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Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Solve direct variation problems. Solve inverse variation problems.
7.8
Variation
Solve direct variation problems.
Solve inverse variation problems. 1 2
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3. Solve direct variation problems.
Objective 1
Solve direct variation problems.
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4. Solve direct variation problems.
Two variables vary directly if one is a constant multiple
of the other.
With direct variation, y varies directly as x if there
exists a constant k such that
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.
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5. Solve direct variation problems. (cont’d)
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.
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6. EXAMPLE 1 If z varies directly as t, and z = 11 when t = 4, find z
Using Direct Variation
EXAMPLE 1
If z varies directly as t, and z = 11 when t = 4, find z
when t = 32.
Solution:
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7. 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.
In the situation of direct variation as a power, y varies directly as the nth power of x if there exists a real number k such that
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8. 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 cm. Find the circumference if the radius is 11 cm.
Solution:
Thus, the circumference of the circle is cm if the radius equals 11 cm.
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9. Solve inverse variation problems.
Objective 2
Solve inverse variation problems.
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10. 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.
y varies inversely as x if there exists a real number k such
that
Also, y varies inversely as the nth power of x if there exists a real number k such that
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11. Using Inverse Variation
EXAMPLE 3
Suppose y varies inversely as the square of x. If y = 5 when x = 2, find y when x = 10.
Solution:
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12. EXAMPLE 4
Using Inverse Variation
If the cost of producing pairs of rubber gloves varies inversely as the number of pairs produced, and pairs can be produced for $0.50 per pair, how much will it cost per pair to produce 10,000 pairs?
Solution:
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