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MAT081 :: Lecture Note :: Week 12
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### Introduction to Measurements

Merriam-Webster Online `[http://w-m.com]` defines measurement as follows.

```   1 : the act or process of measuring
2 : a figure, extent, or amount obtained by measuring : DIMENSION
```

Measurements are used to measure stuff like time, temperature, length, weight, speed, money, power, and so on.

There are two primary measurement systems: US and Metric.

Each measurement system consists of many different units. For example, in the US, lengths are measured using units of inches, feet, yards, and miles. The metric system uses length units based on the meter (kilometer, millimeter, centimeter, and so on). [UNC.edu:: metric prefixes]

Let's use inch as a starting point. An inch is about the width of a man's thumb at the base of the nail and it is used to measure length.

Inches work well when measuring short lengths. Longer lengths are usually measured using other units of measurements such as feet and yards. Example: A football field is `3600 inches` in length, but we usually say it is `100 yards`.

Ratios have been established to convert from one unit of measure to another. The following are some US length ratios.

```     12 inches equals 1 foot
3 feet   equals 1 yard
1760 yards  equals 1 mile
```

Exercise: Convert `36 inches` to `feet`. We are given that `one foot` equals `twelve inches`.

```   Setup two proportions and solve for the unknown.

1  ft.     n  ft.
------  =  ------
12 in.     36 in.

1) multiply both sides by 36 in.

1  ft.      n  ft.
36 in. * ------  =  ------- * 36 in.
12 in.     36  in.

2) solve for 'n'

1  ft.      n  ft.
36 in. * ------  =  ------- * 36 in.
12 in.     36  in.

36 * 1 ft.
----------  =  n ft.
12

36 ft.
------  =  n  ft.
12

3 ft.  =  n  ft.

36 inches equals 3 feet.
```
##### Units of Measurement Resources

NIST.gov:: General Tables of Units of Measurement
National Institute of Standards and Technology
`http://ts.nist.gov/WeightsAndMeasures/Publications/appxc.cfm`

UNC.edu:: How Many?
University of North Carolina
`http://www.unc.edu/~rowlett/units`

##### Time
```   60 seconds per 1 minute
60 minutes per 1 hour
24 hours per 1 day
```

Let's compute how many seconds are in a day.

```   We want to solve the following.

? seconds
---------
1 day

We are given the following.

60 seconds     60 minutes     24 hours
----------     ----------     --------
1 minute        1 hour        1 day

Multiply the three fractions together.  The
units of measurement "cancel" each other out.

60 seconds     60 minutes     24 hours
----------  *  ----------  *  --------
1 minute        1 hour        1 day

60 * 60 * 24  =  86,400

86,400 seconds
--------------
1 day
```

A couple time and money quotes...

"Remember that time is money."
--Benjamin Franklin

"Minutes are worth more than money. Spend them wisely."
--Thomas P. Murphy

##### Money
```   5 pennies per 1 nickel
2 nickels per 1 dime
5 nickels per 1 quarter
4 quarters per 1 dollar
```

A penny is `one hundreth` of a dollar (i.e. `\$0.01`) and a dime is `one tenth` of a dollar (i.e. `\$0.10`).

A nickel is `%5` of a dollar (i.e. `\$0.05`) and a quarter is `%25` of a dollar (i.e. `\$0.25`).

##### Temperature (measure of molecular motion)

Degrees Fahrenheit are used to record surface temperature measurements by meteorologists in the United States. Most of the rest of the world uses degrees Celsius. Most of the scientific world uses Kelvin. [The word "degrees" is not used with Kelvin. In addition, Kelvin starts at "absolute zero;" therefore, there are no negative values. "Absolute zero" is the point at which all molecular motion stops.]

Formula: Fahrenheit to Celsius

```   °C = (°F - 32) / 1.8

or

Tc = (5/9) * (Tf - 32)

where
Tc is temperature in degrees celsius
Tf is temperature in degress fahrenheit
```

Formula: Fahrenheit to Kelvin

```   K = (°F + 459.67) / 1.8
```

The boiling point of sea water is `100 °C`.

The freezing point of water at sea level is `0 °C`.

The average room temperature is `20 °C`.

The average human body temperature approximates `37 °C`.

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Introduction to the Metric System

On 8 January 2007, NASA announced that the Moon will be metric leaving only the "United States, Liberia, and Burma still primarily using English units."

The metric system is also called the International System of Units (denoted by SI).

The following are base metric units of measurement.

```   length................ meter...... m
mass.................. kilogram... kg
time.................. second..... s
electric current...... ampere..... A
temperature........... Kelvin..... K
amount of substance... mole....... mol
luminous intensity.... candela.... cd
volume................ liter...... L  [non-SI]
```
##### Metric Prefixes
```   101  deka-  da
102  hecto- h
103  kilo-  k
106  mega-  M
109  giga-  G
1012 tera-  T
1015 peta-  P
1018 exa-   E
1021 zeta-  Z
1024 yotta- Y

10-1  deci-  d
10-2  centi- c
10-3  milli- m
10-6  micro- u
10-9  nano-  n
10-12 pico-  p
10-15 femto- f
10-18 atto-  a
10-21 zepto- z
10-24 yocto- y
```
```   1 dekameter...... 10 meters
1 hectometer..... 100 meters
1 kilometer...... 1000 meters
...
1 gigameter...... 1,000,000,000 meters

1 decimeter...... 0.1 meter
1 centimeter..... 0.01 meter
1 millimeter..... 0.001 meter
...
1 nanometer...... 0.000000001 meter
```

The metric prefixes work on other base metric units.

```   1 kilogram....... 1000 grams
1 milligram...... 0.001 gram
...
1 nanogram....... 0.000000001 gram

1 hectoliter..... 100 liters
1 milliliter...... 0.001 liter
...
1 nanoliter...... 0.000000001 liter
```

[nano--very, very, very small]

##### Update::2008.11.13

From nano-tech...

```   "Sculpted using nanolithography by University of Michigan
mechanical engineer, John Hart, each Obama face is composed
of 150 million carbon nanotubes and measures half a
millimeter across."
```

NanoObama.com::Vote for Science

to high-tech...

```   "Cisco predicts IP traffic will nearly double every two years,
increasing at a combined annual growth rate of 46 percent from
2007 to 2012. That growth would result in an annual bandwidth
demand on the world's IP networks of approximately 522 exabytes,
or more than half a zettabyte. This demand on networks is
[...]
"On the scalability front, the Cisco ASR 9000 offers up to six
times the capacity of comparable edge-router solutions, with
up to 6.4 terabits per second of total capacity and up to four
times the line-card speed available on the market, with
400 gigabits per slot."
```

NewsFactor.com::Cisco Unveils 'Zettabyte-Era' Router for Online Video

at Scottsdale Community College (SCC).

```   MathBabbler did not have a MAT082 class during the Spring
2008 semester, but during that semester, Dr. Art Decabooter
stepped down as president of SCC.  Notice how Decabooter's
name starts with the metric prefix deca-.
```

Drugs have taught an entire generation of Americans the metric system. -- P. J. O'Rourke

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

NIST.gov:: General Tables of Units of Measurement
National Institute of Standards and Technology
`http://ts.nist.gov/ts/htdocs/230/235/appxc/appxc.htm`

UNC.edu:: How Many?
University of North Carolina
`http://www.unc.edu/~rowlett/units`

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Unit of Measurement: Length Example (miles to kilometers)

Tempe and Show Low are `163` miles apart. Calculate this distance in `kilometers`.

```   Given:

1 mi = 1.609344 km

Solve:

??? kilometers
--------------
163 miles

1) setup a proportion

??? kilometers    1.61 kilometers
-------------- =  ---------------
163 miles         1 mile

2) solve for ??? by multiplying both sides by 163 miles

??? kilometers    1.61 kilometers
163 miles x -------------- =  --------------- x 163 miles
163 miles         1 mile

??? kilometers    1.61 kilometers
163 miles x -------------- =  --------------- x 163 miles
163 miles         1 mile

1.61 x 163 equals 262.43

163 miles is approximately equal to 262.43 kilometers
```

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Unit of Measurement: Length Example (miles to inches)

Tempe and Superior are appoximately `52 miles` apart. Compute how many inches this is.

```   Solve.

? inches
--------
52 miles

Given.

1 mile = 1760 yards
1 yard =    3 feet
1 foot =   12 inches

1) Convert 52 miles into yards.

1760 yards     ? yards
---------- =  --------
1 mile     52 miles

52 miles equals 91,520 yards

2) Convert 91,520 yards into feet.

3 feet     ? feet
------ = ------------
1 yard   91,520 yards

91,520 yards equals 274,560 feet

3) Convert 274,560 feet into inches.

12 inches    ? inches
--------- = ------------
1 foot     274,560 feet

3,294,720 inches equals 274,560 feet

52 miles equals 3,294,720 inches.
```

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Unit of Measurement: Time Example (seconds to day)

Given.

```   60 seconds per 1 minute
60 minutes per 1 hour
24 hours per 1 day
```

Let's compute how many seconds are in a day.

```   We want to solve the following.

? seconds
---------
1 day

We are given the following.

60 seconds     60 minutes     24 hours
----------     ----------     --------
1 minute        1 hour        1 day

Multiply the three fractions together.  The
units of measurement "cancel" each other out.

60 seconds     60 minutes     24 hours
----------  x  ----------  x  --------
1 minute        1 hour        1 day

60 * 60 * 24  =  86,400

86,400 seconds
--------------
1 day
```

A couple time and money quotes...

"Remember that time is money."
--Benjamin Franklin

"Minutes are worth more than money. Spend them wisely."
--Thomas P. Murphy

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Unit of Measurement: Money

The first definition for money given by Merriam-Webster Online is as follows.

```   "1 : something generally accepted as a medium of exchange,
a measure of value, or a means of payment: as a :
officially coined or stamped metal currency"
```

Here is only one of many good money quotes...

Money can't buy you happiness, but it does bring you a more pleasant form of misery.
-- Spike Milligan (01918-02002) { British actor and comic; more... } [money/happiness]

In the United States of America we deal with monetary units of measurement such as pennies, nickels, dimes, quarters and dollars (\$1, \$5, \$10, \$20, \$50, \$100, ...; what about \$2?).

Conversion ratios.

```   5 pennies per 1 nickel   (  5 cents; \$0.05)
2 nickels per 1 dime     ( 10 cents; \$0.10)
5 nickels per 1 quarter  ( 25 cents; \$0.25)
4 quarters per 1 dollar  (100 cents; \$1.00)
```

A penny is `one hundreth` of a dollar (i.e. `1/100` or `\$0.01`) and a dime is `one tenth` of a dollar (i.e. `1/10` or `\$0.10`).

A nickel is `%5` of a dollar (i.e. `\$0.05`) and a quarter is `%25` of a dollar (i.e. `\$0.25`).

Finance.Yahoo.com:: Currency Converter

Currency conversion examples.

That's my .02 cents on the subject.

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Unit of Measurement: Temperature

Temperature is a measure of molecular motion.

Here is a quote about temperature.

A Harvard Medical School study has determined that rectal thermometers are still the best way to tell a baby's temperature. Plus, it really teaches the baby who's boss.
-- Tina Fey (1970-???) { Saturday Night Live comedian; pic }

Degrees Fahrenheit are used to record surface temperature measurements by meteorologists in the United States. Most of the rest of the world uses degrees Celsius. Most of the scientific world uses Kelvin. [The word "degrees" is not used with Kelvin. In addition, Kelvin starts at "absolute zero;" therefore, there are no negative values. "Absolute zero" is the point at which all molecular motion stops.]

Formula: Fahrenheit to Celsius

```   °C = (°F - 32) / 1.8

or

Tc = (5/9) * (Tf - 32)

where
Tc is temperature in degrees celsius
Tf is temperature in degress fahrenheit
```

Formula: Fahrenheit to Kelvin

```   K = (°F + 459.67) / 1.8
```

The boiling point of sea water is `100 °C`.

The freezing point of water at sea level is `0 °C`.

The average room temperature is `20 °C`.

The average human body temperature approximates `37 °C`.

The hottest recorded temperature in the United States is `56.7 °C (134 °F)` in Death Valley, California on 10 July 1913. The hottest temperature ever recorded on Earth is `136 °F` at Al' Aziziyah, Libya in September of 1922.

The coldest recorded temperature on earth was `-128.6 °F` on 31 July 1983 at a Russian research station at Vostok, Antarctica.

The following was copied from ClassZone.com

```   "The Vostok ice core provides the longest continuous record
of Antarctic climatic history. Analysis of the core has been
completed to a depth of 3350 meters, representing approximately
440,000 years of climate history."
```

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Introduction to Geometry

Geometry is a "branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids."

When Moses was alive, these pyramids were a thousand years old.
Here began the history of architecture. Here people learned to
measure time by a calendar, to plot the stars by astronomy and
chart the earth by geometry. And here they developed that most
awesome of all ideas - the idea of eternity.

-- Walter Cronkite (01916-02009) {"the most trusted man in America"; more...} [geometry]

[update::2012.04.16] CBS News marks historic Cronkite anniversary

##### Terminology

The following terms are used extensively in geometry.

```          point:  location (no dimensions, no length, no width, no depth)
space:  set of all points
plane:  set of points on one surface
line:  set of points (at least two) that has only
one dimension, length
line segment:  line having two end-points
ray:  line with one end-point
angle:  two lines that intersect form an angle
vertex:  point where two lines intersect
```

According to Euclid, an plane angle is "the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line."

##### Points

Points are typically labeled.

##### Lines

A line segment is a line that has a starting and ending point. A ray has a starting point, but no ending point.

Let `A` and `B` be two points on the same line.

```                                       <-->
<-----A------------B----->    line:  AB

__
A---------------B        segment:  AB

-->
A---------------B----->   ray:  AB

<--
<-----A--------------B        ray:  AB

```

Lines are often used in pairs and there are four types of pairs: intersecting, parallel, perpendicular, and skewed. Intersecting lines cross each other; parallel lines never touch and are always equal distance apart; perpendicular lines form right angles; and skewed lines never intersect nor are they parallel (they are not in the same plane). [Railroad tracks provide an example of parallel lines.]

```         A
|
|
C--i-----D
|
B

The line segments AB and CD intersect and are perpendicular.
The vertex (i.e. point) where they intersect is labeled 'i'.
```
##### Angles

An angle is formed when two lines intersect.

There are four types of angles: straight, right angle, acute, and obtuse. Angles are measured using degrees (a fraction of a circle [360°]).

```   straight angle ... 180°
right angle ......  90°
acute angle ...... less than 90°
obtuse angle ..... greater than 90° but less than 180°
```

Jason Kidd, professional basketball player, when traded to a new team, was quoted he was going to help turn the team around `360°`.

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Geometry: Polygons

A polygon is a closed plane figure bounded by straight lines (i.e. sides). Common polygons are squares, rectangles, and triangles.

Every polygon contains at two or more angles.

A quadrilateral is a polygon having four sides.

A parallelogram is a quadrilateral where opposite sides are both parallel and equal length.

A rectangle is a parallelogram consisting of four right angles. A square is a rectangle where all four sides are equal length.

```   rhombus -- parallelogram having 4 sides of equal length
trapezoid -- exactly one pair of opposite sides parallel
```

A triangle is a three-sided polygon.

The following is list of some other polygons.

```   type            #sides  #angles
-------------------------------
Triangle........   3       3
Pentagon........   5       5
Hexagon.........   6       6
Heptagon........   7       7
Octagon.........   8       8
Nonagon........    9       9
Decagon........    10      10
```

Two polygons are congruent if they are the same size and shape (i.e. corresponding angles and sides are equal). Two polygons are similar if they have the same shape, but different sizes.

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Geometry: Circle

A circle is a plane figure consisting of points that are equal distance from a center point.

```     radius:  distance from circle center to a circle point
diameter:  distance across a circle through the center
```

The diameter of a circle is equal to two times its radius.

```   d = 2 * r

where d is the diameter and r is the radius
```

The circumference of a circle is its perimeter (i.e. distance around the circle).

```   C = 2 * pi * r

where C is circumference; pi is the value 3.14;

C = pi * d

where d is the diameter
```

The value of `pi` approximates the fraction `22/7`.

The area of a circle is the number of "square units" needed to cover the circle.

A = pi * r2 where A is area; pi is the value 3.14; and r is radius ["Why do they say 'pie are square' when pies are round?"]
```       *   *
*         *
*   r       *
*-----+     *    + is the center of the circle
*           *
*         *
*   *
```

GDT::BAB:: Washington Tree May Be Dying

Demon.co.ukPi

http://gwydir.demon.co.uk/jo/numbers/interest/pi.htm

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Geometry: Perimeter and Area for Polygons

A perimeter is a measure of the distance around a region. [Hint: perimeter]

```   rectangle    P = 2 * length + 2 * width
square       P = 4 * side
triangle     P = side a + side b + side c
```

An area is a measure of the surface of a region.

```   rectangle    A = length * width
square       A = side2
triangle     A = 0.5 * base * height
```
##### Square Yard Example

[Saturday, 24 April 2004, Arizona Republic]
The U.S. Department of Agriculture reported that most of western Nebraska has at least 15 grasshoppers per square yard. Some areas had as many as 370 grasshoppers in a square yard. [Anything over 40 is considered severe.]

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Continue with Geometry: Plane Figures

##### Plane Figures

Recall, a plane is a flat surface that spans to infinity. A plane figure is a geometric form that lies on a plane.

##### Triangles

A triangle has three sides that form three angles. The sum of the three angles total 180°.

The following are special types of triangles.

```   equilateral -- all sides have equal length and
all angles are equal
isoceles -- two sides are same length; angles
opposite the equals size are equal
scalene -- no sides have the same length and no
angles are equal
right -- has a right angle; the side opposite the
right angle is the hypotenuse and the other
two sides are legs
```

If you know the length of two sides of a right triangle, then the length of the third side can be calculated using the Pythagorean Theorem.

```   a2 + b2 = c2

where 'a' and 'b' are the legs and 'c' is the hypotenuse
```

GDT::BAB:: Find X

The perimeter of a triangle is the sum of the length of its sides.

The area of a triangle is one-half the product of its base length and height.

```       1
A = - * b * h
2

+
/|
/ |
/  | h
/   |
/    |
+-----+
b
```

GDT::BAB:: Triangles In Space May Be Signals

```   parallelogram -- opposite sides are parallel and
have equal length
rectangle -- parallelogram with 4 right angles
square -- rectangle with equal length sides
rhombus -- parallelogram having 4 sides of equal length
trapezoid -- exactly one pair of opposite sides parallel
```

{SCC math resources... MathAS::id 5778; key 5778 | OER::book | playbook |
{classroom: quiet | thinking | speed e-raser | wave erase}

### Volume Formulas

Volume is measured in "cubic" units. The volume of a figure is the number of cubes required to completely fill it.

Volumes are used for cubes, rectangular prisms, irregular prisms, cylinders, pyramids, cones, spheres, and ellipsoids.

Math.com::Geometry:: Cylinders, cones and spheres