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.
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