5.8 Modeling Using Variation

* Solve direct variation problems.
* Solve inverse variation problems.
* Solve problems involving joint variation.

1. 5.8Modeling 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
⚫yvaries directlyas x, or yis directly proportionalto x,
if there exists a nonzero real number k, called the
constant of variation, such that
y= kx.
⚫Let nbe a positive real number. Then yvaries directly
as the nth powerof x, or yis directly proportional to
the nth powerof x, if there exists a nonzero real
number ksuch that n
y kx=

4. Inverse Variation
⚫yvaries inverselyas x, or yis inversely proportional
to x, if there exists a nonzero real number k, such that
⚫Let nbe a positive real number. Then yvaries inversely
as the nth powerof x, or yis inversely proportional to
the nth powerof x, if there exists a nonzero real
number ksuch thatk
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 yvaries directly as x, and y= 24 when x= 8, find y
when x= 12.
2.If yvaries inversely as x, and y= 7 when x= 4, find y
when x= 14.

7. Examples
1.If yvaries directlyas x, and y= 24 when x= 8, find y
when x= 12.
2.If yvaries inverselyas x, and y= 7 when x= 4, find y
when x= 14.()24 8k= 3k= ()()3 12y= 36y= 7
4
k
= 28k= 28
14
y= 2y=
Step 1
Step 2
Step 3

8. Joint Variation
⚫Let mand nbe real numbers. Then yvaries jointlyas
the nth power of xand the mth power of zif there exists
a nonzero real number ksuch that
⚫Note: If nor mis negative, then the variable is said to
vary inversely.nm
y kx z=

9. Example
3.If uvaries jointly as vand w, and u= 48 when v= 12 and
w= 8, find uwhen v= 10 and w= 6.
4.If zvaries directly as xand inversely as y
2
, and z= 8
when x= 6 and y= 3, find zwhen x= 10 and y= 4.

10. Example
3.If uvaries jointly as vand w, and u= 48 when v= 12 and
w= 8, find uwhen v= 10 and w= 6.
4.If zvaries directly as xand inversely as y
2
, and z= 8
when x= 6 and y= 3, find zwhen x= 10 and y= 4.()()48 12 8k= 0.5k= ()()()0.5 10 6u= 30u= ()
2
6
8
3
k
= 12k= ()()
2
12 10
4
z= 7.5z= ukvw= 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 nrepresent the number of vibrations per second, T
represent the tension, and Lrepresent the length of the
string.

13. Example (cont.)
Let nrepresent the number of vibrations per second, T
represent the tension, and Lrepresent the length of the
string.kT
n
L
= 225
5
.6
k
= 15
5
.6
.2
k
k
=
= .2196
4.3
.65
n=

14. Classwork
⚫College Algebra2e
⚫5.8: 4-22 (even); 5.6: 30-34, 40-50 (even); 5.5: 56-70
(even)
⚫5.8ClassworkCheck
⚫Quiz 5.6