6.8 1

6.8 Solving (Rearranging) Formulas & Types of Variation

Rearranging formulas containing rational expressions Variation

Variation Inverse Joint Combined

6.8 2

Electronics:“Solving” formulas for different variable

Solve the electronic resistance formula for the variable r1

What’s the LCD?

Rr

Rrr

RrRrr

RrRrrr

RrRrrr

rRrrrR

rrR

2

21

221

2121

1221

2121

21

111

111

6.8 3

Astronomy:“Solving” formulas for different variable

Solve for the heightvariable h in the satellite escapevelocity equation

What’s the LCD?

2

22

222

222

22

22

2

2

2

2

2

2

2

2

2

V

RVgRh

RVgRhV

gRhVRV

gRhRV

hRRhR

g

R

V

hR

g

R

V

6.8 4

Acoustics (the Doppler Effect):“Solving” formulas for different variable

Solve for the speedvariable s in the doppler effect equation

What’s the LCD?

fg

fv

gf

fvs

fvgfs

fvsgfs

sgfvfs

sgvsf

vsvs

sgf

vs

sgf

)(

6.8 5

Direct VariationThe words “y varies directly with x”

or “y is directly proportional to x” mean that y = kx for some nonzero constant k

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

Express the verbal model in symbols:

“A varies directly with the square of p”.A = kp2

Find the constant of variation, if A = 18 when p = 3 18 = k(3)2 so 18 = 9k therefore k = 2Real: Distance of a lightning bolt varies directly with the time between seeing the flash and hearing the thunder. m = (1/5)s

6.8 6

Inverse VariationThe words “y varies inversely with x”

or “y is inversely proportional to x” mean that y = k/x for some nonzero constant k

The constant k is called the constant of variation

Express the verbal model in symbols:“z varies inversely with the cube of t”.z = k/t3

Find the constant of variation, if t = 2 when z = 10 10 = k/23 so 10 = k/8 therefore k = 80Real: Loudness of sound varies inversely with the square of the distance from the sound. L = k/d2

6.8 7

Joint VariationThe words “y varies jointly with x and z”

or “y is jointly proportional to x and z” mean that y = kxz for some nonzero constant k

The constant k is called the constant of variation

Express the verbal model in symbols:“M varies inversely with the cube of n and jointlywith x and the square of z”.M = kxz2/n3

Find the constant of variation, if M = 3 when z=10, x=2, n=1 3 = k(2)(10)2/13 so 3 = 200k therefore k = 3/200

6.8 8

Solving Variation Problems(at least two sets of values)

1. Translate the verbal model into an equation.2. Substitute the first set of values into the equation from step

1 to determine the value of k.3. Substitute the value of k into the equation from step 1.4. Substitute the remaining set of values into the equation

from step 3 and solve for the unknown.

ELECTRONICS The power (in watts) lost in a resistor (in the form of heat) is directly proportional to the square of the current (in amperes) passing through it. The constant of proportionality is the resistance (in ohms). What power is lost in a 5-ohm resistor carrying a 3-ampere current? powerlostofwatts

w

w

w

kcw

45

45

)9(5

)3(5 2

2

6.8 9

Heating up the Gas (mixed variation)The pressure of a certain amount of gas is

directly proportional to the temperature (measured in degrees Kelvin) and inversely proportional to the volume.

A sample of gas at a pressure of 1 atmosphere occupies a volume of 1 cubic meter at a temperature of 273° Kelvin. When heated, the gas expands to twice its volume, but the pressure remains constant.

To what temperature is it heated? KtoheatedwasgasThe

T

T

ksok

kfindfirstV

TkP

546

5462273

11

273

1

1

2731

,

6.8 10

What Next? Exponents and Radicals - Section 7.1