This document discusses exponential functions and their applications to modeling compound interest, population growth, radioactive decay, and dating artifacts. It provides definitions of compound interest, continuous compounding, and exponential decay models. Examples are given to illustrate using these models to calculate amounts with compound interest over time, population sizes over generations, radioactive material half-life decay, and estimating the ages of artifacts based on their remaining radioactive carbon.

1. 5-3: Applications of
Exponential Functions
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Create and use exponential
models

2. 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.
(1 )t
A P r= +
An exponential grow function

3. 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

4. ExampleIf you invest
$9000 at 4%
interest
compounded
monthly, use
the compound
interest model
to find how
much you
have after 12
years.

5. 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

6. 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. rt
A Pe=

7. 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

8. Important Idea
Compounding Amount*
Annual $14409.29
Monthly $14533.06
Weekly $14541.99
Continuous $14544.67
*$9000 @ 4% for 12 yrs.

9. 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.

10. Important Idea
(1 )t
A P r= +
( ) (1 )x
f x P r= +
Exponentia
l Grow
Model:
Compound
Interest
Model: The models are
the same.

11. 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?

12. Try This
A newly formed lake is
stocked with 900 fish. After 6
months, biologists estimate
there are 1710 fish.
Assuming exponential
growth, how many fish will
there be after 24 months?
11729

13. Definition
Exponential Decay is of the
form where( ) (1 )x
f x P r= −
( )f x is the amount at time x,
P is the beginning amount
(at t=0) and r is the decay
rate.

14. 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.

15. 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.

16. Definition
The amount of a radioactive
substance that remains is
given by
( ) (.5)
x
hf x P=
where is the remaining
amount, P is the initial
amount and h is the half life
of the substance.
( )f x

17. Example
An isotope of strontium-90
has a half-life of 25 years.
How much of an 18mg.
sample be left after 15
years?

18. 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.

19. 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.

20. 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

21. 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

22. Lesson Close
Exponential models are
widely used to solve
compound interest,
population growth and
resource depletion
problems.