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Chapter 1 Introduction and Mathematical Concepts 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 • 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 Examples • 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 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 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 Examples • Express as real numbers • 1.59 x 10-2 • .0159 Scientific Notation Sample Problems • 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 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 • 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 • which instrument gives us more sigfigs? • that’s the one you want to use (but it probably costs a lot more!) Rules for Significant Figures • 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 Significant Figure Sample Problem • How many significant figures do each of • Answers: these have? – 15.008 – 146000 – 0.00025760 – 2357 – 6.0 x 105 – 15.008 = 5 – 146000 = 3 – 0.00025760 = 5 – 2357 = 4 – 6.0 x 105 = 2 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 sign 2. 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 units Le 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 Length 3 . 281 feet 979 . 0 meters 3212 feet 1 meter 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. miles Speed 65 1 1 hour miles 5280 feet 1 hour feet 65 95 hour mile 3600 s second feet Speed 95 1 second feet 1 meter meters 95 29 second 3.281 feet second 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 Sine, Cosine and Tangent sin ho h Abbrev. SOH CAH TOA cos ha h tan ho ha 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) A B b) A C c) B C d) B A e) C B 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) A B b) A C c) B C d) B A e) C B 1.4 Trigonometry Trig Example: Solve for ho tan ho ha tan 50 ho 67 . 2 m h o tan 50 67 . 2 m 80 . 0 m 1.4 Trigonometry When solving for the angle, you must use the inverse trig function sin 1 ho h cos 1 tan 1 ha h ho h a 1.4 Trigonometry Solve for the angle tan 1 ho h a tan 1 2 . 25 m 9 . 13 14 . 0 m 1.4 Trigonometry Pythagorean theorem: When you know 2 sides of a right triangle and you are trying to solve for the 3rd 2 2 2 h ho h a 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 described by 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 Arrows are used to represent vectors. The direction of the arrow gives the direction of the vector. By convention, the length of a vector arrow is proportional to the magnitude of the vector. Examples 4 lb 8 lb 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 If the vectors are facing the SAME DIRECTION we can simply add their magnitudes 3m 5m 8m 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 a and b ? 1.6 Vector Addition and Subtraction 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 2.00 m 6.00 m 1.6 Vector Addition and Subtraction R 2 . 00 m 6 . 00 m 2 2 R 2 . 00 2 m 6 . 00 m 6 . 32 m 2 2 R 2.00 m 6.00 m 1.6 Vector Addition and Subtraction tan 2 . 00 6 . 00 tan 1 2 .00 6 . 00 18 . 4 6.32 m 2.00 m 6.00 m 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 B AB A A B AB 1.6.4. Consider the two vectors represented in the drawing. Which of the following options is the correct way to subtract graphically vectors a and b ? 1.7 The Components of a Vector x and y are called the x vector component and the y vector component of r . 1.7 The Components of a Vector of A are two perpendicu The vector components lar vectors A x and A y that are parallel to the x and y axes, and add together vectoriall y so that A A x A y . 1.7 The Components of a Vector Example A displacement vector has a magnitude of 175 m and points at an angle of 50.0 degrees relative to the x axis. Find the x and y components of this vector. sin y r y r sin 175 m sin 50 . 0 134 m 112 m cos x r x r cos 175 m cos 50 . 0 r 112 m xˆ 134 m yˆ 1.7.2. Vector r 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 r ? a) rx +22 km/h ry +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 1.8 Addition of Vectors by Means of Components Notice in this example the 2 vectors being added are not perpendicular. We have to add BY COMPONENTS C AB A A x xˆ A y yˆ B B x xˆ B y yˆ 1.8 Addition of Vectors by Means of Components C A x xˆ A y yˆ B x xˆ B y yˆ A x B x xˆ A y B y yˆ C x Ax B x C y Ay B y 1.8.1. V ector A has scalar com ponents Ax = 35 m /s and Ay = 15 m /s. V ector B has scalar com ponen ts B x = 22 m /s and B y = 18 m /s. D eterm ine the scalar com ponents of vector C = A B. a) Cx 13 m/s Cy 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