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?

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.

New stuff!

Inverse VariationInverse Variation

“y varies inverselyvaries inversely with x.”

k is the constant of variationconstant of variation.

x

ky

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

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

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

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).

• 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?

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

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.

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 =

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.

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

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.

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

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

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

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

• 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)

AssignmentAssignment

p. 307p. 307

3-33 every third 3-33 every third problem, problem,