Download Mathematics (P)review

Mathematics (P)review
The star Proxima Centauri is 23,400,000,000,000 miles
away from Earth. If we could travel in a spaceship at 5000
miles/hour, it would take over 534,000 years to get there.
Scientific Notation eliminates all unnecessary place holders
by making use of powers of 10
E.g.
2000 = 2 x 103
The star Proxima Centauri is 2.34 x 1013 miles away from
Earth. If we could travel in a spaceship at 5 x 103
miles/hour, it would take over 5.34 x 105 years to get there.
How to Write a Number in Scientific Notation
Scientific notation is the mantissa multiplied by appropriate power of ten (10)
534,000 = 5.34 x 105
Step 1: Find the mantissa by moving the decimal so that there is
only one digit to the left (eliminate unnecessary digits).
718,000 = 7.18 x 105
0.00942 = 9.42 x 10-3
Step 2: Find the appropriate power of 10:
• when moved to the left, power = + number of places moved
• when moved to the right, power = - number of places moved
Astronomical Scales
Distance to nearest star: Proxima Centauri: ~ 4 light-years
1 light-year (L.Y.) – distance light travels in one year’s time
Distance = speed x time (d = v x t)
1 L.Y. = speed of light x one year
1 L.Y. = (186,000 mi/sec) x (31,500,000 sec)
1 L.Y. = (1.86 x 105 mi/sec) x (3.15 x 107 sec)
1 L.Y. = (1.86 x 3.15) x (105 x 107)
1 L.Y. = 5.86 x 1012 miles
Dist. to Proxima Centauri = 4 L.Y. = 2.34 x 1013 miles
Proportions
Proportions (ratios) are used to compare two quantities
Astronomical Unit – average Earth-Sun distance
Distance to Pr oxima Cen
=
Distance to Sun
2.34 x 1013 miles
=
9.3 x 10 7 miles
= 252,000
The distance to the nearest star (Proxima Cen) is 252,000
times the distance to the Sun
Proportions can be used to establish a scale for a model
!
!
Scale
Quantifies the relationship between a model (e.g. map) and
the subject that it represents.
Model: Sun and Earth are separated by 1 foot; how far away will
Proxima Centauri be?
"Distance to Proxima Cen%
=
$#
'
&Model
Distance to Sun
"Distance to Proxima Cen%
$#
'&
Distance to Sun
Reality
"Distance to Proxima Cen%
= 252,000
$#
'
&Model
1 foot
[Distance to Proxima Cen]Scale = 252,000 feet = 47.7 miles!
Useful Powers of Ten
Ten (10) = 101
Hundred (100) = 102
Thousand (1000) = 103
Kilo-
Million (1,000,000) = 106
Mega-
Billion (1,000,000,000) = 109
Giga-
Average distance between the Earth & Sun
93 million miles = 93 x 106 miles or 93 Mega-miles
Useful Powers of Ten
Tenth (0.1) = 10-1
Hundredth (0.01) = 10-2
Centi-
Thousandth (0.001) = 10-3
Milli-
Millionth (0.000001) = 10-6
Micro-
Billionth (0.000000001) = 10-9
Nano-
Average thickness of human hair
75 x 10-6 meters = 75 micrometers
Basic Algebra Review
Substitution Method
Insert known values into an equation and calculate the value for the desired unknown
Determine the value of y if x = 4.
y = 2x + 7
y = 2(4) + 7 = 8 + 7
y = 15
!
!
!
!
!
Determine the value of y if x = 3.2 x 106 and z = 1.4 x 103.
!x 2
y =
z
(3.2 x 10 6 ) 2
1.024 x 1013
y =
=
3
1.4 x 10
1.4 x 10 3
y = 7.31 x 10 9
!
Basic Algebra Review
Inverse Operations
To get an unknown value by itself, apply inverse operations to both sides of an equation.
The primary inverse operations are as follows:
Addition
Subtraction
Multiplication
Division
Raising to a Power
Taking a Root
Determine the value of y
y2
- 7= 2
4
y2
= 9
4
!
!
!
y 2 = 36
y= 6
Add 7 to both sides
Multiply both sides by 4
Take the square root of both sides
Data Analysis Using a Graph
Pressure (psi)
70
Depth (ft)
Pressure (psi)
20
24
40
31
60
43
80
49
100
62
50
30
20
40
60
80
100
Depth (ft)
To get the rate at which the pressure increases with depth:
(1) Draw in a line that fits the data points the best
(2) Get the coordinates of two arbitrary points on the line
(3) Use the coordinates to find the slope of the line
P1 = (13, 20)
m =
P2 = (90, 54)
"y
(54 - 20)
=
= 0.44 psi/ft
"x
(90 - 13)
Measurement Systems
Every measurement consists of a value and a unit
It is 215 miles from Boston to New York City
Having one system of units allows everyone to be on the same page
Requirements for any measurement system include:
- A standardized basis
- Easy to convert measurements within the system
The metric system is both the scientific standard and the world
standard (including U.S. though the British Imperial System is
use in everyday experience)
British Imperial System
Length/Distance
12 inches = 1 foot; 3 feet = 1 yard; 2 yards = 1 fathom
1 mile = 5280 feet = 1760 yards = 880 fathoms = 63,360 inches
• other units of length include: digit, nail, palm, hand, shaftment,
span, cubit, rod, chain, furlong
Mass
1 Pound = 16 ounces (originally 12 ounces ≡ Troy pound)
1 Ton = 20 hundredweights
(1 Hundredweight = 120 lbs)
• 1 Ton = 2240 lbs (2000 lbs is the American “short-ton”)
Volume
Gallon – dry: 268.8 cubic inches; liquid: 231 cubic inches
Pint – American: 16 oz; British: 20 oz.
The Metric System
Seven basic properties of nature are identified within the SI System and
each has an assigned base unit.
Basic Physical Property
Distance
Base Unit
meter (m)
Mass
kilogram (kg)
Time
second (s)
Temperature
Kelvin (K)
Electric Current
Ampere (A)
Amount of a Substance
Mole (mol)
Intensity of Light
Candela (cd)
Extension of Base Units
(within metric system)
Base Unit
Conversion Factor
Gigameter (Gm)
109 m
Megameter (Mm)
106 m
Kilometer (km)
1000 m
Hectometer (Hm)
100 m
Decameter (Dm)
10 m
Meter (m)
1m
Decimeter (dm)
0.1 m
Centimeter (cm)
0.01 m
Millimeter (mm)
0.001 m
Micrometer (µm)
10-6 m
10-9 m
Nanometer (nm)
Extension of Base Units
(within metric system)
Base Unit
Conversion Factor
Gigagram (Gg)
109 g
Megagram (Mg)
106 g
Kilogram (kg)
1000 g
Hectogram (Hg)
100 g
Decagram (Dg)
10 g
Gram (g)
1g
Decigram (dg)
0.1 g
Centigram (cg)
0.01 g
Milligram (mg)
0.001 g
Microgram (µg)
10-6 g
10-9 g
Nanogram (ng)
Conversion of Units - Metric
Multiply the given value by the ratio of the conversion factor (CF) of the
given unit to the CF of the desired unit
Known Measurement
Desired Measurement's Power of 10
=
Desired Measurement
Known Measurement's Power of 10
Problem: Convert 12 km to centimeters
12 km
10 -2
=
x cm
10 3
!
12 " 10 3 = x " 10 -2
5
x
=
12
"
10
cm = 1,200,000 cm
!
!
Conversion Between Different Systems
Problem: Convert 4 km to miles
• Set up a proportion using the proper conversion factors:
1 mile = 1.6 km
4 km
1.6 km
=
x miles
1 miles
• Cross multiply:
(4) · (1) km = 1.6 · x miles
• Solve for unknown:
x = (4) ÷ (1.6) = 2.5 miles
Physical Properties: Other Units of Measure
Velocity (v) – how fast something is moving
Base Unit: meters-per-second (m/s)
distance
velocity =
time
velocity
Acceleration (a) – rate something changes velocity a =
time
2
Base Unit: meters-per-square-second (m/s )
Force (F) – push or a pull on an object
Base Unit: Newton (N)
F = mass ! acceleration
Luminosity (L) – light energy emitted over time
Base Unit: Watts (W)
Energy
L=
time
Significant Figures
Measure the length of the double ended arrow.
cm
1
2
3
4
5
6
11
7
8
12
9
10
11
12
13
14
13
cm
Length of the arrow = 11.743576498 cm
Rule of significant figures states we are allowed only one (1) estimate.
Amounts to one decimal place after the smallest marked division.