: P0 = initial amount of the substance (at time t = 0), P(t) = amount of the substance left after time, t = time, k = decay rate.
The half-life is the amount of time it takes for a substance to decay to half of the original amount.
Decay Rate and Half-Life
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
The decay rate k and the half-life T are related by
kT = ln 2 or or
Note that the relationship between decay rate and half-life is the same as that between growth rate and doubling time.
k ln2
TT
ln2
k
Example
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Carbon Dating. The radioactive element carbon-14 has a half-life of 5750 years. The percentage of carbon-14 present in the remains of organic matter can be used to determine the age of that organic matter. Archaeologists discovered that the linen wrapping from one of the Dead Sea Scrolls had lost 22.3% of its carbon-14 at the time it was found. How old was the linen wrapping?
k ln2
5750k 0.00012
Now we have the function P t P0e 0.00012t .