Download Solve direct variation problems.

Chapter 7
Section 8
7.8 Variation
Objectives
1
Solve direct variation problems.
2
Solve inverse variation problems.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Solve direct variation problems.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 7.8-3
Solve direct variation problems.
Two variables vary directly if one is a constant multiple
of the other.
Direct Variation
y varies directly as x if there exists a constant k such that
y  kx.
In these equations, y is said to be proportional to x. The constant k in
the equation for direct variation is a numerical value. This value is
called the constant of variation.
Some simple examples of variation include:
Direct Variation:
The harder one pushes on a car’s gas
pedal, the faster the car goes.
Inverse Variation:
The harder one pushes on a car’s brake
pedal, the slower the car goes.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 7.8-4
Solve direct variation problems. (cont’d)
Solving a Variation Problem
Step 1: Write the variation equation.
Step 2: Substitute the appropriate given values and solve for k.
Step 3: Rewrite the variation equation with the value of k from Step 2.
Step 4: Substitute the remaining values, solve for the unknown, and
find the required answer.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 7.8-5
EXAMPLE 1 Using Direct Variation
If z varies directly as t, and z = 11 when t = 4, find z when t = 32.
Solution:
Let z  kt
and 11  k  4
11
then, k 
4
11
z t
4
11
z   32
4
z  88
Thus, z  88 when t  32.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 7.8-6
Solve direct variation problems. (cont’d)
The direct variation equation y = kx is a linear equation. Other kinds of
variation involve other types of equations.
Direct Variation as a Power
y varies directly as the nth power of x if there exists a real number
n
k such that
y  kx .
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 7.8-7
EXAMPLE 2 Solving a Direct Variation Problem
The circumference of a circle varies directly as the radius. A circle with
a radius of 7 cm has a circumference of 43.96 cm. Find the
circumference if the radius is 11 cm.
Solution:
C  2 r  k
43.96  2  7  k
C  2 111
C  69.08 cm
43.96
k
2  7
k 1
Thus, the circumference of the circle is 69.08 cm if the radius
equals 11 cm.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 7.8-8
Objective 2
Solve inverse variation problems.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 7.8-9
Solve inverse variation problems
Unlike direct variation, where k > 0 and k increases as y increases.
Inverse variation is the opposite. As one variable increases, the
other variable decreases.
Inverse Variation
y varies inversely as x if there exists a real number k such
that
k
y .
x
Also, y varies inversely as the nth power of x if there exists a real
number k such that
k
y n.
x
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 7.8-10
EXAMPLE 3 Using Inverse Variation
Suppose y varies inversely as the square of x. If y = 5 when x = 2,
find y when x = 10.
Solution:
k
y 2
x
k
5 2
2
20
y 2
10
1
y
5
k  20
1
Thus, y  when x  10
5
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 7.8-11
EXAMPLE 4 Using Inverse Variation
If the cost of producing pairs of rubber gloves varies inversely as the
number of pairs produced, and 5000 pairs can be produced for $0.50
per pair, how much will it cost per pair to produce 10,000 pairs?
Solution:
k
c
x
2500
c
1000
k
0.50 
5000
c  $0.25
k  2500
Thus, it will cost $0.25 per pair to produce 10,000 pairs.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 7.8-12