5.1 inverse & joint variation
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5.1 Inverse & Joint Variation. p.303 What is direct variation? What is inverse variation? What is joint variation?. Just a reminder…. Direct Variation Use y=kx. Means “y varies directly with x.” k is called the constant of variation. New stuff!. Inverse Variation - PowerPoint PPT PresentationTRANSCRIPT
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5.1 Inverse & Joint Variation5.1 Inverse & Joint Variation
p.303p.303
What is direct variation?What is direct variation?
What is inverse variation?What is inverse variation?
What is joint variation?What is joint variation?
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Just a reminder…Just a reminder…
Direct Variation
Use y=kx.
Means “y varies directlyvaries directly with x.”
k is called the constant of variationconstant of variation.
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New stuff!
Inverse VariationInverse Variation
“y varies inverselyvaries inversely with x.”
k is the constant of variationconstant of variation.
x
ky
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Ex: tell whether x & y show direct variation, inverse variation, or neither.
a. xy=4.8
b. y=x+4
c. 5.1
yx
Hint: Solve the equation for y and take notice of
the relationship.
xy
8.4
Inverse Variation
Neither
xy 5.1
Direct Variation
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Ex: The variables x & y vary inversely, and y=8 when x=3.
• Write an equation that relates x & y.
k=24• Find y when x= -4.
y= -6
x
kyuse :
38
k
x
ky
xy
24
4
24
y
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4. x = 4, y = 3
The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x = 2.
y = ax Write general equation for inverse variation.
Substitute 3 for y and 4 for x.4 3 = a
12 = a Solve for a.
12xThe inverse variation equation is y =
When x = 2, y =122 = 6.
ANSWER
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MP3 PlayersThe number of songs that can be stored on an MP3 player varies inversely with the average size of a song. A certain MP3 player can store 2500 songs when the average size of a song is 4 megabytes (MB).
Write a model that gives the number n of songs that will fit on the MP3 player as a function of the average song size s (in megabytes).
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• Make a table showing the number of songs that will fit on the MP3 player if the average size of a song is 2MB, 2.5MB, 3MB, and 5MB as shown below. What happens to the number of songs as the average song size increases?
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STEP 1 Write an inverse variation model.an = s Write general equation for inverse variation.
a2500 =4 Substitute 2500 for n and 4 for s.
10,000 = a Solve for a.
A model is n = s10,000ANSWER
STEP 2 Make a table of values.
From the table, you can see that the number of songs that will fit on the MP3 player decreases as the average song size increases.
ANSWER
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The table compares the area A (in square millimeters) of a computer chip with the number c of chips that can be obtained from a silicon wafer.
Computer Chips
• Write a model that gives c as a function of A.• Predict the number of chips per wafer when the area of a chip is 81 square millimeters.
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SOLUTION
STEP 1 Calculate the product A c for each data pair in the table.
58(448) = 25,984
62(424) = 26,288
66(392) = 25,872
70(376) = 26,320
Each product is approximately equal to 26,000. So, the data show inverse variation. A model relating A and c is: A c = 26,000 , or c =A
26,000
STEP 2 Make a prediction. The number of chips per wafer for a chip with an area of 81 square millimeters is
8126,000 321c =
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Joint Variation
• When a quantity varies directly as the product of 2 or more other quantities.
• For example: if z varies jointly with x & y, then z=kxy.
• Ex: if y varies inversely with the square of x, then y=k/x2.
• Ex: if z varies directly with y and inversely with x, then z=ky/x.
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Examples: Write an equation.
• y varies directly with x and inversely with z2.
• y varies inversely with x3.
• y varies directly with x2 and inversely with z.
• z varies jointly with x2 and y.
• y varies inversely with x and z.
2z
kxy
3x
ky
z
kxy
2
ykxz 2
xz
ky
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The variable z varies jointly with x and y. Also, z = –75 when x = 3 and y = –5. Write an equation that relates x, y, and z. Then find z when x = 2 and y = 6.SOLUTION
STEP 1 Write a general joint variation equation.
z = axy
–75 = a(3)(–5)
Use the given values of z, x, and y to find the constant of variation a.
STEP 2
Substitute 75 for z, 3 for x, and 25 for y.
–75 = –15a Simplify.5 = a Solve for a.
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STEP 3Rewrite the joint variation equation with the value of a from Step 2.
z = 5xy
STEP 4Calculate z when x = 2 and y = 6 using substitution.
z = 5xy = 5(2)(6) = 60
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Write an equation for the given relationship.
Relationship Equationa. y varies inversely with x.
b. z varies jointly with x, y, and r.
z = axyr
y = ax
c. y varies inversely with the square of x.
y = ax2
d. z varies directly with y and inversely with x.
z = ayx
e. x varies jointly with t and r and inversely with s.
x = atrs
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10. x = 4, y = –3, z =24
SOLUTION
STEP 1Write a general joint variation equation.z = axy
24 = a(4)(– 3)
Use the given values of z, x, and y to find the constant of variation a.
STEP 2
Substitute 24 for z, 4 for x, and –3 for y.
24 = –12a Simplify.
Solve for a.= a– 2
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STEP 3Rewrite the joint variation equation with the value of a from Step 2.
z = – 2 xy
STEP 4Calculate z when x = – 2 and y = 5 using substitution.
z = – 2 xy = – 2 (– 2)(5) = 20
z = – 2 xy ; 20ANSWER
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• What is direct variation?What is direct variation?
y varies directly with x (y = kx)y varies directly with x (y = kx)
• What is inverse variation?What is inverse variation?
y varies inversely with x (y = k/x)y varies inversely with x (y = k/x)
• What is joint variation?What is joint variation?
A quantity varies directly as the product of A quantity varies directly as the product of two or more other quantities ( y = kxy)two or more other quantities ( y = kxy)
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AssignmentAssignment
p. 307p. 307
3-33 every third 3-33 every third problem, problem,