Mathematical Modeling and Variation
MATH 160, Precalculus
J. Robert Buchanan
Department of Mathematics
Fall 2011
J. Robert Buchanan
Mathematical Modeling and Variation
Objectives
In this lesson we will learn to:
write mathematical models for direct variation,
write mathematical models for direct variation as an nth
power,
write mathematical models for inverse variation,
write mathematical models for joint variation.
J. Robert Buchanan
Mathematical Modeling and Variation
Direct Variation
The general linear model is represented by the equation
y = mx + b.
If b = 0 so that y = mx we say that y varies directly with (or is
directly proportional to) x.
J. Robert Buchanan
Mathematical Modeling and Variation
Direct Variation
The general linear model is represented by the equation
y = mx + b.
If b = 0 so that y = mx we say that y varies directly with (or is
directly proportional to) x.
Direct Variation
The following statements are equivalent.
1
y varies directly with x.
2
y is directly proportional to x.
3
y = k x for some nonzero constant k .
The constant k is called the constant of proportionality or the
constant of variation.
J. Robert Buchanan
Mathematical Modeling and Variation
Example
State sales tax is based on retail price. An item that sells for
$189.99 has a sales tax of $11.40. Find a mathematical model
that gives the amount of sales tax y in terms of the retail price
x. Use the model to find the sales tax on a $639.99 purchase.
J. Robert Buchanan
Mathematical Modeling and Variation
Example
State sales tax is based on retail price. An item that sells for
$189.99 has a sales tax of $11.40. Find a mathematical model
that gives the amount of sales tax y in terms of the retail price
x. Use the model to find the sales tax on a $639.99 purchase.
Assuming that y = kx then
11.40 = k (189.99)
⇐⇒
J. Robert Buchanan
k=
11.40
= 0.06.
189.99
Mathematical Modeling and Variation
Example
State sales tax is based on retail price. An item that sells for
$189.99 has a sales tax of $11.40. Find a mathematical model
that gives the amount of sales tax y in terms of the retail price
x. Use the model to find the sales tax on a $639.99 purchase.
Assuming that y = kx then
11.40 = k (189.99)
⇐⇒
k=
11.40
= 0.06.
189.99
The sales tax on the new purchase will be
y = (0.06)(639.99) = 38.40.
J. Robert Buchanan
Mathematical Modeling and Variation
Direct Variation as an nth Power
Sometimes a variable changes directly with a power of another
variable.
Example
The volume V of a cube varies directly with the 3rd power of
the length of an edge of the cube.
J. Robert Buchanan
Mathematical Modeling and Variation
Direct Variation as an nth Power
Sometimes a variable changes directly with a power of another
variable.
Example
The volume V of a cube varies directly with the 3rd power of
the length of an edge of the cube.
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.
3
y = k x n for some constant k .
J. Robert Buchanan
Mathematical Modeling and Variation
Example
Consider the solid particles which may be found in a river,
stream, or creek. The diameter of the largest particle that the
water can move varies directly with the square of the velocity of
the water.
1
Write a mathematical model relating the diameter of the
largest particle the water can move to the velocity of the
water.
2
By what factor does the diameter decrease if the velocity of
the water is halved?
J. Robert Buchanan
Mathematical Modeling and Variation
Example
Consider the solid particles which may be found in a river,
stream, or creek. The diameter of the largest particle that the
water can move varies directly with the square of the velocity of
the water.
1
Write a mathematical model relating the diameter of the
largest particle the water can move to the velocity of the
water.
d = kv 2
2
where d is the diameter and v is the velocity.
By what factor does the diameter decrease if the velocity of
the water is halved?
J. Robert Buchanan
Mathematical Modeling and Variation
Example
Consider the solid particles which may be found in a river,
stream, or creek. The diameter of the largest particle that the
water can move varies directly with the square of the velocity of
the water.
1
Write a mathematical model relating the diameter of the
largest particle the water can move to the velocity of the
water.
d = kv 2
2
where d is the diameter and v is the velocity.
By what factor does the diameter decrease if the velocity of
the water is halved?
kv 2 = d
v 2
1
d
k
= (kv 2 )
=
2
4
4
J. Robert Buchanan
Mathematical Modeling and Variation
Inverse Variation
Sometimes a variable increases while another decreases.
J. Robert Buchanan
Mathematical Modeling and Variation
Inverse Variation
Sometimes a variable increases while another decreases.
Inverse Variation
The following statements are equivalent.
1
y varies inversely as x.
2
y is inversely proportional to x.
k
y = for some constant k .
x
3
J. Robert Buchanan
Mathematical Modeling and Variation
Inverse Variation
Sometimes a variable increases while another decreases.
Inverse Variation
The following statements are equivalent.
1
y varies inversely as x.
y is inversely proportional to x.
k
3
y = for some constant k .
x
k
In some situations y = n and we say that y varies inversely
x
with the nth power of x.
2
J. Robert Buchanan
Mathematical Modeling and Variation
Example
The frequency of vibration of a piano string varies directly as
the square root of the tension on the string and inversely as the
length of the string. The middle A string has a frequency of 440
vibrations per second. Find the frequency of a string that has
1.25 times as much tension and is 1.2 times as long.
J. Robert Buchanan
Mathematical Modeling and Variation
Example
The frequency of vibration of a piano string varies directly as
the square root of the tension on the string and inversely as the
length of the string. The middle A string has a frequency of 440
vibrations per second. Find the frequency of a string that has
1.25 times as much tension and is 1.2 times as long.
A suitable mathematical model is
√
T
,
L
f is frequency, T is tension, and L is length. For the middle A
string
√
TA
440LA
440 = k
⇐⇒
k= √
.
LA
TA
f =k
J. Robert Buchanan
Mathematical Modeling and Variation
Example
The frequency of vibration of a piano string varies directly as
the square root of the tension on the string and inversely as the
length of the string. The middle A string has a frequency of 440
vibrations per second. Find the frequency of a string that has
1.25 times as much tension and is 1.2 times as long.
A suitable mathematical model is
√
T
,
L
f is frequency, T is tension, and L is length. For the middle A
string
√
TA
440LA
440 = k
⇐⇒
k= √
.
LA
TA
f =k
For the new string
√
√
√
1.25TA
440LA 1.25TA
440 1.25
f =k
= √
=
= 409.9
1.2LA
1.2
TA 1.2LA
J. Robert Buchanan
Mathematical Modeling and Variation
vibrations per second.
Joint Variation
Sometimes a variable changes directly with two (or more) other
variables.
J. Robert Buchanan
Mathematical Modeling and Variation
Joint Variation
Sometimes a variable changes directly with two (or more) other
variables.
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 = k x y for some constant k .
J. Robert Buchanan
Mathematical Modeling and Variation
Joint Variation
Sometimes a variable changes directly with two (or more) other
variables.
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 = k x y for some constant k .
In some situations z = k x n y m and we say that z varies jointly
with the nth power of x and the mth power of y .
J. Robert Buchanan
Mathematical Modeling and Variation
Example
The maximum load that can be safely supported by a horizontal
beam varies jointly as the width of the beam and the square of
its depth, and inversely as the length of the beam.
1
Write down a mathematical model for this situation.
2
Determine the changes in the maximum safe load when
width and depth of the beam are doubled.
J. Robert Buchanan
Mathematical Modeling and Variation
Example
The maximum load that can be safely supported by a horizontal
beam varies jointly as the width of the beam and the square of
its depth, and inversely as the length of the beam.
1
Write down a mathematical model for this situation.
k W D2
L
where M is the maximum safe load and W , D, L are
respectively width, depth, and length for the beam.
Determine the changes in the maximum safe load when
width and depth of the beam are doubled.
M=
2
J. Robert Buchanan
Mathematical Modeling and Variation
Example
The maximum load that can be safely supported by a horizontal
beam varies jointly as the width of the beam and the square of
its depth, and inversely as the length of the beam.
1
Write down a mathematical model for this situation.
k W D2
L
where M is the maximum safe load and W , D, L are
respectively width, depth, and length for the beam.
Determine the changes in the maximum safe load when
width and depth of the beam are doubled.
M=
2
k W D2
L
k (2W ) (2D)2
L
= M
k W D2
= 8
= 8M
L
The maximum safe load is increased by a factor of 8.
J. Robert Buchanan
Mathematical Modeling and Variation
Homework
Read Section 1.10.
Exercises: 35, 39, 43, 47, . . . , 71, 75
J. Robert Buchanan
Mathematical Modeling and Variation