Lesson 9: Combined and Joint Variation

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Chapter 2 Variation and Graphs

• Lesson 9 Combined and Joint Variation
• Mrs. Parziale

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Equations with more than two variables
Examples
Volume of a Rectangular Solid Volume of a
Cylinder
directly
Joint Variation One quantity varies
as the product of two or more independent
variables but not inversely as any variable.
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Combined Variation When direct and inverse
variations occur
together
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Table 1 Table 1
a c
1 3
2 6
3 9
4 12
5 15
Table 2 Table 2
b c
1 12
2 3
3 1.33
4 .75
5 0.5
Part 1 a and c In the first table
determine 1) Type of variation 2)
Equation
Direct
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Table 2 Table 2
b c
1 12
2 3
3 1.33
4 .75
5 0.5
Part 2 b and c In the second table
determine 1) Type of variation 2)
Equation
Inverse
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1) Words to describe the
relationship 2) Final
equation
Part 3 Putting it together
C varies directly as a and inversely as b2
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Example 2 X. Perri Menter performed an experiment
to determine how the pressure of a liquid on an
object is related to the depth of the object and
the density of the liquid. She placed an object
at a depth of 25 inches into various solutions
that had different densities. She measured the
pressure on the object and obtained the data in
Table 1.
Table 1 Table 1
Density D (pounds per inches cubed) Pressure P (pounds per inches squared)
0 5
0.5 15.6
1.2 37.5
1.5 46.9
2.7 84.4
3.8 118.8
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Part 1 Look at D and P what is happening to
the values From the data determine 1)
Type of variation 2) Equation
Direct
Table 1 Table 1
Density D (pounds per inches cubed) Pressure P (pounds per inches squared)
0 5
0.5 15.6
1.2 37.5
1.5 46.9
2.7 84.4
3.8 118.8
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Table 2 Table 2
Depth d (inches) Pressure P (pounds per inches squared)
0 5
25 46.9
50 93.4
75 140.6
100 187.5
125 234.4
Part 2 Look at d and P From the second table
determine 1) Type of variation 2)
Equation
Direct
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Part 3 Putting it together
1) Words to describe the relationship
2) Final equation
P varies directly as Density and depth
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Example 2 A baseball pitchers earned run
average ( ERA) varies directly as the number of
earned runs allowed and inversely as the number
of innings pitched. Write a general equation to
model this situation.
Let E earned run average
Let R number of earned runs
Let I number of innings pitched
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Problem In a recent year, a pitcher had an ERA
of 2.56, having given up 72 earned runs in 253
innings. How many earned runs would the pitcher
have given up if he had pitched 300 innings,
assuming that his ERA remained the same?