Introduction and Mathematical Concepts

Introduction and Mathematical Concepts

Chapter 1

Objectives

• To express real numbers in scientific notation and solve using a calculator

• To determine the number of sig figs in a given number

• To solve equations using only variables

• Scientific notation: useful for small and large #’s

• Use a coefficient raised to the power of 10

• Coefficient must be between 1 and 10• Use a power of 10 depending on how

many times you moved the decimal place

Scientific Notation

• Example: –10,000,000 becomes 1 x107

–0.0000025 becomes 2.5 x10-6

• Greater than 1= positive power• Less than 1= negative power

Scientific Notation

• Express in scientific Notation

• .00000000001874

• 1.874 x 10-11

Scientific Notation Examples

• Express in scientific Notation

• 200,000,000,000

• 2 x 1011

Scientific Notation Examples

• Express in scientific Notation

• .00076321

• 7.6321 x 10-4

Scientific Notation Examples

• Express in scientific Notation

• 984.73

• 9.8473 x 102

• Explanation: Why is this example odd?

Scientific Notation Examples

• Express as real numbers

• 1.34 x 105

• 134,000

Scientific Notation Examples

• Express as real numbers

• 6.0 x 10-3

• .0060

Scientific Notation Examples

• Express as real numbers

• 5.223 x 108

• 522,300,000

Scientific Notation Examples

Objectives

• To express real numbers in scientific notation and solve using a calculator

• To determine the number of sig figs in a given number

• To solve equations using only variables

Scientific Notation and Calculators

• When using a calculator DO NOT type in x10 meaning DO NOT Push the buttons [x], [1] & [0].

• Depending on your calculator you will use the [EE], [Exp] or [x10n] button when a number is in scientific notation

• What is the answer to…

• 642 x (4.0 x 10-5)

• 2.6 x 10-2 or .02568

Scientific Notation

• What is the answer to…

• 1.7 x 105 / 3.88 x 107

• 4.4 x 10-3 or .0044

Scientific Notation

• What is the answer to…

• 2.9 x 10-5 x (8.1 x 102)

• 2.3 x 10-2 or .023

Scientific Notation

• What is the answer to…

• 6.02 x 1023 / (7.2 x 108)

• 8.4 x 1014

Scientific Notation

• What is the answer to…

• 5.40 x10-18 / 769

• 7.02 x 10-21

Scientific Notation

• What is the answer to…

• (1.0 x 107) x (1.0 x 10-6)

• 10

Scientific Notation

• Express as real numbers

• 1.59 x 10-2

• .0159

Scientific Notation Examples

• Put the following numbers in scientific notation:–4500000–0.00234–168000000000–0.00000000036

Scientific Notation Sample Problems

• Answers:–4.5 x106

–2.34 x10-3

–1.68 x1011

–3.6 x10-10

Scientific Notation Sample Problems

Objectives

• To express real numbers in scientific notation and solve using a calculator

• To determine the number of sig figs in a given number

• To solve equations using only variables

• Significant figures: (also called significant digits) of a number are those digits that carry meaning contributing to its precision. This includes all digits except:

• leading and trailing zeros which are merely placeholders to indicate the scale of the number.

• spurious digits introduced

Significant Figures

• which instrument gives us more sigfigs?• that’s the one you want to use (but it probably

costs a lot more!)

• 1) The #’s 1-9 are always significant• Examples:

–2467–4 sig. figs.–2.344678, –7 sig. figs.

Rules for Significant Figures

• 2) A zero between 2 significant figures is always significant

• Examples:–23045, –5 sig. figs.–450000001, –9 sig. figs. Because all zeros are

between significant figures

Rules for Significant Figures

• 3) Place holders are not significant, so a zero in a decimal before a significant figure is not counted

• Example:–0.023, –2 sig. figs. Zeroes come before a

significant figure–0.00000004, –1 sig. fig., all zeroes come before a

significant figure

Rules for Significant Figures

• 4) Zeroes after a significant figure and after a decimal are counted.

• Example:– 0.00230,– 3 sig. figs. The zero after the 3 is after a

significant number and a decimal so it is counted

– 23.34000, – 7 sig. figs.

Rules for Significant Figures

• 5) Zeroes after a significant figure when there is no decimal are place holders so they are not counted.

• Example:–1200–2 sig. figs. Zeros are place holders–145670, –5 sig. figs., last decimal is a place

holder

Rules for Significant Figures

• 6) If you count to have an exact quantity, you can use unlimited significant figures.

• Example:–You counted 18 sheep, you could write

the number as 18.0000000 and have 9 significant figures

–You can use as many significant figures as you want because you know you have exactly 18 sheep

Rules for Significant Figures

• How many significant figures do each of these have?–15.008–146000–0.00025760–2357–6.0 x 105

• Answers:–15.008 = 5–146000 = 3–0.00025760 = 5–2357 = 4 –6.0 x 105 = 2

Significant Figure Sample Problem

Objectives

• To express real numbers in scientific notation and solve using a calculator

• To determine the number of sig figs in a given number

• To solve equations using only variables

Solving Equations Using Variables

• 2 Rules: The variable you are solving for must…

1. Be by itself on one side of the equal sign2. In the numerator

• Look to worksheet for examples

Objectives

• We will be able to convert between different metric units given a conversion chart

• We will be able to convert between units using dimensional analysis

• We will be able to solve right triangle problems using sine, cosine, and tangent

1.2 Units

SI unitsLe Système International d’Unités(What the rest of the world uses

Length: meter (m)

Mass: kilogram (kg)

Time: second (s)

1.2 Units

The Standard Platinum-Iridium Meter Bar kept at 0°C

1.2 Units

The standard platinum-iridium kilogram

1.3 The Role of Units in Problem Solving

REMEMBER THIS!

The Great Mighty Kids Have Dropped Over

Dead Converting Metrics Many Nights

Past Friday

1.2.8. How many meters are there in 12.5 kilometers?

a) 1.25m

b) 125m

c) 1250m

d) 12 500m

e) 125 000m

1.3 The Role of Units in Problem Solving

Reasoning Strategy: Converting Between Units

1. In all calculations, write down the units explicitly.

2. Treat all units as algebraic quantities. When identical units are divided, they are eliminated algebraically.

Objectives

• We will be able to convert between different metric units given a conversion chart

• We will be able to convert between units using dimensional analysis

• We will be able to solve right triangle problems using sine, cosine, and tangent

1.3 The Role of Units in Problem Solving

Example 1 The World’s Highest Waterfall

• The highest waterfall in the world is Angel Falls in Venezuela, with a total drop of 979.0 m. Express this drop in feet.

• Since 3.281 feet = 1 meter, it follows that

feet 3212meter 1

feet 281.3meters 0.979 Length

1.3 The Role of Units in Problem Solving

Example 2 Interstate Speed Limit

Express the speed limit of 65 miles/hour in terms of meters/second.Use 5280 feet = 1 mile and 3600 seconds = 1 hour and 3.281 feet = 1 meter.

second

feet95s 3600

hour 1 mile

feet 5280hourmiles 6511

hourmiles 65 Speed

secondmeters29

feet 3.281meter 1

secondfeet951

secondfeet95 Speed

Objectives

• We will be able to convert between different metric units given a conversion chart

• We will be able to convert between units using dimensional analysis

• We will be able to solve right triangle problems using sine, cosine, and tangent

1.4 Trigonometry

The sides of a right triangle

1.4 Trigonometry

hhosin

hhacos

a

o

hh

tan

Sine, Cosine and Tangent

Abbrev. SOH CAH TOA

1.4.3. Referring to the triangle with sides labeled A, B, and C as shown, which of the following ratios is equal to the sine of the angle ?

a)

b)

c)

d)

e)

BA

CA

CB

AB

BC

1.4.4. Referring to the triangle with sides labeled A, B, and C as shown, which of the following ratios is equal to the tangent of the angle ?

a)

b)

c)

d)

e)

BA

CA

CB

AB

BC

1.4 Trigonometry

m2.6750tan oh

m0.80m2.6750tan oh

a

o

hh

tan

Trig Example: Solve for ho

1.4 Trigonometry

hho1sin

hha1cos

a

o

hh1tan

When solving for the angle, you must use the inverse trig function

1.4 Trigonometry

a

o

hh1tan 13.9

m0.14m25.2tan 1

Solve for the angle

1.4 Trigonometry

222ao hhh

Pythagorean theorem: When you know 2 sides of a right triangle and you are trying to solve for the 3rd

Objectives

• We will be able to identify vectors and scalars when given a quantity

• We will be able to solve problems that involve adding or subtracting vectors that are parallel, perpendicular and skewed

1.5 Scalars and Vectors

A scalar quantity is one that can be describedby a single number:

temperature, speed, mass

A vector quantity deals inherently with both magnitude and direction:

velocity, force, displacement

1.5.2. Which one of the following quantities is a vector quantity?

a) the age of the pyramids in Egypt

b) the mass of a watermelon

c) the sun's pull on the earth

d) the number of people on board an airplane

e) the temperature of molten lava

1.5.4. Which one of the following situations involves a vector quantity?

a) The mass of the Martian soil probe was 250 kg.

b) The overnight low temperature in Toronto was 4.0 C.

c) The volume of the soft drink can is 0.360 liters.

d) The velocity of the rocket was 325 m/s, due east.

e) The light took approximately 500 s to travel from the sun to the earth.

1.5 Scalars and Vectors

By convention, the length of a vectorarrow is proportional to the magnitudeof the vector.

Examples 8 lb4 lb

Arrows are used to represent vectors. Thedirection of the arrow gives the direction ofthe vector.

Objectives

• We will be able to identify vectors and scalars when given a quantity

• We will be able to solve problems that involve adding or subtracting vectors that are parallel, perpendicular and skewed

1.6 Vector Addition and Subtraction

Often it is necessary to add or subtract one vector with another.We will use the tail to head method

The combined vector is called the RESULTANT

1.6 Vector Addition and Subtraction

5 m 3 m

8 m

If the vectors are facing the SAME DIRECTION we can simply add their magnitudes

1.6 Vector Addition and Subtraction

Using the tail to head method, the resultant goes from the TAIL of the 1st vector to the HEAD of the 2nd vector

1.6.3. Consider the two vectors represented in the drawing. Which of the following options is the correct way to add graphically vectors

and ?a b

1.6 Vector Addition and Subtraction

2.00 m

6.00 m

If 2 vectors are PREPENDICULAR then we will use the PYTHAGOREAN THEOREM to solve for the 3rd side.

Solve for the 3rd side and the angle

1.6 Vector Addition and Subtraction

2.00 m

6.00 m

222 m 00.6m 00.2 R

R

m32.6m 00.6m 00.2 22 R

1.6 Vector Addition and Subtraction

2.00 m

6.00 m

6.32 m

00.600.2tan

4.1800.600.2tan 1

1.6 Vector Addition and Subtraction

When a vector is multiplied by -1, the magnitude of the vector remains the same, but the direction of the vector is reversed.

1.6 Vector Addition and Subtraction

A

B

BA

A

B

BA

1.6.4. Consider the two vectors represented in the drawing. Which of the following options is the correct way to subtract graphically vectors and ?a b

1.7 The Components of a Vector

. ofcomponent vector theandcomponent vector thecalled are and r

yx

yx

1.7 The Components of a Vector

.AAA

AA

A

yx

that soy vectoriall together add and

axes, and the toparallel are that and vectors

larperpendicu twoare of components vector The

yxyx

1.7 The Components of a Vector

Example

A displacement vector has a magnitude of 175 m and points atan angle of 50.0 degrees relative to the x axis. Find the x and ycomponents of this vector.

rysin

m 1340.50sinm 175sin ry

rxcos m 1120.50cosm 175cos rx

yxr ˆm 134ˆm 112

1.7.2. Vector has a magnitude of 88 km/h and is directed at 25 relative to the x axis. Which of the following choices indicates the horizontal and vertical components of vector ?

rx ry

a) +22 km/h +66 km/h

b) +39 km/h +79 km/h

c) +79 km/h +39 km/h

d) +66 km/h +22 km/h

e) +72 km/h +48 km/h

r

r

1.8 Addition of Vectors by Means of Components

BAC

yxA ˆˆ yx AA

yxB ˆˆ yx BB

Notice in this example the 2 vectors being added are not perpendicular. We have to add BY COMPONENTS

1.8 Addition of Vectors by Means of Components

yx

yxyxCˆˆ

ˆˆˆˆ

yyxx

yxyx

BABA

BBAA

xxx BAC yyy BAC

Cx Cy

a) 13 m/s 3 m/s

b) 57 m/s 33 m/s

c) 13 m/s 33 m/s

d) 57 m/s 3 m/s

e) 57 m/s 3 m/s

x

y x

y

1.8.1. Vector A has scalar components = 35 m/s and = 15 m/s. Vector B has scalar components = 22 m/s

and = 18 m/s. Determine the scalar components of vectorC = A B.

AA B

B