Applications of Exponential Functions

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Applications of Exponential Functions
Learning Goals

• Create and use exponential models

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Definition
Compound Interest Model
If P dollars is invested at interest rate r
(expressed as a decimal) per time period t, then
A is the amount after t periods.
An exponential grow function
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Try This
If you invest 9000 at 4 interest compounded
annually, use the compound interest model to find
how much you have after the end of 12 years.
14409.29
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Example
If you invest 9000 at 4 interest compounded
monthly, use the compound interest model to find
how much you have after 12 years.
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Try This
If you invest 9000 at 4 interest compounded
weekly, use the compound interest model to find
how much you have after the end of 12 years.
There are 52 weeks in a year.
14541.99
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Definition
The Continuous Compounding Model
If P dollars are invested at an annual interest
rate of r compounded continuously then A is the
amount after t years.
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Try This
If you invest 9000 at 4 interest compounded
continuously, use the continuous compounding
model to find how much you have after the end of
12 years.
14544.67
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Important Idea
9000 _at_ 4 for 12 yrs.
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Example
The world population in 1950 was about 2.5
billion people and has been increasing at
approximately 1.85 per year. Write the equation
that gives the world population in year x when x
corresponds to 1950.
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Important Idea
Exponential Grow Model
Compound Interest Model
The models are the same.
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Example
At the beginning of an experiment, a culture
contains 1000 bacteria. Five hours later there
are 7600 bacteria. Assuming the bacteria are
growing exponentially, how many bacteria will
there be after 24 hours?
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Try This
A newly formed lake is stocked with 900 fish.
After 5 months, biologists estimate there are
1710 fish. Assuming exponential growth, how many
fish will there be after 24 months?
11729
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Definition
Exponential Decay is of the form
where
is the amount at time x, P is the
beginning amount (at t0) and r is the decay rate.
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Example
Each day 15 of the chlorine in a swimming pool
evaporates. Use the Exponential Decay Model to
predict how many days are required for 60 of the
chlorine to evaporate.
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Definition
The half life of a radioactive substance is the
time required for a given quantity of the
substance to decay to one-half its original mass.
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Definition
The amount of a radioactive substance that
remains is given by
where is the remaining amount, P is the
initial amount and h is the half life of the
substance.
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Example
An isotope of strontium-90 has a half-life of 25
years. How much of an 18mg. sample be left after
15 years?
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Try This
Carbon 14 is a radio-active substance with a half
life of 5730 years. How much of a 5 mg. sample
will be left after 300 years?
4.82 mg.
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Example
In 1988, the Vatican authorized the British
Museum to date a cloth relic know as the Shroud
of Turin. This cloth which first surfaced in the
14th century, contains the negative image of a
human body that was widely believed to be that of
Jesus.
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The report of the museum showed that the fibers
of the cloth contained about 93 of their
original carbon-14. The half-life of carbon-14
is 5730 years. Estimate the age of the shroud.
Adapted from Calculus, Howard Anton, 6th ed
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Try This
If a turtle shell is found and tested to have 45
of its original carbon-14, how old is the shell?
The half-life of carbon-14 is 5730 years.
About 6599 years
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Lesson Close
Exponential models are widely used to solve
compound interest, population growth and resource
depletion problems.