### Formulas, Applications, and Variation

```6.8 Solving (Rearranging) Formulas
& Types of Variation
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
Rearranging formulas containing rational expressions
Variation
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


Variation
Inverse
Joint
Combined
6.8
1
Electronics:
“Solving” formulas for different variable


Solve the electronic
resistance formula
for the variable r1
What’s the LCD?
1 1 1
 
R r1 r2
1 1 1
     Rr1r2
 R r1 r2 
r1r2  Rr2  Rr1
r1r2  Rr1  Rr2
r1 r2  R   Rr2
Rr2
r1 
r2  R
6.8
2
Astronomy:
“Solving” formulas for different variable


Solve for the height
variable h in the
satellite escape
velocity equation
What’s the LCD?
V2
2g

R2 R  h
V 2
2g  2
 2 
 R R  h 
Rh
R
V 2 R  h   2 gR2
V 2 R  V 2 h  2 gR2
V 2 h  2 gR2  V 2 R
2 gR2  V 2 R
h
V2
6.8
3
Acoustics (the Doppler Effect):
“Solving” formulas for different variable


Solve for the speed
variable s in the
doppler effect
equation
What’s the LCD?
sg
f 
sv
sg 

f 
s  v 
sv

f s  v   sg
fs  fv  sg
fs  sg   fv
s( f  g )   fv
fv
fv
s

f g g f
6.8
4
Direct Variation
The 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 = 2
Real: Distance of a lightning bolt varies directly with the time
6.8
between seeing the flash and hearing
the thunder. m = (1/5)s
5
Inverse Variation
The 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 = 80
Real: Loudness of sound varies inversely with the square of
the distance from the sound. L6.8= k/d2
6
Joint Variation
The 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 jointly
with 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
7
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
6.8
resistor carrying a 3-ampere current?
w  kc2
w  5(3) 2
w  5(9)
w  45
45 watts of lost power
8
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?
6.8
T
P  k
first , find k
V
273
1
1 k
so k 
1
273
1 T
1

273 2
546  T
The gas was heated to 546 K
9
What Next?

Exponents and Radicals - Section 7.1
6.8
10
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