Transcript Slide 1

Fundamentals
When scientists talk about a mathematical
model for a real-world phenomenon, they
often mean an equation that describes
the relationship between two quantities.
• For instance, the model may describe how
the population of an animal species varies
with time or how the pressure of a gas varies
as its temperature changes.
Fundamentals
In this section, we study
a kind of modeling called
variation.
1.11 Modeling Variation
Modeling Variation
Two types of mathematical models
occur so often that they are given
special names.
Direct Variation
Direct Variation
Direct variation occurs when
one quantity is a constant multiple
of the other.
• So, we use an equation of the form y = kx
to model this dependence.
Direct Variation
If the quantities x and y are related by
an equation y = kx for some constant k ≠ 0,
we say that:
• y varies directly as x.
• y is directly proportional to x.
• y is proportional to x.
The constant k is called the constant
of proportionality.
Direct Variation
Recall that the graph of an equation
of the form
y = mx + b
is a line with:
• Slope m
• y-intercept b
Direct Variation
So, the graph of an equation y = kx
that describes direct variation is
a line with:
• Slope k
• y-intercept 0
E.g. 1—Direct Variation
During a thunderstorm, you see the lightning
before you hear the thunder because light
travels much faster than sound.
• The distance between you and the storm
varies directly as the time interval between
the lightning and the thunder.
E.g. 1—Direct Variation
(a) Suppose that the thunder from
a storm 5,400 ft away takes 5 s
to reach you.
•
Determine the constant of proportionality
and write the equation for the variation.
E.g. 1—Direct Variation
(b) Sketch the graph of this equation.
•
What does the constant of proportionality
represent?
(c) If the time interval between the lightning
and thunder is now 8 s, how far away is
the storm?
E.g. 1—Direct Variation
Example (a)
Let d be the distance from you to the storm
and let t be the length of the time interval.
• We are given that d varies directly as t.
• So,
d = kt
where k is a constant.
E.g. 1—Direct Variation
Example (a)
To find k, we use the fact that t = 5
when d = 5400.
• Substituting these values in the equation,
we get:
5400 = k(5)
5400
k
 1080
5
E.g. 1—Direct Variation
Example (a)
Substituting this value of k in the equation
for d, we obtain:
d = 1080t
as the equation for d as a function of t.
E.g. 1—Direct Variation
Example (b)
The graph of the equation d = 1080t is
a line through the origin with slope 1080.
• The constant
k = 1080 is
the approximate
speed of sound
(in ft/s).
E.g. 1—Direct Variation
Example (c)
When t = 8, we have:
d = 1080 ∙ 8 = 8640
• So, the storm is 8640 ft ≈ 1.6 mi away.
Inverse Variation
Inverse Variation
Another equation that is frequently used
in mathematical modeling is
y = k/x
where k is a constant.
Inverse Variation
If the quantities x and y are related by
the equation
k
y
x
for some constant k ≠ 0,
we say that:
• y is inversely proportional to x.
• y varies inversely as x.
Inverse Variation
The graph of y = k/x for x > 0 is shown
for the case k > 0.
• It gives a picture of
what happens when y
is inversely proportional
to x.
E.g. 2—Inverse Variation
Boyle’s Law states that:
• When a sample of gas is compressed
at a constant temperature, the pressure
of the gas is inversely proportional to
the volume of the gas.
E.g. 2—Inverse Variation
(a) Suppose the pressure of a sample
of air that occupies 0.106 m3 at 25°C
is 50 kPa.
•
Find the constant of proportionality.
•
Write the equation that expresses
the inverse proportionality.
E.g. 2—Inverse Variation
(b)If the sample expands to
a volume of 0.3 m3, find
the new pressure.
E.g. 2—Inverse Variation
Example (a)
Let P be the pressure of the sample of gas
and let V be its volume.
• Then, by the definition of inverse proportionality,
we have:
k
P
V
where k is a constant.
E.g. 2—Inverse Variation
Example (a)
To find k, we use the fact that P = 50
when V = 0.106.
• Substituting these values in the equation,
we get:
k
50 
0.106
k = (50)(0.106) = 5.3
E.g. 2—Inverse Variation
Example (a)
Putting this value of k in the equation
for P, we have:
5.3
P
V
E.g. 2—Inverse Variation
Example (b)
When V = 0.3, we have:
5.3
P
 17.7
0.3
• So, the new pressure is about 17.7 kPa.
Joint Variation
Joint Variation
A physical quantity often depends on more
than one other quantity.
If one quantity is proportional to two or more
other quantities, we call this relationship
joint variation.
Joint Variation
If the quantities x, y, and z are related by
the equation
z = kxy
where k is a nonzero constant,
we say that:
• z varies jointly as x and y.
• z is jointly proportional to x and y.
Joint Variation
In the sciences, relationships between
three or more variables are common.
• Any combination of the different types of
proportionality that we have discussed is possible.
• For example, if
x
zk
y
we say that z is proportional to x
and inversely proportional to y.
E.g. 3—Newton’s Law of Gravitation
Newton’s Law of Gravitation says that:
Two objects with masses m1 and m2 attract
each other with a force F that is jointly
proportional to their masses and inversely
proportional to the square of the distance r
between the objects.
• Express the law as an equation.
E.g. 3—Newton’s Law of Gravitation
Using the definitions of joint and inverse
variation, and the traditional notation G for
the gravitational constant of proportionality,
we have:
m1m2
F G 2
r
Gravitational Force
If m1 and m2 are fixed masses,
then the gravitational force between them
is:
F = C/r2
where C = Gm1m2 is a constant.
Gravitational Force
The figure shows the graph of this
equation for r > 0 with C = 1.
• Observe how
the gravitational
attraction decreases
with increasing
distance.