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MAT081 :: Lecture Note :: Week 10
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### Start of the Week Exercises

```   a) A pecan pie is cut into five pieces.  Four-fifths of the pie is
consumed.  Herb eats one-eigth of the remaining pie.  How much
pie is left after Herb treats himself?

b) Write the whole number 4 as a fraction.

c) Solve:  4 + (7/8)

d) Solve:  4 * (7/8)

e) Solve:  4 - (7/8)

f) Solve:  4 ÷ (7/8)
```

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

### 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."
```