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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 4

Joint Variation

Find a value when one variable varies jointly with two or more others.

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Key Ideas Easy Medium Hard

y = kxz

k = yxz

What to Remember

The formula

Joint variation multiplies the variables.

y = kxz means y varies jointly with x and z.

Find k

Rearrange to k = yxz.

Substitute the given values, then reduce.

A power

y = kx²z varies jointly with the square of x and with z.

A square root stays under the radical sign until it simplifies.

A different variable

If the constant is 1, the formula may already be V = lwh.

Rearrange for the variable the question asks for.

Level 1

Easy

y = kxz

Example
y varies jointly with x and z. y = 24 when x = 3 and z = 2. Find y when x = 5 and z = 1.

y = kxz

k = yxz

xz = 3×2

xz = 6

k = 246

k = 4

y = 4xz

y = 4×5×1

y = 20

Answer: y = 20

1
y varies jointly with x and z. y = 36 when x = 3 and z = 4. Find y when x = 2 and z = 3.
Show Answer

y = kxz

k = yxz

xz = 3×4

xz = 12

k = 3612

k = 3

y = 3xz

y = 3×2×3

y = 18

Answer: y = 18

2
y varies jointly with x and z. y = 40 when x = 5 and z = 2. Find y when x = 4 and z = 3.
Show Answer

y = kxz

k = yxz

xz = 5×2

xz = 10

k = 4010

k = 4

y = 4xz

y = 4×4×3

y = 48

Answer: y = 48

3
y varies jointly with x and z. y = 18 when x = 2 and z = 3. Find y when x = 5 and z = 2.
Show Answer

y = kxz

k = yxz

xz = 2×3

xz = 6

k = 186

k = 3

y = 3xz

y = 3×5×2

y = 30

Answer: y = 30

4
y varies jointly with x and z. y = 48 when x = 4 and z = 3. Find y when x = 2 and z = 5.
Show Answer

y = kxz

k = yxz

xz = 4×3

xz = 12

k = 4812

k = 4

y = 4xz

y = 4×2×5

y = 40

Answer: y = 40

5
y varies jointly with x and z. y = 60 when x = 5 and z = 4. Find y when x = 3 and z = 2.
Show Answer

y = kxz

k = yxz

xz = 5×4

xz = 20

k = 6020

k = 3

y = 3xz

y = 3×3×2

y = 18

Answer: y = 18

Level 2

Medium

A square in the joint formula

Example
y varies jointly with the square of x and with z. y = 40 when x = 2 and z = 5. Find y when x = 3 and z = 1.

y = kx²z

k = yx²z

x² = 4

x²z = 4×5

x²z = 20

k = 4020

k = 2

y = 2x²z

x² = 9

y = 2×9×1

y = 18

Answer: y = 18

1
y varies jointly with the square of x and with z. y = 36 when x = 3 and z = 2. Find y when x = 2 and z = 4.
Show Answer

y = kx²z

k = yx²z

x² = 9

x²z = 9×2

x²z = 18

k = 3618

k = 2

y = 2x²z

x² = 4

y = 2×4×4

y = 32

Answer: y = 32

2
y varies jointly with the square of x and with z. y = 16 when x = 2 and z = 2. Find y when x = 4 and z = 1.
Show Answer

y = kx²z

k = yx²z

x² = 4

x²z = 4×2

x²z = 8

k = 168

k = 2

y = 2x²z

x² = 16

y = 2×16×1

y = 32

Answer: y = 32

3
y varies jointly with the square of x and with z. y = 48 when x = 2 and z = 3. Find y when x = 1 and z = 5.
Show Answer

y = kx²z

k = yx²z

x² = 4

x²z = 4×3

x²z = 12

k = 4812

k = 4

y = 4x²z

x² = 1

y = 4×1×5

y = 20

Answer: y = 20

4
y varies jointly with the square of x and with z. y = 18 when x = 3 and z = 1. Find y when x = 2 and z = 3.
Show Answer

y = kx²z

k = yx²z

x² = 9

x²z = 9×1

x²z = 9

k = 189

k = 2

y = 2x²z

x² = 4

y = 2×4×3

y = 24

Answer: y = 24

5
y varies jointly with the square of x and with z. y = 100 when x = 5 and z = 2. Find y when x = 2 and z = 3.
Show Answer

y = kx²z

k = yx²z

x² = 25

x²z = 25×2

x²z = 50

k = 10050

k = 2

y = 2x²z

x² = 4

y = 2×4×3

y = 24

Answer: y = 24

Level 3

Hard

A square root, or a missing dimension

Example
b varies jointly with the square root of x and with y. b = 12 when x = 16 and y = 3. Find b when x = 4 and y = 5.

b = k×square root of x×y

k = bsquare root of x×y

square root of 16 = 4

4×3 = 12

k = 1212

k = 1

b = 1×square root of x×y

square root of 4 = 2

b = 1×2×5

b = 10

Answer: b = 10

1
b varies jointly with the square root of x and with y. b = 18 when x = 9 and y = 2. Find b when x = 4 and y = 6.
Show Answer

b = k×square root of x×y

k = bsquare root of x×y

square root of 9 = 3

3×2 = 6

k = 186

k = 3

b = 3×square root of x×y

square root of 4 = 2

b = 3×2×6

b = 36

Answer: b = 36

2
b varies jointly with the square root of x and with y. b = 20 when x = 4 and y = 5. Find b when x = 9 and y = 2.
Show Answer

b = k×square root of x×y

k = bsquare root of x×y

square root of 4 = 2

2×5 = 10

k = 2010

k = 2

b = 2×square root of x×y

square root of 9 = 3

b = 2×3×2

b = 12

Answer: b = 12

3
b varies jointly with the square root of x and with y. b = 24 when x = 16 and y = 3. Find b when x = 9 and y = 4.
Show Answer

b = k×square root of x×y

k = bsquare root of x×y

square root of 16 = 4

4×3 = 12

k = 2412

k = 2

b = 2×square root of x×y

square root of 9 = 3

b = 2×3×4

b = 24

Answer: b = 24

4
b varies jointly with the square root of x and with y. b = 30 when x = 25 and y = 2. Find b when x = 4 and y = 5.
Show Answer

b = k×square root of x×y

k = bsquare root of x×y

square root of 25 = 5

5×2 = 10

k = 3010

k = 3

b = 3×square root of x×y

square root of 4 = 2

b = 3×2×5

b = 30

Answer: b = 30

5
The volume of a rectangular prism varies jointly with its length, width, and height, and the constant of variation is 1. The volume is 360 cubic meters when the length is 9 meters and the width is 5 meters. Find the height.
Show Answer

V = lwh

h = Vlw

lw = 9×5

lw = 45

h = 36045

h = 8

The height is 8 meters.

Answer: 8 meters

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