Modeling Using Variation

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Modeling Using Variation
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Direct Variation

• If a situation is described by an equation in the
  form
• y kx
• where k is a constant, we say that y varies
  directly as x. The number k is called the
  constant of variation.

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Solving Variation Problems

• Write an equation that describes the given
  English statement.
• Substitute the given pair of values into the
  equation in step 1 and find the value of k.
• Substitute the value of k into the equation in
  step 1.
• Use the equation from step 3 to answer the
  problem's question.

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Direct Variation with Powers

• 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.

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Text Example
The amount of garbage, G, varies directly as the
population, P. Allegheny County, Pennsylvania,
has a population of 1.3 million and creates 26
million pounds of garbage each week. Find the
weekly garbage produced by New York City with a
population of 7.3 million.
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Text Example cont.
The amount of garbage, G, varies directly as the
population, P. Allegheny County, Pennsylvania,
has a population of 1.3 million and creates 26
million pounds of garbage each week. Find the
weekly garbage produced by New York City with a
population of 7.3 million.
Solution
Step 2 Use the given values to find k.
Allegheny County has a population of 1.3 million
and creates 26 million pounds of garbage weekly.
Substitute 26 for G and 1.3 for P in the direct
variation equation. Then solve for k. 26
1.3k 26/1.3 k 20 k
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Text Example cont.
The amount of garbage, G, varies directly as the
population, P. Allegheny County, Pennsylvania,
has a population of 1.3 million and creates 26
million pounds of garbage each week. Find the
weekly garbage produced by New York City with a
population of 7.3 million.
Solution
Step 3 Substitute the value of k into the
equation. G kP Use the equation from
step 1. G 20P Replace k, the constant
of variation, with 20.
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Text Example cont.
The amount of garbage, G, varies directly as the
population, P. Allegheny County, Pennsylvania,
has a population of 1.3 million and creates 26
million pounds of garbage each week. Find the
weekly garbage produced by New York City with a
population of 7.3 million.
Solution
Step 4 Answer the problem's question. New
York City has a population of 7.3 million. To
find its weekly garbage production, substitute
7.3 for P in G 20P and solve for G. G 20P
Use the equation from step 3. G 20(7.3)
Substitute 7.3 for P. G 146 The weekly
garbage produced by New York City weighs
approximately 146 million pounds.
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Inverse Variation

• If a situation is described by an equation in the
  form
• y k/x
• where k is a constant, we say that y varies
  inversely as x. The number k is called the
  constant of variation.

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Example

• Describe in words the variation shown by the
  given equation

SolutionH varies directly as T and inversely as
Q
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Joint Variation

• Joint variation is a variation in which a
  variable varies directly as the product of two or
  more other variables. Thus, the equation y kxz
  is read "y varies jointly as x and z."

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Example

• An objects weight on the moon, M, varies
  directly as its weight on Earth, E. A person who
  weighs 75 kilograms on Earth weighs 12 kilograms
  on the moon. What is the moon weight of a person
  who weighs 80 kilograms on Earth?
• Solution
• M kE
• 12 k75
• k .16
• M .16E
• M .16 80
• M 12.8 kilograms

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Modeling Using Variation