8.1 Model Inverse and Joint
Variation
Inverse variation
Two
variables x and y show inverse variation if they are related as
follows:
Constant
of variation
The nonzero constant a in a
variation equation
Joint
variation
When a quantity varies directly with
the product of two or more other quantities
INVERSE
VARIATION
Two
variables x and y show inverse variation if they are related as
follows: y =_____
The constant a is
the constant of variation, and y is said to __vary inversely__
with x.
Example 1
Tell whether x and y
show direct variation, inverse variation, or neither.
|
Given
Equation |
Rewritten Equation |
Type of Variation |
|
a.
|
_y = 9x_ |
_Direct_ |
|
b. xy = 3 |
|
_Inverse_ |
Example 2
The variables x and y
vary inversely, and y = 3 when x = 6. Write an equation that
relates x and y. Find y when x = -9.
|
|
Write general equation for inverse
variation. |
||
|
3 |
Substitute for y and for x. |
||
|
__18__ = a |
Solve for a. |
The inverse variation equation is y = When x = -9, y =
Example 3
Determine whether m and n show
inverse variation. If they do, write a model that gives n as a function
of m. Find n when m = 45.
|
m |
5 |
10 |
15 |
20 |
25 |
|
n |
45 |
22.5 |
15 |
11.25 |
9 |
Calculate
the product m · n for each data pair in the
table.
|
5(45) =225 |
10(22.5) =225 |
15(15) = 225 |
|
20(11.25) =225 |
25(9) = 225 |
|
Each product is equal to _225_.
So, the data _do show_ inverse variation. A
model relating m and n is
225
m .
m · n = _225_ or n =______
The value of n when m = 45 is n =
JOINT VARIATION
Joint variation occurs when a
quantity varies directly with the product of _two or more_ other
quantities. In the equation below, a is a nonzero
constant.
z = __axy__
z varies jointly with x and y.
Example 4
The variable z varies jointly
with x and y. Also, z = -84 when x = -4 and y = 3. Write an equation that relates x, y,
and z. Find z when x = 5 and y = 2.
Write the general joint variation
equation. Use the given values of z, x, and y to find the
constant of variation a.
|
z = axy |
|
|
_-84_ =a(__-4__)(__3__) |
Substitute
for z, x, and y. |
|
_-84_ = __-12__a |
Simplify. |
|
_7_ = a |
Solve
for a. |
The joint variation equation is z =__7xy__
Calculate z when x = 5 and y = 2 using substitution.
z = __7xy__ = __7(5)(2)__
= __70__
Example 5
Write an equation for the
relationship.
|
Relationship |
Equation |
||||||||||||||
|
a.
m
varies jointly with n, p, and
q. |
m = __anpq__ |
||||||||||||||
|
b.
r varies inversely with s. |
a
s |
||||||||||||||
|
c.
x varies
inversely with the cube of y. |
axy = x a y 3 |
||||||||||||||
|
d. k varies jointly with x
and y and inversely with m. |
= au = k m |
||||||||||||||
|
e. t varies directly with u and inversely with w. |
t w |
