Modeling Using Variation

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Modeling Using Variation
Section 3.7

• Objectives
• Solve direct variation problems.
• Solve inverse variation problems.
• Solve combined variation problems.

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Example of Direct Variation

• Because light travels faster than sound, during
  a thunderstorm we see lightning before we hear
  thunder. The formula
• d 1080t describes the distance, in feet, of
  the storms center if it takes t seconds to hear
  thunder after seeing lightning.
• Thus if t 1, d 1080
• if t 2, d 2160
• and if t 3, d 3240.
• This distance is said to vary directly.

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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.

As x increases, y increases. As x decreases, y
decreases.
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Writing a Direct Variation Equation

• If v varies directly as r, then

Solution v kr
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Solving Variation Problems

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

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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.
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.
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.
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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Practice 1

• The pressure, P, of water on an object below the
  surface varies directly as its distance, D below
  the surface. If a submarine experiences a
  pressure of 25 pounds per square inch 60 feet
  below the surface, how much pressure will it
  experience 330 feet below the surface.

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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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Practice 2

• The distance, s, that a body falls from rest
  varies directly as the square of the time, t, of
  the fall. If skydivers fall 64 feet in 2
  seconds, how far will they fall in 4.5 seconds?

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Inverse Variation

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

As x increases, y decreases. As x decreases, y
increases.
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Combined Variation Example

• Describe in words the variation shown by the
  given equation

SolutionH varies directly as T and inversely as
Q