This document discusses modeling using variation, including direct variation, inverse variation, and joint variation. It provides examples of solving variation problems by writing the general relationship as an equation using the constant of variation k, substituting given values to find k, and then substituting values into the original equation to find the unknown. Examples include finding y when x is a given value for direct and inverse variation problems, and finding u when v and w are given values for a joint variation problem. The final example involves finding the number of vibrations per second of a guitar string when the tension and length are given values using direct and inverse variation.

1. 5.8 Modeling Using Variation
Chapter 5 Polynomial and Rational Functions

2. Concepts and Objectives
⚫ The objectives of this section are
⚫ Solve direct variation problems.
⚫ Solve inverse variation problems.
⚫ Solve problems involving joint variation.

3. Direct Variation
⚫ y varies directly as x, or y is directly proportional to x,
if there exists a nonzero real number k, called the
constant of variation, such that
y = kx.
⚫ Let n be a positive real number. Then y varies directly
as the nth power of x, or y is directly proportional to
the nth power of x, if there exists a nonzero real
number k such that n
y kx
=

4. Inverse Variation
⚫ y varies inversely as x, or y is inversely proportional
to x, if there exists a nonzero real number k, such that
⚫ Let n be a positive real number. Then y varies inversely
as the nth power of x, or y is inversely proportional to
the nth power of x, if there exists a nonzero real
number k such that
k
y
x
=
n
k
y
x
=

5. Solving Variation Problems
1. Write the general relationship among the variables as
an equation. Use the constant k.
2. Substitute given values of the variables and find the
value of k.
3. Substitute this value and the remaining values into the
original equation and solve for the unknown.

6. Examples
1. If y varies directly as x, and y = 24 when x = 8, find y
when x = 12.
2. If y varies inversely as x, and y = 7 when x = 4, find y
when x = 14.

7. Examples
1. If y varies directly as x, and y = 24 when x = 8, find y
when x = 12.
2. If y varies inversely as x, and y = 7 when x = 4, find y
when x = 14.
( )
24 8
k
=
3
k =
( )( )
3 12
y =
36
y =
7
4
k
=
28
k =
28
14
y =
2
y =
Step 1
Step 2
Step 3

8. Joint Variation
⚫ Let m and n be real numbers. Then y varies jointly as
the nth power of x and the mth power of z if there exists
a nonzero real number k such that
⚫ Note: If n or m is negative, then the variable is said to
vary inversely.
n m
y kx z
=

9. Example
3. If u varies jointly as v and w, and u = 48 when v = 12 and
w = 8, find u when v = 10 and w = 6.
4. If z varies directly as x and inversely as y2, and z = 8
when x = 6 and y = 3, find z when x = 10 and y = 4.

10. Example
3. If u varies jointly as v and w, and u = 48 when v = 12 and
w = 8, find u when v = 10 and w = 6.
4. If z varies directly as x and inversely as y2, and z = 8
when x = 6 and y = 3, find z when x = 10 and y = 4.
( )( )
48 12 8
k
=
0.5
k =
( )( )( )
0.5 10 6
u =
30
u =
( )
2
6
8
3
k
=
12
k =
( )( )
2
12 10
4
z =
7.5
z =
u kvw
=
2
kx
z
y
=

11. Example
5. The number of vibrations per second (the pitch) of a
steel guitar string varies directly as the square root of
the tension and inversely as the length of the string. If
the number of vibrations per second is 5 when the
tension is 225 kg and the length is .6 m, find the number
of vibrations per second when the tension is 196 kg and
the length is .65 m.

12. Example
5. The number of vibrations per second (the pitch) of a
steel guitar string varies directly as the square root of
the tension and inversely as the length of the string. If
the number of vibrations per second is 5 when the
tension is 225 kg and the length is .6 m, find the number
of vibrations per second when the tension is 196 kg and
the length is .65 m.
Let n represent the number of vibrations per second, T
represent the tension, and L represent the length of the
string.

13. Example (cont.)
Let n represent the number of vibrations per second, T
represent the tension, and L represent the length of the
string.
k T
n
L
=
225
5
.6
k
=
15
5
.6
.2
k
k
=
=
.2 196
4.3
.65
n = 

14. Classwork
⚫ College Algebra 2e
⚫ 5.8: 4-22 (even); 5.6: 30-34, 40-50 (even); 5.5: 56-70
(even)
⚫ 5.8 Classwork Check
⚫ Quiz 5.6