W A T K I N S - J O H N S O N C O M P A N Y Semiconductor Equipment Group

1
Chabot Mathematics
1.1 Expressions Real No.s
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
2
Basic Terminology

• A LETTER that can be any one of various numbers
  is called a VARIABLE.
• If a LETTER always represents a particular number
  that NEVER CHANGES, it is called a CONSTANT

A B are CONSTANTS
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Algebraic Expressions

• An ALGEBRAIC EXPRESSION consists of variables,
  numbers, and Math-Operation signs.
• Some Examples

• When an equal sign is placed between two
  expressions, an EQUATION is formed ?

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Translate English ? Algebra

• Word Problems must be stated in ALGEBRAIC form
  using Key Words

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Example ? Translation

• Translate this Expression
• Eight more than twice the product of 5 and an
  Unknown number
• SOLUTION
• LET n the UNknown Number

Eight more than twice the product of 5 and a
number.
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Evaluate Algebraic Expressions

• When we REPLACE A VARIABLE with a number, we are
  SUBSTITUTING for the variable.
• The calculation that follows is called EVALUATING
  the expression
• Note the normal result for a Evaluation is
  usually a SINGLE NUMBER with NO LETTERS

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Example ? Evaluating

• Evaluate

• SOLUTION

8xz - y 823 - 7
Substituting
48 - 7
Multiplying
41
Subtracting
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Exponential Notation
The expression an, in which n is a counting
number (1, 2, 3, etc.) means n factors In
an, a is called the base and n is called the
exponent, or power. When no exponent appears, it
is assumed to be 1. Thus a1 a.
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Order of Operations (PEMDAS)

• Perform operations in this order
• Grouping symbols parentheses ( ), brackets ,
  braces , absolute value , and radicals v
• Exponents from left to right, in order as they
  occur.
• Multiplication/Division from left to right, in
  order as they occur.
• Addition/Subtraction from left to right, in order
  as they occur.

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Example ? Order of Ops

• Evaluate

• SOLUTION

2(x 3)2 12 x2
2(2 3)2 12 22
Substituting
Working within parentheses
Simplifying 52 and 22
Multiplying and Dividing
Subtracting
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Formulas

• A FORMULA is an equation that uses letters to
  represent a RELATIONSHIP between two or more
  quantities.
• Example ? The area, A, of a circle of radius r is
  given by the formula

r
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Example ? Temp Conversion

• The formula for converting the temperature in
  degrees Celsius (C) to degrees Fahrenheit (F)

• Use the formula to convert 86F to the Celsius
  form

Substitute 86 for F
Solve for C
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Temperature Conversion cont.
Solve for C.

• Thus 30 C is the Equivalent of 86 F

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Mathematical Modeling

• A mathematical model can be a formula, or set
  of formulas, developed to represent a
  real-world situation.

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Example ? Math Model

• Mei-Li is 5ft 7in tall with a Body Mass Index
  (BMI) of approximately 23.5. What is her weight?
• SOLUTION
• Familiarize. The body mass index I depends on a
  persons height and weight. The BMI formula

• where W is the weight in pounds and H is the
  height in inches.

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BMI Example cont

1. Carry Out

1. Translate. Solve the formula for W

5 ft 7 in. 67 in.
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BMI Example cont

1. State Answer

1. Check

• Mei-Li weighs about 150 Pounds

?
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Set of Real Numbers

• SET a Collection of Objects
• Braces are used to indicate a set. For example,
  the set containing the numbers 1, 2, 3, and 4 can
  be written 1, 2, 3, 4.
• The numbers 1, 2, 3, and 4 are called the members
  or elements of this set.

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Set Notation
Roster notation 2, 4, 6, 8 Set-builder
notation
x x is an even number between 1
and 9
The set of all x
such that
x is an even number between 1 and 9
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Sets of Real Numbers

• Natural Numbers (Counting Numbers)
• Numbers used for counting 1, 2, 3,
• Whole Numbers
• The set of natural numbers with 0 included 0,
  1, 2, 3,
• Integers
• The set of all whole numbers AND their opposites
  ,-3, -2, -1, 0, 1, 2, 3,

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Sets of Real Numbers, x

• Rational Numbers (Integer Fractions)
• Maybe expressed as a FRACTION of two INTEGERS
• Terminating Decimals (e.g. 7/16)
• Repeating NonTerminating Decimals (e.g., 2/7)
• Irrational Numbers
• Can NOT be expressed as a Fraction of two
  Integers
• NONterminating, NONreapeating decimals(e.g., p
  3.1459)

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• Real No. Family Tree

Irrational numbers
Integers
Zero 0
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Real Number Nest
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Real Number Line

• An Infinite line whose points have been assigned
  number-coordinates is called the real number line

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3 Sectors of the Number Line

1. The negative real numbers are the CoOrds to the
   left of the origin O
2. The real number zero is the CoOrd of the origin O
3. The positive real numbers are the CoOrds to the
   right of the origin O

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InEqualities

• lt means is less than
• means is less than or equal to
• gt means is greater than
• means is greater than or equal to
• For any two numbers on a number line, the one
  to the left is said to be less than the one to
  the right.

1 lt 3 since 1 is to the left of 3
1 0 1 2 3
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Chabot Mathematics
1.2 Operations with Real No.s
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
28
Absolute Value

• The ABSOLUTE VALUE of a number is its distance
  from zero on a number line.
• The symbol x to represents the absolute value
  of a number x.
• Example ? 5 -5 5

?5 units from 0??
?5 units from 0??

• Distance is ALWAYS Positive

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Find Absolute Value

• Finding Absolute Value
• If a number is negative, make it positive
• If a number is positive or zero, leave it alone
• Example ? Find the absolute value of each number.
• a) -4.5 b) 0 c) -3
• Solution
• a) -4.5 The dist of -4.5 from 0 is 4.5, so
  -4.5 4.5
• b) 0 The distance of 0 from 0 is 0, so 0
  0.
• c) -3 The distance of 3 from 0 is 3, so
  -3 3

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Real No. Addition Rules

• Positive numbers Add the numbers. The result
  is positive.
• Negative numbers Add absolute values. Make the
  answer negative.
• A positive and a negative number Subtract the
  smaller absolute value from the larger. Then
• If the positive number has the greater absolute
  value, make the answer positive.
• If the negative number has the greater absolute
  value, make the answer negative.
• If the numbers have the same absolute value, the
  answer is 0.
• One number is zero The sum is the other number

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Inverse Property of Addition

• For any real number a, the opposite, or additive
  inverse, of a, (which is -a) is such that
• a (-a) -a a 0
• Example ? Find the opposite, or additive inverse
  a) 8 b) -13
• Solution ?
• a) 8 -(8) -8
• b) -13 -(-13) 13

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Subtraction

• Subtraction The difference a - b is the unique
  number c for which a b c.
• That is, a - b c if c is a number such that a
  b c
• Subtracting by Adding the Opposite
• For any real numbers a and b
• a - b a (-b)
• We can subtract by adding the opposite (additive
  inverse) of the number being subtracted (the
  Subtrahend)

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Real No. Multiplication

• The Product of a Positive and a Negative Number
• To multiply a positive number and a negative
  number, multiply their absolute values. The
  answer is negative 3(-2) -6
• The Product of Two Negative Numbers
• To multiply two negative numbers, multiply their
  absolute values. The answer is positive
  (-13)(-11) 143

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Real No. Division

• The quotient a ? b or a/b where b ?? 0, is that
  unique real number c for which a b c.
• Example ? Divide

• SOLUTION

Because -5 (-9 ) 45
Because -8 (8) -64
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Multiply Divide Rules

• To multiply or divide two real numbers
• Multiply or divide the absolute values.
• If the signs are the same, then the answer is
  positive.
• If the signs are different, then the answer is
  negative.
• DIVISION by ZERO
• NEVER divide by ZERO. If asked to divide a
  nonzero number by zero, we say that the answer is
  UNDEFINED. If asked to divide 0 by 0, we say
  that the answer is INDETERMINATE.

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Real Number Properties

• COMMUTATIVE property of addition and
  multiplication
• a b b a and ab ba
• ASSOCIATIVE property
• a (b c) (a b) c and a(bc) (ab)c
• DISTRIBUTIVE property
• a(b c) ab ac

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Real Number Properties

• ADDITIVE IDENTITY property
• a 0 0 a a
• ADDITIVE INVERSE property
• -a a a (-a) 0
• MULTIPLICATIVE IDENTITY property
• a 1 1 a a
• MULTIPLICATIVE INVERSE property
• a (1/a) (1/a) a 1 (a ? 0)

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Simplify Expressions

• A TERM is a number, a variable, a product of
  numbers and/or variables, or a product or
  quotient of two numbers and/or variables.
• Terms are SEPARATED by ADDITION signs. If there
  are SUBTRACTION signs, we can find an equivalent
  expression that uses addition signs.
• COLLECTING LIKE TERMS is based on the
  DISTRIBUTIVE law

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Like (or Similar) Terms

• Terms in which the variable factors are exactly
  the same, such as 9x and -5x, are called like, or
  similar terms.
• Like Terms UNlike Terms
• 7x and 8x 8y and 9y2
• 3xy and 9xy 5ab and 4ab2

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Example ? Combine Terms

• a) 7x 3x b) 4a 5b 2 a - 6 - 5b
• SOLUTION
• 7x 3x (7 3)x 10x
• 4a 5b 2 a - 6 - 5b
• 4a 5b 2 a (-6) (-5b)
• 4a a 5b (-5b) 2 (-6)
• 4a a 5b (-5b) 2 (-6)
• 5a 0 (-4)
• 5a - 4

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Example ? Simplify

• Remove parentheses and simplify
• 6(m 3) 5m 4(n 5) 8(n 4)
• SOLUTION
• 6(m 3) 5m 4(n 5) 8(n 4)
• 6m 18 5m 4n 20 8n 32 Distribute
  
• m 18 4n 52 Collect like terms
  within brackets
• m 18 4n 52 Removing brackets
• m 4n 34 Collecting like terms

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TERMS ? factors

• Factors are the pieces of a Multiplication
  Chain e.g., if
• Then y has four factors 7, u, v, w
• TERMS are the pieces of an ADDITION Chain ?
• Then z has Three TERMS 7a, 3b, -5c

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Chabot Mathematics
1.3 GraphingEquations
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
44
Points and Ordered-Pairs

• To graph, or plot, points we use two
  perpendicular number lines called axes. The point
  at which the axes cross is called the origin.
  Arrows on the axes indicate the positive
  directions
• Consider the pair (2, 3). The numbers in such a
  pair are called the coordinates. The first
  coordinate, x, in this case is 2 and the second,
  y, coordinate is 3.

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Plot-Pt using Ordered Pair

• To plot the point (2, 3) we start at the origin,
  move horizontally to the 2, move up vertically 3
  units, and then make a dot
• x 2
• y 3

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Example ? Plot the point (-4,3)

• Starting at the origin, we move 4 units in the
  negative horizontal direction. The second number,
  3, is positive, so we move 3 units in the
  positive vertical direction (up)
• x -4 y 3

4 units left
3 units up
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Example ? Read XY-Plot

• Find the coordinates of pts A, B, C, D, E, F, G

• Solution Point A is 5 units to the right of the
  origin and 3 units above the origin. Its
  coordinates are (5, 3). The other coordinates are
  as follows
• B (2, 4)
• C (3, 4)
• D (3, 2)
• E (2, 3)
• F (3, 0)
• G (0, 2)

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XY Quadrants

• The horizontal and vertical axes divide the
  plotting plane into four regions, or quadrants

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Graphing Equations

• Definitions
• An ordered pair (a, b) is said to satisfy an
  equation with variables a and b if, when a is
  substituted for x and b is substituted for y in
  the equation, the resulting statement is true.
• An ordered pair that satisfies an equation is
  called a solution of the equation
• Frequently, the numerical values of the variable
  y can be determined by assigning appropriate
  values to the variable x. For this reason, y is
  sometimes referred to as the dependent variable
  and x as the independent variable.

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Eqn Graph ? Bottom Line

• ANY and ALL points (ordered pairs) on a Math
  Graph are SOLUTIONS to the Equation that
  generated the Graph

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Graph of an Equation

• The graph of an equation in two variables, such
  as x and y, is the set of all ordered pairs (a,
  b) in the coordinate plane that satisfy the
  equation ?

y 2x 6 -2x y 62x - y 6 0
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Graphing by Plotting Points

• Graph y x2 3
• SOLUTION ? Make T-table

Ordered Pair
X col
Calc y
(x, y)
y x2 3
x
(3, 6)
y (3)2 3 9 3 6
3
Pick x
(3, 6) is a solutionto y x2 3 6 32 3
?
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Graph by Pt-Plot y x2 3

• Pick enough xs for T-table

(x, y)
y x2 3
x
(3, 6)
y (3)2 3 9 3 6
3
(2, 1)
y (2)2 3 4 3 1
2
(1, 2)
y (1)2 3 1 3 2
1
(0, 3)
y 02 3 0 3 3
0
(1, 2)
y 12 3 1 3 2
1
(2, 1)
y 22 3 4 3 1
2
(3, 6)
y 32 3 9 3 6
3
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Graph by Pt-Plot y x2 3

• Plot (x,y) Points listed in T-table and connect
  the dots to complete the plot
• Note that most Graphs are curves so connect
  dots with curved lines

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Graph by Pt-Plot y x2 2x 6

• Construct T-table

• Plot-Pts Connect-Dots

x y (x, y)
?3 9 (?3, 9)
?2 2 (?2, 2)
0 ?6 (0, ?6)
1 ?7 (1, ?7)
2 ?6 (2, ?6)
3 ?3 (3, ?3)
4 2 (4, 2)
5 9 (5, 9)
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GRAPH BY PLOTTING POINTS

• Step1. Make a representative T-table of
  solutions of the equation.
• Step 2. Plot the solutions as ordered pairs in
  the Cartesian coordinate plane.
• Step 3. Connect the solutions in Step 2 by a
  smooth curve

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Domain Range by Graphing

• Graph y x2. Then State the Domain Range of
  the equation
• Select integers for x, starting with -2 and
  ending with 2. The T-table

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Example ? Domain Range

• Now Plot the Five Points and connect them with a
  smooth Curve

(-2,4)
(2,4)
(-1,1)
(1,1)
(0,0)
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Example ? Domain Range

• The DOMAIN of a function is the set of ALL first
  (or x) components of the Ordered Pairs that
  appear on the Graph
• Projecting on the x-axis ALL the x-components of
  ALL POSSIBLE ordered pairs displays the DOMAIN of
  the function just plotted

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Example ? Domain Range

• Domain of y x2 Graphically

• This Projection Pattern Reveals a Domain of

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Example ? Domain Range

• The RANGE of a function is the set of all second
  (or y) components of the ordered pairs. The
  projection of the graph onto the y-axis shows
  the range

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Example ? Domain Range

• The projection of the graph onto the y-axis is
  the interval of the y-axis at the origin or
  higher, so the range is

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Chabot Mathematics
1.4 SolveLinear Eqns
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
64
Solution For an Equation

• Any NUMBER-REPLACEMENT for the VARIABLE that
  makes an equation true is called a SOLUTION of
  the equation.
• To Solve an equation means to find ALL of its
  Solutions.

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Equivalence and Addition Principle

• Equivalent Equations
• Equations with the SAME SOLUTIONS are called
  EQUIVALENT equations
• Addition Principle
• For any real numbers u, v, and w, u v is
  equivalent to u w v w
• e.g. u 3, v 3, w 7 ?3 3 and 37 37

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Multiplication Principle

• The Multiplication Principle For any real
  numbers r, s, and t with t ? 0, r s is
  equivalent to rt st

• Example
• Solve for x

Multiply Both Sides by 4/3
?

• Solution

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Linear Equations

• A linear equation in one variable, such as x, is
  an equation that can be written in the standard
  form

• where a and b are real numbers with a ? 0

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Solution to Linear Equations

• Procedure for solving linear equations in one
  variable
• Eliminate Fractions if Needed, Clear Fractions
  by Multiplying both sides of the equation by the
  least common denominator (LCD) of all the
  fractions
• Simplify Simplify both sides of the equation by
  removing parentheses and other grouping symbols
  (if any) and combining like terms

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Solution to Linear Equations

1. Isolate the Variable Term Add appropriate
   expressions to both sides, so that when both
   sides are simplified, the terms containing the
   VARIABLE are on ONE SIDE and all constant terms
   are on the other side.
2. Combine Terms Combine terms containing the
   variable to obtain one term that contains the
   variable as a factor.

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Solution to Linear Equations

1. Isolate the Variable Divide both sides by the
   coefficient of the variable to obtain the
   solution.
2. Check the Solution Substitute the solution into
   the original equation

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Example ? Solve Linear Eqn

• Solve for x

• SOLUTION (No Fractions to clear)

Step 2
72
Example ? Solve Linear Eqn

• Solve for x

• SOLUTION

Step 3
Step 4
Step 5
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Example ? Solve Linear Eqn

• Solve for x

• SOLUTION

Step 6 Check x 3
?

• State ? The Solution is x 3

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Chabot Mathematics
1.5 Problem Solving
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
75
Mathematical Model

• A mathematical model is an equation or inequality
  that describes a real situation.
• Models for many APPLIED (or Word) problems
  already exist and are called FORMULAS
• A FORMULA is a mathematical equation in which
  variables are used to describe a relationship

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Formula Describes Relationship
Relationship
Mathematical Formula
Perimeter of a triangle
Area of a triangle
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Example ? Geometry of Cone
Relationship
Mathematical Formulae
Volume of a cone
Surface area of a cone
(slant-sides bottom)
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Example ? F ? C
Relationship
Mathematical Formulae
Celsius to Fahrenheit
Fahrenheit to Celsius
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Example ? Mixtures
Relationship
Mathematical Formula
Percent Acid, P
80
Solving a Formula

• Sometimes the formula is solved for a Different
  variable than the one we need
• Example ? A mathematical model tell us that
  voltage, V, in a circuit is equal to current, I,
  times resistance, R V I R
• To determine the amount of resistance in a
  circuit, it would help to first solve the
  formula for R.

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Example ? Solve V IR for R

• We solve this formula for R by
• treating V and I as CONSTANTS (having fixed
  values)
• treating R as the only variable.
• Begin by writing the formula so that the variable
  for which we are solving, R, is on the left side.

I R V
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Example ? Solve V IR for R

• Finally, use Algebra properties to isolate the
  variable R.

Divide both sides by I.
Isolated R ? Done
83
Example ? Trapezoid Base, B
b
This formula gives the relationship between the
height, h, and two bases, B and b, of a
trapezoid and its area, A.
h
B
84
Example ? Trapezoid Base, B
Mult. Prop. of Equality.
Assoc. Prop.
Inverse Prop.
Identity Prop.
85
Example ? Trapezoid Base, B
Distributive Prop.
Add. Prop. Of Equality
Inverse Prop.
Divide by h.
86
Example ? Prismatic Volume
The volume of a triangular prism is given by
If the volume of a triangular cylinder is 880
cm3, the base is 10 cm, and the length is 22
cm, then find the height.
h
l
b
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Example ? Prismatic Volume
for h.
First, solve the equation
88
Example ? Prismatic Volume

• Second, find the height, h, by substituting the
  given values of V, b, and l into this formula

• State ? The height of the triangular prism is 8
  cm

89
Example ? from a Pie Chart

• The pie chart shown at right represents the
  distribution of grades in MTH2 (Calculus) last
  year. Use the information in the chart to
  estimate how many Bs will be given in a new
  class of size 70 students.

A 16
F 8
D 12
B 28
C 36
90
Example ? from a Pie Chart

• According to the chart, 28 OF the students
  should get a grade of B. Let x represent the
  number of students getting a B.

91
WhiteBoard Work

• Problems From 1.n Exercise Sets
• NONE Today ? Lecture PPT took Entire Class Time

• Identity Symbol

• Difference of Two Squares Identity

92
All Done for Today
TheDefined asSymbol
93
Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu

94
Set Membership Notation
95
Tool For XY Graphing

• Called Engineering Computation Pad
• Light Green Backgound
• Tremendous Help with Graphing and Sketching
• Available in Chabot College Book Store
• I use it for ALL my Hand-Work

Graph on this side!