Transcript Section 3.5 - Warren County Public Schools

Section 3.5
Exponential and Logarithmic
Models
Important Vocabulary



Bell-shaped curve The graph of a
Gaussian model.
Logistic curve A model for describing
populations initially having rapid growth
followed by a declining rate of growth.
Sigmoidal curve Another name for a
logistic growth curve.
5 Most Common Types of Models

The exponential
growth model

Y = aebx

The exponential decay
model

Y = ae-bx


The Gaussian model

The logistic growth
model


Logarithmic models

y = ae-(x-b)^2/c
Y = a/(1 + be-rx)
Y = a + b ln x and
y = a + b log10 x
Section 3.5 Mathematical
Models
Exponential Growth




Suppose a population is
growing according to the
model P = 800e0.05t , where t
is given in years.
(a) What is the initial size of
the population?
(b) How long will it take this
population to double?
Carbon Dating Model

To estimate the age of dead organic
matter, scientists use the carbon dating
model R = 1/1012 e−t/8245 , which denotes
the ratio R of carbon 14 to carbon 12
present at any time t (in years).
Exponential Decay

The ratio of carbon 14
to carbon 12 in a
fossil is R = 10−16.
Find the age of the
fossil.
Gaussian Models


The Gaussian model is
commonly used in
probability and statistics
to represent populations
that are normally
distributed .
On a bell-shaped curve,
the average value for a
population is where the
maximum y-value of the
function occurs.
Gaussian Models


The test scores at Nick
and Tony’s Institute of
Technology can be
modeled by the following
normal distribution y =
0.0798e-(x-82)^2/50 where x
represents the test
scores.
Sketch the graph and
estimate the average test
score.
Logistic Growth Models


Give an example of a
real-life situation that
is modeled by a
logistic growth model.
Use the graph to help
with the explanation.
Logistic Growth Examples

The state game commission
released 100 pheasant into a
game preserve. The agency
believes that the carrying
capacity of the preserve is 1200
pheasants and that the growth of
the flock can be modeled by
p(t) = 1200/(1 + 8e-0.1588t) where
t is measured in months.
How long will it take the flock to
reach one half of the preserve’s
carrying capacity?
Logarithmic Models

The number of kitchen
widgets y (in millions)
demanded each year is
given by the model
y = 2 + 3 ln(x + 1) ,
where x = 0 represents
the year 2000 and x ≥ 0.
Find the year in which the
number of kitchen
widgets demanded will be
8.6 million.
Homework
P. 232 – 236
28, 29, 32, 37, 39, 41
