 Making
2.5
Models
Using Variation
Fundamentals
 
When scientists talk about a mathematical model for a realworld 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.
  Direct Variation
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.
A line is an example of a direct variation – y=mx+b
Direct Variation
  So, the
graph of an equation y = kx
that describes direct variation is
a line with:
Slope k
  y-intercept 0
 
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.
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?
Direct Variation
  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.
Direct Variation
  To
find k (the constant of proportionality), we
use the fact that t = 5 when d = 5400.
 
Substituting these values in the equation,
we get:
5400 = k(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.
Direct Variation
  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).
Direct Variation
  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
where k is a constant.
Inverse Variation
 
If the quantities x and y are related by
the equation
for some constant k ≠ 0,
we say that:
 
 
y is inversely proportional to x.
y varies inversely as x.
Inverse Variation
 
Ex 35. Loudness of Sound
 
The loudness of sound is inversely proportional to the
square of the distance, d from the source of the sound. A
person who is 10 ft away from a lawnmower experiences a
sound level of 70dB
How loud is the lawn mower when the person is 100 ft
away?
 
Inverse Variation
 
 
 
 
 
 
 
L = loudness
L=k/d2
Now plug in values to solve k to get the constant of
proportionality.
K=L/d2 = 70/(10)2=70*100=7,000
Now use k and d to find L:
L=(7000)/(100)2 = .7DB
Since 20dB is a whisper, you would imagine that you
cannot hear this lawn mower at 100 ft
  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
we say that z is proportional to x
and inversely proportional to y.
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.
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:
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.
So gravitational force is inversely proportional to the
radius squared between the two masses.