
How
to Solve Word Problems Programmatically

Chapter
4




Chapter 4. Unit Systems




4.1

US System



Abbrev.

Name

Derivation

Illustrative Examples




Length Dimension

in

inch


between two knuckles of an average finger

ft

foot

12 in

average man's
foot

yd

yard

3 ft

from floor to
standard door knob

mi

mile

5280 ft

four times
around a traditional track field




Weight Dimension

oz

ounce


onefourth cube
of butter

lb

pounds

16 oz

four cubes of
butter

ton

tons

2000 ton

a lot of butter




Area Dimension

in^{2}

square inch


square that
surrounds a quarter

ft^{2}

square feet

144 in^{2}

average size of
floor tile

yd^{2}

square yard

9 ft^{2}

an average card
table

acre

acre

4840 yd^{2}

just over 90
yards of a football field

section

section

640 acres

1 square mile

township

township

36 sections

6 miles by 6
miles




Volume Dimension

tsp

teaspoon


standard spoon
size

tbsp

tablespoon

3 tsp

large spoon size

floz

fluid ounce

6 tsp

2 large spoon
sizes

cup

cup

8 floz

small size
carton of milk

pt

pint

2 cup

standard size of
carton for cream

qt

quart

2 pt

medium size of
carton for milk

gal

gallon

4 qt

large size of
carton for milk





Common US Units

4.2

Metric System



Abbrev.

Name

Derivation

Illustrative Examples




Length Dimension

mm

meter


width of one dime

cm

centimeter

10 mm

width of ten
dimes

m

meter

10 cm

one half height
of 6 foot 6 in basketball player

km

kilometer

10 m

almost three
times around high school track field




Weight Dimension

mg

milligram



g

gram

1000 mg

water in small
thumbnail (1 ml)

kg

kilogram

1000 g

water in a
common size of bottled water (1 liter)

tonne

metric ton

1000 kg

water in a cubic
meter container




Area Dimension

mm^{2}

square
millimeter


size of surface
of a pin head

cm^{2}

square
centimeter

100 mm^{2}

size of a small
fingernail

dm^{2}

square
deciimeter

100 cm^{2}

size of a small
fingernail

m^{2}

square meter

100 dm^{2}

just larger than
an average card table

km^{2}

square kilometer

1,000,000 m^{2}

about one third
of a square mile

barn

barn

10^{28} m^{2}

cross section of nuclei of atom




Volume Dimension

mL

milliliter

1 cm^{3}

small thumbnail
of water

L

liter

1000 mL

common size of
bottled water





Common Metric
Units
The metric system emphasizes the use of the decimal system
of numbers. Notice that the
derivations of related units come from factors of 10. This makes unit conversions and
calculations more convenient, it has become an international standard,
especially for scientific publications.
Common conversions from other systems (including the US System) to the
metric system can be found at websites such as:
http://www.convertit.com/Go/Bioresearchonline/Measurement/Units.ASP
(Conversions)
http://www.chezcrowe.com/conv_app.htm
(Conversion)

4.3

Unit Prefixes



Name

Prefix

Unit Identity

Inverse Unit Identity

yotta

Y

10^24 m/Ym

10^24 Ym/m

zetta

Z

10^21 m/Zm

10^21 Zm/m

exa

E

10^18 m/Em

10^18 Em/m

peta

P

10^15 m/Pm

10^15 Pm/m

tera

T

10^12 m/Tm

10^12 Tm/m

giga

G

10^9 m/Gm

10^9 Gm/m

mega

M

10^6 m/Mm

10^6 Mm/m

kilo

k

10^3 m/km

10^3 km/m

hecto

h

100 m/hm

.01 hm/m

deca

da

10 m/dam

.1 dam/m

1




deci

d

0.1 m/dm

10 dm/m

centi

c

0.01 m/cm

100 cm/m

milli

m

10^3 m/mm

10^3 mm/m

micro

µ

10^6 m/µm

10^6 µm/m

nano

n

10^9 m/nm

10^9 nm/m

pico

p

10^12 m/pm

10^12 pm/m

femto

f

10^15 m/fm

10^15 fm/m

atto

a

10^18 m/am

10^18 am/m

zepto

z

10^21 m/zm

10^21 zm/m

yocto

y

10^24 m/ym 
10^24 ym/m 

Unit Prefixes

Since the metric system is conveniently based on the
decimal system, we can use prefixes to create units related to a given unit
by factors of ten. For example, a
kilogram (using the prefix "kilo") is a new unit derived from the
unit gram by multiplying by 1000. The
above table gives the unit identities that convert between such units.

4.4

Using Natural Phenomena



[meter~Length(Thing)]
Calculated from curvature of the earth



[liter~Volume(Thing)]
Cubic decimeter
[kilogram~Mass(Thing)]
Liter of water
[gram~Mass(Thing)]
Cubic centimeter of water

The metric system begins with a definition of the unit "meter" for the
dimension "length" based on the natural phenomena of the curvature
of the earth which a person can determine using trigonometry to a reasonable
amount of accuracy anywhere on the earth.
Later scientists gained more accuracy by using natural concepts of
light. The choice of meter came close
to the commonly used unit of a yard.
The metric system defines the unit "liter" for
the dimension "volume" as a cubic decimeter. This particular choice of how a liter
depends on a meter results in a volume unit close to the unit
"quart". Now using the
natural phenomena of water, the metric system defines the unit
"kilogram" of the dimension "mass" as the mass of water contained
in one liter. This means that a gram
of water fills a cube having onecentimeter edges.
Water is an abundantly available phenomena. In addition to the mass unit Liter, we
use water to define units of temperature (degrees Centigrade) and energy
(calories). Following the decimal
goals of the metric system, the freezing point of water determines zero
degrees Centigrade and the boiling point of water determines 100 degrees
Centigrade. The unit kelvin (K)
uses the same increment (1°C = 1 K) but
shifts the value 0 to absolute zero. Originally, scientists measured energy with the unit calorie
defined as the amount of energy to raise one gram of water one degree
Centigrade.


Today, they use the unit joules to measure energy and the
relationship between calories and joules for a particular substance (in
addition to just water) is the specific heat of that substance. For more information see http://library.thinkquest.org/C004970/thermo/specific.htm?tqskip1=1

4.5

SI System (System International)



Abbreviation

Unit

Dimension


SEVEN
BASE UNITS


m

meter

length

kg

kilogram

mass

s

second

time

A

ampere

electric current

K

kelvin

temperature

mol

mole

amount of matter

cd

candela

luminous intensity


ANGLE
UNITS


rad

radians

plane angle

sr

steradians

solid angle


DERIVED
UNITS


m^{2}

meters squared

area

m^{3}

meters cubed

volume

m/s

meters per second

velocity

m/s^{2}

meters per second squared

acceleration

rad/s

radians per second

angular velocity

rad/s^{2}

radians per second squared

angular acceleration

kg/m^{3}

kilograms per meters cubed

density

cd/m^{2}

candela per meter squared

luminance

A/m

ampere per meter

magnetic field strength


DERIVED
UNITS
WITH NEW
SYMBOLS


N = kg*m/s^{2}

newton

force

Pa = N/m^{2}

pascal

pressure

J = N*m

joule

quantity of energy

W = J/s

watt

power

V = W/A

volt

voltage

= V/A

ohm

electric resistance

C = A*s

coulomb

electric charge

F = C/V

farad

electric capacitance

Hz = cycles/s

hertz

frequency

Wb = V*s

weber

magnetic flux

T = Wb/m^{2}

tesla

magnetic flux density

H = Wb/A

henry

inductance

lm = cd*sr

lumen

flux of light

lx = lm/m^{2}

lux

illumination





SI System

Common SI Units
The seven base units create the foundation from which we
derive all other SI units. Notice
that each unit combines previously defined units using simple division and
multiplications. In fact, most of
units relate back to the first three base units (meters, kilograms, and
seconds) through simple algebraic substitutions. A 75 watt light bulb refers to a 75 kg*m^{2}/s^{3}
light bulb where we have the chain of derivations: W=J/s=N*m/s= (kg*m/s^{2})*m/s=
kg*m^{2}/s^{3}. Even
the base unit "ampere" relates back to the first three units
through the relationship between magnetic forces and electronic currents;
briefly, an ampere represents the amount of current in two straight parallel
wires set 1 meter apart to produce a magnetic force of 2*10^7 newtons. (See http://physics.nist.gov/cuu/Units/units.html)
The angle units represent supplementary units that use
length and area to measure plane and solid angles respectfully. You can find more information concerning
the SI System of Units at sites such as:
http://metre.info/ (Extensive
Details of SI or Modern Metric System)
http://www.metricmethods.com/metricmoments.html

4.6

UM System (Unified Math)



Abbreviation

Unit

Dimension


NINE BASE
UNITS


m

meter

length

kg

kilogram

mass

s

second

time

A

ampere

electric current

K

kelvin

temperature

mol

mole

amount of matter

cd

candela

luminous intensity

dol

dollar

monetary value

ins

instance

occurance


EXAMPLE
DERIVED
UNITS


dol/m

dollars per lineal meter

price per length

ins/s

instance per second

event frequency

s/ins

seconds per instance

duration

dol/ins

dollars per instance

price

dol/unit

dollars per unit

unit price


UM System

The Unified Math® system of units adds two new base units
to the SI system, allowing us to handle business and statistical problems.

4.7

Measuring Instruments




4.8

Calibration



The process of calibrating a measuring instrument often
involves adjusting the size of one dimension to match a predefined value of
another dimension. Take for example a
kitchen scale that measures weight based on a spring inside the measuring
instrument. Such a device quite often
has a knob (shown on the back of the one above) that allows the user to
adjust the pointer connected to the spring so that it lines up with the 0
mark on the scale.

4.9

Impact of Units on Formulas




Distance = Velocity * Time
d = v * t
(if d miles, v miles/hour, t hour)
d = v * t / 60 (if
d miles, v miles/hour, t minutes)
d = 1.61 * v * t (if d
kilometers, v miles/hour, t hour)

Units can change form of
equation

The choice of units determines the form of the
equation. It is not sufficient just
to say that distance equals the velocity (speed) times the time. We need to also specify the units of each
quantity in the equation.


4.10

Activities and Explorations




Activities:

Play “Unit Scavenger Hunt” game
Compile Lists of Measuring Instruments
Have teams create a list of measuring instruments. They can do this by browsing through the
Internet or retail stores making a list of measuring instruments and the
scales encountered.

Explorations:
Exercises:

Do Related Problems In Your Textbook


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Copyright ©
2004 Dr. Ranel E. Erickson
