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MAT081 :: Lecture Note :: Week 12
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Overview

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 

   Answer.

      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.

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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 
    equivalent to downloading 125 billion DVD movies per month."
   [...]
   "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-.

BAB::Metric Prefixes: Is it Deca- or Deka-?

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

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Some Measurement Hyperlinks

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

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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

   Answer:

      163 miles is approximately equal to 262.43 kilometers

[check answer] Google.com::Calculator:: 163mi to km

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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

   Answer.

      52 miles equals 3,294,720 inches.

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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 

   Answer.

      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

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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.

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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."

YouTube.com::How Hot Can It Get?

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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°.

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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       
   Quadrilateral...   4       4       
   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.

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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;
   and r is radius

   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

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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.]

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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

Quadrilaterals
   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

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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

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