PAP Algebra 2 NOTES 9.4

TLW…Simplify and work problems dealing with direct, inverse, and joint variations.

K = Constant of VariationDirect

Variation

The equation is:

How to find k:

Inverse Variation

The equation is:

How to find k:

Y=kx x

ky

x

yk xyk

K = Constant of VariationJoint Variation

Joint variation is the same as direct variation with two or more quantities. Example of Joint Variation

y = 7x2z3, here y

varies jointly with x2 and z3 (k=7).

Given: y varies directly with x and y = 12 when x = -4

1. What is the constant

of variation (k) ?

kxy 412 k

k 3

2. Find y when x = 5

xy 3

53y

15y

5

6,2

7

x

ky

275

6 k

2.410

42k

Find k in each Inverse Variation

(5,10) and (2,y) are from the same inverse variation. Find y.

x

ky

510

k

k50

xy50

2

50y

25y

x and y vary inversely where x=-2 & y=1/2 Write a function to model inverse variation.

Find y if x=1/2

x

ky

22

1

k

k 1

xy

1

211

y

2y

If z varies directly with x and inversely with y, the equation for this is:

kz x

y

Suppose that z varies jointly with x and y when x = 2 and y = 3, then z = 60.

A. Write the function that models the variation.

Find the value of “k” first ))(( yxkz

)3)(2(60 kk10

Now you can write the equation

xyz 10B. Find the value of z when x = 4 and y = -10

xyz 10)10)(4(10 z 400z

XX YY

33

-2-2

99

44

Is the relationship between the variables direct or inverse?

6

1

2

1

4

1

2

1

Direct : y = kx so k = y/x

Inverse: y = k/x so k = xy

Find “k” using both equations. If the value of “k” remains the same each time, then it represent that variation.

Neither

18

1

8

1

18

1

8

2

1

2

1

2

9

2

XX YY

11 33

-5-5 -15-15

22 66

1010 3030

Is the relationship between the variables direct or inverse?

Find “k” using both equations. If the value of “k” remains the same each time, then it represent that variation.

Direct

xy 3

Direct : y = kx so k = y/x

Inverse: y = k/x so k = xy

3k

Describe the combined variation that is modeled by the formula:

Molar mass varies directly with density and temperature

essure

eTemperaturDensity

Pr

))((4.62Molar Mass

and inversely with pressure

BTW… k = 62.4

Describe the combined variation that is modeled by the equation:

y varies directly with w2 and √ x

3

2

3

2

z

xwy

and inversely with z3

BTW… k = 2/3

If m varies directly as the square root of y, inversely as p5, and directly as n, what happens to m when y is

multiplied by 9, p is multiplied by 2 and n is quadrupled?:

5p

ynmEquation: m is multiplied by

52

94

p

ynnew

532

34

p

yn 532

12

p

yn

58

3

p

yn

8

3BTW… k = 1

A drama club is planning a bus trip to NYC to see a Broadway play. The cost per person for the bus rental varies inversely as the number of people going on the trip. It will cost $50

per person if 20 people go on the trip. To the nearest dollar, how much will it cost per person if 100 people go on the trip?

p

kc

2050

k

1000k

pc1000

personc /10$

100

1000c

Suppose that m and r vary inversely and that r=4/9 when m=6.Write a function that models the inverse variation

and find r when m=2

r

km

94

6k

9

24k

rm 9

24

r924

2

9

242 r

2

1

9

24r

3

4

18

24r

The amount of oil used by a ship travelling at a uniform speed varies jointly with the distance and the square of the speed. The ship uses 500 barrels of oil in travelling

50mi at 30mph. Write an equation that models this variation and find k. Also find how many barrels of oil

are used when the ship travels 10mi at 45mph.2kdro

23050500 k

k45000500

2

90

10 dr

barrelso 225

2451090

1o

90

1k

20259

1o

Homework #4 Due next class

You next quiz will cover Direct, Inverse and Joint Variation problems