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8.4

• Variation and Problem Solving

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Direct Variation

• y varies directly as x, or y is directly
  proportional to x, if there is a nonzero constant
  k such that y kx.
• The family of equations of the form y kx are
  referred to as direct variation equations.
• The number k is called the constant of variation
  or the constant of proportionality.

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Direct Variation

• If y varies directly as x, find the constant of
  variation k and the direct variation equation,
  given that y 5 when x 30.
• y kx
• 5 k30
• k 1/6

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Direct Variation
Example

• If y varies directly as x, and y 48 when x 6,
  then find y when x 15.
• y kx
• 48 k6
• 8 k
• So the equation is y 8x.
• y 815
• y 120

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Direct Variation
Example

• At sea, the distance to the horizon is directly
  proportional to the square root of the elevation
  of the observer. If a person who is 36 feet
  above water can see 7.4 miles, find how far a
  person 64 feet above the water can see. Round
  your answer to two decimal places.

Continued.
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Direct Variation
Example continued
We substitute our given value for the elevation
into the equation.
So our equation is
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Inverse Variation

• y varies inversely as x, or y is inversely
  proportional to x, if there is a nonzero constant
  k such that y k/x.
• The family of equations of the form y k/x are
  referred to as inverse variation equations.
• The number k is still called the constant of
  variation or the constant of proportionality.

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Inverse Variation
Example

• If y varies inversely as x, find the constant of
  variation k and the inverse variation equation,
  given that y 63 when x 3.
• y k/x
• 63 k/3
• 633 k
• 189 k

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Powers of x

• y can vary directly or inversely as powers of x,
  as well.
• y varies directly as a power of x if there is a
  nonzero constant k and a natural number n such
  that y kxn.

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Powers of x
Example

• The maximum weight that a circular column can
  hold is inversely proportional to the square of
  its height.
• If an 8-foot column can hold 2 tons, find how
  much weight a 10-foot column can hold.

Continued.
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Powers of x
Example continued
We substitute our given value for the height of
the column into the equation.
So our equation is
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Variation and Problem Solving
Example Kathy spends 1.5 hours watching
television and 8 hours studying each week. If
the amount of time spent watching TV varies
inversely with the amount of time spent studying,
find the amount of time Kathy will spend watching
TV if she studies 14 hours a week.
1.) Understand
Read and reread the problem.
2.) Translate
We are told that the amount of time watching TV
varies inversely with the amount of time spent
studying.
Let T the number of hours spent watching
television.
Let s the number of hours spent studying.
Continued
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Variation and Problem Solving
Example continued
3.) Solve
To find k, substitute T 1.5 and s 8.
We now write the variation equation with k
replaced by 12.
Replace s by 14 and find the value of T.
Continued
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Variation and Problem Solving
Example continued
3.) Interpret
Kathy will spend approximately 0.86 hours (or 52
minutes) watching TV.