Math 1014: Precalculus with Transcendentals
Ch. 3: Polynomials and Rational Functions
Sec. 3.7 Modeling Using Variation
I.
Direct Variation
A. Definition
y = kx , where k is a nonzero
constant, we say that y varies directly as x or y is directly proportional to x . The
If a situation is described by an equation in the form
number
k is called the constant of variation or the constant of proportionality.
B. Examples
Solve the variation:
II.
1.
y varies directly as x . y = 45 when x = 5 . Find y when x = 13 .
2.
C varies jointly as A and T . C = 175 when A = 2100 and T = 4 . Find C when
A = 2400 and T = 6 .
Inverse Variation
A. Definition
k
where k is a nonzero
x
constant, we say that y varies inversely as x or y is inversely proportional to x . The
If a situation is described by an equation in the form
number
y=
k is called the constant of variation or the constant of proportionality.
B. Example
y varies inversely as x . y = 6 when x = 3 . Find y when x = 9 .
III. Direct Variation with Powers
A. Definition
y varies directly as the nth power of x if there exists some nonzero constant k , such
that
y = kx n . We also say that y is directly proportional to the nth power of x .
B. Example
Write an equation that expresses the relationship if
of
x varies jointly as y and the square
z . Then solve the equation for y .
IV. Combined Variation
A. In combined variation direct variation and inverse variation occur at the same time.
B. Example
1. Write an equation that expresses the relationship if
x varies directly as the cube
root of z and inversely as y . Then solve the equation for y .
2. Solve the variation.
a varies directly as b and inversely as the square of c . a = 7 when b = 9 and
c = 6 . Find a when b = 4 and c = 8 .