1.10 Variation
Objective
• Write mathematical models for direct
variation, direct variation as an nth power,
inverse variation, and joint variation.
Direct Variation
• Examine the graph at the right:
The linear equation graph at
the right shows that as the x
value increases, so does the y
value increase for the
coordinates that lie on this line
.
• For instance, if x = 2, y = 4.
If x = 6 (multiplied by 3),
then y = 12 (also multiplied by
3).
• This is a graph of direct variation. If the value
of x is increased, then y increases as well.
Both variables change in the same manner.
If x decreases, so does the value of y. We say
that y varies directly as the value of x.
• A direct variation between 2 variables, y and x,
is a relationship that is expressed as:
y = kx
• where the variable k is called the constant of
proportionality.
Direct Variation
•
•
•
•
•
The following statements are equivalent.
1. y varies directly as x.
2. y is directly proportional to x.
3. y = kx for some nonzero constant k.
k is the constant of variation or the
constant of proportionality.
Example 1
• The simple interest on an investment is
directly proportional to the amount of the
investment. By investing $2500 in a
certain bond issue, you obtained an
interest payment of $187.50 at the end of
1 year. Find a mathematical model that
gives the interest I for this bond issue at
the end of 1 year in terms of the amount
invested P.
I  kP
187.50  k 2500
187.50
k
2500
.075  k
Model I  .075k
Direct Variation as an nth Power
• Another type of direct variation relates one
variable to a power of another variable.
A r
2
• The area A is directly proportional to the square
of the radius r.
Direct Variation as an nth Power
• The following statements are equivalent.
• 1. y varies directly as the nth power of x.
• 2. y is directly proportional to the nth power of
x.
n
• 3. y  kx
Example 2
• Neglecting air resistance, the distance s
an object falls varies directly as the square
of the duration t of the fall. An object falls
a distance of 144 feet in 3 seconds.
• Write an equation relating distance s and
duration t.
• How far will an object fall in 6 seconds?
s  kt
2
144  k (3)
144
k
9
Model s  16t 2
2
s  16t 2 Substitute 6 for t
s  16(6)  576 feet
2
Inverse Variation
(The opposite of direct variation)
• In an inverse variation, the values of
the two variables change in an opposite
manner - as one value increases, the
other decreases.
For instance, a biker traveling at 8 mph
can cover 8 miles in 1 hour. If the
biker's speed decreases to 4 mph, it will
take the biker 2 hours (an increase of
one hour), to cover the same distance.
• Inverse variation: when one variable
increases,
the other variable decreases.
• Notice the shape of the graph of inverse
variation.
If the value of x is increased, then y
decreases.
If x decreases, the y value increases. We say
that y varies inversely as the value of x.
• An inverse variation between 2 variables, y and
x, is a relationship that is expressed as:
»y = k / x
• where the variable k is called the constant of
proportionality.
Inverse Variation
•
•
•
•
The following statements are equivalent.
1. y varies inversely as x.
2. y is inversely proportional to x.
3. y = k / x for some constant k.
Example 3
• A company has found that the demand for
its product varies inversely as the price of
the product. When the price is $2.75, the
demand is 600 units.
• Write an equation relating demand d and
price p.
• Approximate the demand when the price is
$3.25.
k
d
p
k
2.75
600(2.75)  k
600 
1650  k
1650
Model d 
p
1650
d
 508 units
3.25
Joint Variation
• Sometimes more than one variable is
involved in a direct variation problem. In
these cases, the problem is referred to as
a joint variation. The formula remains
the same, with the additional variables
included in the product.
Joint Variation
•
•
•
•
The following statements are equivalent.
1. z varies jointly as x and y.
2. z is jointly proportional to x and y.
3. z = kxy for some constant k.
Example 4
• The maximum load that can be safely
supported by a horizontal beam is jointly
proportional to the width of the beam and
the square of its depth, and inversely
proportional to the length of the beam.
Determine the change in the maximum
safe load und the following conditions.
• The width of the beam is doubled.
• The depth of the beam is doubled.
wd 2
ML 
l
Double width
2 wd 2
, ML is doubled
l
Double depth
w(2d ) 2 4 wd 2

, ML is 4 times bigger
l
l
• The works of Leonardo Da
Vinci display the artist's
keen interest in proportions
in the human body. The
facial features of the Mona
Lisa show numerous
applications of the golden
section (golden ratio) which
Leonardo referred to as the
"Divine Proportion" for its
esthetically pleasing nature.
• As a student of anatomy, Leonardo
was fascinated with the mathematical
relationships found in the human
structure. His drawing, Vitruvian Man,
was a study of the proportions of the
human body. It is said that Leonardo
believed the workings of the human
body to be an analogy for the workings
of the universe.Leonardo made a
series of observations concerning the
human body. Among them are
statements such as:
• the length of a man's outstretched
arms is equal to his height.
• the length of the ear is one-third the
length of the face.
• the distance
from the bottom of the chin to the nose
is one-third of the length of the head.
• the length of the hand is one-tenth
of a man's height.
QUESTION 1:
Leonardo observed that the length of the human face varies directly
as the length of the human chin. If a person whose face length is 9
inches has a chin length of 1.2 inches, what is the length of a person's
face whose chin length is 1.5 inches?
QUESTION 2:
Leonardo also observed that the human height varies directly as the
width of the human shoulders. If a person 70 inches tall has a
shoulder width of 18 inches, what is the height of a person whose
shoulder width is 16.5 inches?