Modeling Functions Using Variation

Section 3.6

Direct Variation

Let x and y represent two quantities. The following are equivalent statements:

y = kx, where k is a nonzero constant. y varies directly with x. y is directly proportional to x.

The constant k is called the constant of variation or the constant of proportionality.

Write an equation that describes variation. V varies directly with x³

V=kx³ z varies directly with m

z=km y is directly proportional to the square

root of x

y=k x

Write an equation to represent the variation.

W is directly proportional to the cube of H. W = 128 and H = 4

3kHW 34128 k

k64128

k2Plug the k-value back into original.

32HW

Inverse Variation

Let x and y represent two quantities. The following are equivalent statements:

y= , where k is a nonzero constant. y varies inversely with x. y is inversely proportional to x.

The constant k is called the constant of variation or the constant of proportionality.

NOTE: For inverse variation we see that when the x increases, the y decreases or visa versa.

x

k

Write an equation that describes variation. f varies inversely with c

f =

A varies inversely with tA=

y is inversely proportional to the cube of xy=

c

k

t

k

3x

k

Write an equation to represent the variation.

y varies inversely with x. y = 400 and x = 1000.

x

ky

1000400

k

k000,400

xy

000,400

Joint Variation

When one quantity is proportional to the product of two or more other quantities, it is called Joint Variation.

I is directly proportional to P, r and t.

For example: I = Prt which represents the simple interest formula where:

I is the interest in dollars

P is the principal (initial) dollars

r is the interest rate (decimal form)

t is the time in years

Write an equation to describe the variation.

A is directly proportional to both b and h. A = 10 when b = 5 and h = 4.

kbhA

4510 k

k2010

k2

1

bhA2

1

Write an equation to represent the variation.

F varies inversely with both λ and L. F = 20π when λ = 1 and L = 100

L

kF

100120

k

k2000

LF

2000

Combined Variation

When the variation is mixed, this is called Combined Variation.

P is directly proportional to T and inversely proportional to V.

For example: P = k which represents the combined gas law in chemistry.

P is pressureT is temperatureV is volumek is a gas constant

V

T

Example of Direct Variation

Find a mathematical model describing the monthly long distance bill if the cost is directly proportional to the number of minutes, and a customer that used 160 minutes is billed $40.

1. Declare your variables: c = is the cost in dollars m = the number minutes

2. Set up your direct variation equation and solve for the constant of variation:C = km

40 = k(160) ¼ = k

3. Solution: c = ¼ m

This equation describes the relationship between number of minutes and cost. They are directly proportional.

Example of Inverse VariationIn New York City, the number of potential buyers in the housing

market is inversely proportional to the price of a house. If there are 12,500 potential buyers for a $2 million condominium, how many potential buyers are there for a $5 million condominium?

1. Declare your variables: p = price of a house b = potential buyers

2. Set up your inverse variation equation and solve for the constant of variation: 12500 =

= k

3. Inverse variation equation: b =2000000

k

p

X 10105.2

10105.2 X

p

kb

How many potential buyers are there for a $5,000,000 condo?

4. Solution: b =

b = 5000

There are 5000 potential buyers for condos at $5,000,000.

5000000

105.2 10X

p

Xb

10105.2

Levi’s makes jeans in a variety of price ranges for men. The Silver Tab Baggy jeans sell for about $30, and they are trying to figure out how much to sell the Offender jeans for. The demand for Levi’s jeans is inversely proportional to the price. If 400,000 pairs of the Silver Tab jeans were bought, and they have 150,000 pairs of Offender jeans to sell, what should they list the price at for the Offenders?

p

kd

30000,400

k

k000,000,12

pd

000,000,12

p

000,000,12000,150

000,000,12000,150 p

$80 for Offenders80p

The gas in the headspace of a soda bottle has a volume of 9.0mL, pressure of 2 atm, and a temperature of 298 K. If the soda bottle is stored in a refrigerator, the temperature drops to approximately 279 K. What is the pressure of the gas in the headspace once the bottle is chilled?

V

TkP

9

2982 k

k298

18

V

TP

298

18

9

279

298

18P

87.1P

About 1.87 atm