Chabot Mathematics
Using Units
Of Measure
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Units Introduction
 People measure quantities
through comparisons with
standards.
 Every measured quantity has an associated
“unit” Which is the name of the Standard.
 Need to define sensible and practical "units"
and "standards" that scientists & engineers
everywhere can agree upon
 Even though there exist an almost infinite
number of different physical quantities, we
need no more than a handful of “base”
standards.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
SI System of Units
 Système International d'Unités
(International System of Units)
 A Completely Consistent
Set of Basic Units
• Requires NO
Conversion factors
– e.g., 5280 ft = 1 mile
• Defined by UNCHANGING
Physical Phenomena
– Except for one...
Chabot College Mathematics
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http://www.bipm.org/en/si/
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
SI System History
 In 1960 The 11th General
Conference on Weights and Measures
(GCWM) adopted the name SI System,
for the recommended practical system
of units of measure.
 The 1960 GCWM Specified Seven
well-defined “Base” units which,
by convention, are regarded as
DIMENSIONALLY INDEPENDENT
http://www.bipm.org/en/si/
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
 From this List
Observe
SI Base Units
SI Base Units
Base quantity
length
mass
Name
Symbol
meter
m
kilogram
kg
time
second
s
electric current
ampere
A
thermodynamic
temperature
kelvin
K
amount of substance
mole
mol
candela
cd
luminous intensity
 All but the kg are
defined by Physical
Phenomena
• Examine the Defs
Chabot College Mathematics
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• Very common Units
– Mass (kg)
– Length (m)
– Time (s)
• Some Not so
Common Units
– Current (A)
– Temperature (K)
• Some Uncommon
Units
– Substance amt (mol)
– Luminous Int (cd)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Meter Defined
 Length or
Distance
(meter)
 “The path traveled by light in
vacuum during a time interval
of 1/299792458 of a second.”
1 meter
Laser
photon
Chabot College Mathematics
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1/299792458 s
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
kilogram Defined
 Mass
(kilogram)
 a cylinder of PLATINUMIRIDIUM alloy maintained
under vacuum conditions
by the International
Bureau of Weights and
Measures in Paris
If The ProtoType Were Cubic, its
Edge Length would be About 36.2
mm (1.42”); quite small
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Second Defined
 Time
(Second)
 The duration of 9 192 631 770
periods of the radiation
corresponding to the transition
between the two hyperfine
levels of the ground state of
the cesium 133 atom
• This is the Definition of an
“Atomic” Clock
– more than 200 atomic clocks are
located in metrology institutes and
observatories in more than 30
countries around the world
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Amp Defined
 Electric
Current
(ampere)
 That constant current which,
if maintained in two straight
parallel conductors of infinite
length, of negligible circular
cross-section, and placed
1 m apart in a vacuum, would
produce between these
conductors a force equal to
2 x 10−7 Newton per metre of
length.
• What’s a Newton?→ 1kg-m/(s2)
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Kelvin (Temperature) Defined

 Thermodynamic
temperature
(Kelvin)
The unit of thermodynamic
temperature, is the fraction
1/273.16 of the
thermodynamic temperature of
the triple point of water.
 273.16K = 0.0098 °C
 Room Temperature
(72 °F) is about 295.5
Kelvins
 NO “Degree” Sign
Used with the
Kelvin Unit
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
mole (amt of Substance) Defined
 Amount of  The mole is the amount of
substance of a system which
Substance
contains as many elementary
(mole)
entities as there are atoms in
0.012 kilograms of carbon 12.
 1 mole = 6.023x1023
entities
• entities must be specified
and may be atoms,
molecules, ions, electrons,
other particles, or specified
groups of such particles.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Luminous Intensity Defined
 The luminous intensity, in a given
 Light
direction, of a source that emits
Brightness
monochromatic radiation (one(candela)
color light) of frequency 540 x
1012 Hertz (555 nm) and that has
a radiant intensity in that direction
555nm
color
of 1/683 watt per steradian
 The are 4 (12.57)
Steradians in a Sphere
• 1 Str = 7.96% of the
Sphere Surface
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Units Have Evolved
 Candela Predecessor based
on a Flame
• Hence the Name
 Temperature Based on Freezing points
• Water
• Platinum
 Second Based on the
Sidereal (standard) day
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Units Have Evolved
 History of the Meter (or Metre)
• One ten millionth of the distance
from the North pole to the equator.
• The distance between two
fine lines engraved near the
ends of a platinum-iridium bar
• 1 650 763.73 wavelengths of a particular
orange-red light emitted by atoms of
krypton-86 (86Kr).
• The length of the path traveled by light in a
vacuum during a time interval of
1/299 792 458 of a second.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
SI Derived Units
 The Seven Base Units May be
Algebraically Combined to Produce
“Derived Units”
units of distance
• e.g.:
Units of velocity 
units of time
meters
 Several Derived

seconds
Units have Special
Usefulness and are
 m/s
thus Given their
OWN Names
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Some Derived Units
Derived quantity
Name
Symbol
Expression
in terms of
other SI units
Expression
in terms of
SI base units
plane angle
radian (a)
rad
-
m·m-1 = 1 (b)
solid angle
steradian (a)
sr (c)
-
m2·m-2 = 1 (b)
frequency
hertz
Hz
-
s-1
force
newton
N
-
m·kg·s-2
pressure, stress
pascal
Pa
N/m2
m-1·kg·s-2
energy, work,
quantity of heat
joule
J
N·m
m2·kg·s-2
power, radiant
flux
watt
W
J/s
m2·kg·s-3
electric charge,
quantity of
electricity
coulomb
C
-
Chabot College Mathematics
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s·A
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Some (more) Derived Units
Derived quantity
Name
Symbol
Expression
in terms of
other SI units
Expression
in terms of
SI base units
electric potential
difference,
electromotive
force
volt
V
W/A
m2·kg·s-3·A-1
capacitance
farad
F
C/V
m-2·kg-1·s4·A2
electric
resistance
ohm

V/A
m2·kg·s-3·A-2
electric
conductance
siemens
S
A/V
m-2·kg-1·s3·A2
magnetic flux
Weber
Wb
V·s
m2·kg·s-2·A-1
magnetic flux
density
tesla
T
Wb/m2
kg·s-2·A-1
inductance
henry
H
Wb/A
m2·kg·s-2·A-2
Celsius
temperature
degree Celsius
°C
luminous flux
lumen
lm
cd·sr (c)
illuminance
lux
lx
lm/m2
Chabot College Mathematics
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-
K
m2·m-2·cd = cd
m2·m-4·cd = m2·cd
Bruce Mayer,
PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
SI prefixes – A form of ShortHand
Factor
Name
Symbol
Factor
Name
Symbol
1024
yotta
Y
10-1
Deci
d
1021
zetta
Z
10-2
Centi
c
1018
exa
E
10-3
milli
m
1015
peta
P
10-6
micro
µ
1012
tera
T
10-9
nano
n
109
giga
G
10-12
pico
p
106
mega
M
10-15
femto
f
103
kilo
k
10-18
atto
a
102
hecto
h
10-21
zepto
z
101
deka
da
10-24
yocto
y
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Derived Units Family Tree
No Special Names
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Old (and Tired) Unit Sets
 MKS
• Stands for Meter-Kilogram-Second in the
Most Common Units
– Predecessor to The SI System
 CGS
• Means Centimeter-Gram-Second
– Still Widely Used
 IPS, FPM, FPH
• Inch-Pound-Sec, Foot-Lb-Min, Ft-Lb-Hour
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
American Engineering System,
AES – Still in (declining) Use
Fundamental Dimension
length
foot (ft)
mass
pound (lbm)
force
pound (lbf)
time
second (sec)
electric charge [Q]
coulomb (C)
absolute temperature
degree Rankine (oR)
luminous intensity
candela (cd)
amount of substance
mole (mol)
Chabot College Mathematics
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Base Unit
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Conservation of Units
 Principle of conservation of units:
• Units on the LEFT side of an equation
MUST be the SAME as those on the
RIGHT side of an Equation
 Then Have Dimensional Homogeneity
• Needed to Prevent
“Apples & Oranges” Confusion
– e.g., I Buy 100 ft of Wire at One Store and 50 m
at another; how much total Wire do I have?
(It’s NOT “150”)
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Unit Conversion by Chain-Link
 To Determine the Amount of Wire I have
I Need to Convert to Consistent
(Homogeneous) Units
 Start by Thinking About the Definition of “1”
• AnyThing divided by ITSELF = “1”
• Now Consider a “minute” 60 Seconds  1 minute
therefore
1 min
1
60 sec
or
60 sec
1
1 min
 Read as “60 Seconds per minute”
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Chain-Link Unit Conversion
 Units can also be Multiplied and Divided
in a manner similar to Numbers
• This how we get, say, “Square Feet”
– e.g.; Consider an 8ft x 10ft Engineer’s Cubicle in
Dilbert-Land. How Much WorkSpace Does the
Engineer Have?
WrkSpc  8ft x 10 ft  8x10 ftxft   80 ft
 Now Back to the Wire
• Want to Know how many FEET of Wire
I have in Total
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
2
Chain-Link Unit Conversion cont.
 Check in Table 16.8 and Find
“3.2808 ft per meter”
• Multiply the 50m by this special Value of 1
3.2808 feet
50 meter 1  50 meter 
 164.04 feet
1 meter
 Can “Cancel” The Units by Division
 So then the Total Wire = 264 ft
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Chain Link Examples
 A World-Class Sprinter can Run 100m in 10s.
• How Fast is this in MPH?
100 m 3.2808 ft 1 mile
60 s 60 min
miles




 22.37
10 s
1m
5280 ft 1 min
1 hr
hr
 Gasoline In Seoul Costs 1840 Korean-Won
(W) for one Liter of Regular Unleaded
• How Much is this in $ per Gallon
– Find Currency Exchange Rate → $1 = 1150 W
1840 W
1$
28.317 Liter
1 ft 3
$



 6.06
3
1 liter
1150 W
1 ft
7.48 Gal
Gal
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Several Forms of “1”
 Unit Conversion Factors
1 mile = 5280 feet
1 meter = 3.281 feet
1 foot = 12 inches
1 yard = 3 feet
1 lb = 4.448 Newtons
1 m2 = 1973.5 Circular inch
1 cup = 48 TeaSpoon
$1 = 0.787 €
1 Btu = 1054.4 Joule
1 Watt = 1 Joule/sec
1 HorsePower = 2545 Btu/hr
1 km = 1000 meters
1 furlong = 220 yards
°F = 1.8x°C + 32
1 Acre = 43,560 ft2
$1 = 16,030 Viet Nam Dong
1 hour = 60 min
1 min = 60 sec
1 gallon = 3.785 liters
1 ft3 = 7.4805 gallons
1 Pascal = 1 Newton/m2
1 HorsePower = 550 ft-lb/s
1 lb = 16 ounces
$1 = 10.825 Mexican Pesos
 ANYTHING Divided by ItSelf = 1 
43560 ft 2
550 ft  lbs  s
1054.4 J
10.825 MXNs
1
1
1
1 acre
1 HP
1 Btu
1 USD
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Units – Exponent Properties
a1 = a
0 as an exponent
a0 = 1
Negative
Exponents
(flippers)
an
1
 n,
a
a n bm  a 
 n,  
m
b
a
b
The Product Rule
a m  a n  a mn .
The Quotient Rule
am
 a mn .
n
a
The Power Rule
(am)n = amn
The Product to
a Power Rule
(ab)n = anbn
The Quotient to
a Power Rule
Chabot College Mathematics
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n
n
a a
   n.
b b
n
b
 
a
n
This summary assumes that no
denominators are 0 and that 00 is not
considered. For any integers m and n
1 as an exponent
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Raising units to POWERS
 Start again with 1
1  1  1    1
2
3
 Can do the SAME
Thing with Units.
 And 123 = 1728 so
n
 Thus have 1728
“cubic inches” per
“Cubic Foot”
 What’s a “Cubic
Yard” in “Cubic
Feet”?
 So have 27 cubic-ft
per cubic-yd
• NOT “9”
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
7 inches, Water Column
 An ENGR10 Guests
Speaker noted that
Natural Gas is
delivered by PG&E
to home at a
pressure of 4-7
“inches of Water
Column”
 A U-Tube
Manometer can
measure pressure
Differences in
Inches of Water
Column
 This is a unit of
pressure, Just Like
Pascals or psig
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
7 inches, Water Column
 To Calc the “in-WC”
pressure we need to
know some
Engineering Physics
 From ENGR36
Pg  h
 For Liquid Water at
Room Temperature
and Pressure
 w  9790 N m
 Now find 7 in-WC in
psig
• Where
– γ ≡ Liquid SPECIFIC
WEIGHT
– h ≡ liquid Column
Height
Chabot College Mathematics
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3
Natural Gas @ 9.5 inWC
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
7 inches, Water Column
 Convert out the N & m
N
N  in
Pg   w h w  9790 3  7in  68530 3
m
m
N  in
1lb
 1m   1ft 
Pg  68530 3 



m
4.448N  3.281ft  12in 
3
3
lb  in
1m3
1ft 3
lb
Pg  15407 3 

 0.2524 2
3
3
m
35.32ft 1728in
in
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
White Board Examples cont.

The USA FDA recommends that Adults
consume 2200 Calories per Day
•
What then is the “Power Rating”
of a Grown Human Being?
– Note that there are TWO
types of “Calories”
1. The Amount of Heat Required to Raise the
Temperature of 1 GRAM of water by 1 °C (or 1 Kelvin)
 Often Called the Gram-CAL; This is what is in the Text
2. The Amount of Heat Required to Raise the
Temperature of 1 KILOgram of water by 1 °C
 Often Called the kgCAL or kiloCal; This is what
you read on the side of Food Packaging
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Tire Pressure
 Many AutoMobile
Tires have a
Maximum Pressure
Rating of About 44
psig.
 Convert 44 psi to
kiloPascals (kPa)
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Ton of Refrigeration
 During his
Presentation Mr.
Ian McClaren of
SouthLand
Industries
described the
“Ice Storage”
Cooling System Behind Bldg-1800.
 He Noted that the Cooling Power
of this system was Rated in “Tons”
 What is a “Ton” of Cooling Power
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Ton of Refrigeration
 A TON of the refrigeration
is defined, roughly, as the
COOLING effect of melting 2000 lbs of
water ICE over a 24 HOUR Period
• From PHYS4C (or ASHRAE HandBook)
find that the “Latent Heat of Fusion” for ice
is 333.55 kJ/kg
 On WhtBoard Convert a
“Ton of Refrigeration” to
• kW and Btu/hr
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
White Board Examples
 A 2003 Chevy z06 corvette
• Has a 5.7 Liter V8 Engine
– What is the Engine Displacement in cubic-inches?
• Develops 410 HP
– What is the Power in Watts?
 A the Maximum recommended pressure
for many 65R15 tires is 44 psi (lbs per sqinch; NOT lbs)
• What is this Max Pressure in kPa?
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
All Done for Today
How to
Spend
the
Calories
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Tire Pressure: 44 psi → kPa
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-01_sec_1-1a_Using_Units.pptx