Provide Computational Solutions to Power Engineering Problems SAMPLE. Learner Guide. Version 3

Size: px

Start display at page:

Download "Provide Computational Solutions to Power Engineering Problems SAMPLE. Learner Guide. Version 3"

Ezra Maxwell
5 years ago
Views:

1 Provide Computational Solutions to Power Engineering Problems Learner Guide Version 3 Training and Education Support Industry Skills Unit Meadowbank Product Code: 5793

2 Acknowledgments The TAFE NSW Training and Education Support Industry Skills Unit, Meadowbank would like to acknowledge the support and assistance of the following people in the production of this learner resource guide: Project Managers: Steve Parkinson Kerry Barlow TAFE NSW Reviewer: David Cassidy Enquiries Enquiries about this and other publications can be made to: Training and Education Support Industry Skills Unit, Meadowbank Meadowbank TAFE Level 3, Building J, See Street, MEADOWBANK NSW 2114 Tel: Fax: The State of New South Wales, Department of Education and Training, TAFE NSW, Training and Education Support Industry Skills Unit, Meadowbank, Copyright of this material is reserved to TAFE NSW Training and Education Support Industry Skills Unit, Meadowbank. Reproduction or transmittal in whole or in part, other than for the purposes of private study or research, and subject to the provisions of the Copyright Act, is prohibited without the written authority of TAFE NSW, Training and Education Support Industry Skills Unit, Meadowbank. ISBN TAFE NSW (Training & Education Support, Industry Skills Unit Meadowbank) 2013

3 Table of Contents Introduction Using This Learner Guide Prior Knowledge and Experience Assessment... 7 Section1. Arithmetic... 9 Section 2. Algebra Section 3. Approximations, Errors, Significant figures... 9 Section 4. Geometry and Trigonometry Section 5. Trigonometric functions Section 6. Coordinate Geometry Section 7. Simultaneous Equations Section 8. Matrices Section 9. Quadratics Section. Exponential and Logarithmic Functions Section 11. Vectors Section 12. Complex Numbers Assessment Samples TAFE NSW (Training & Education Support, Industry Skills Unit Meadowbank) 2013

4 Section 1 - Arithmetic TABLE OF CONTENTS 1. ARITHMETIC Rational and Irrational numbers Surds Manipulating Surds Indices Base : Scientific and Engineering Notation Scientific and Engineering Notation Scientific Notation Engineering Notation S.I. Units Harder Conversions Areas and Volume Related Rates Liquid Measure Review Answers TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013 Page 9 of 670

5 1. ARITHMETIC At the completion of this section you should be able to: Evaluate expressions involving square roots and cube roots Convert Units of physical quantities expressed in SI units Show awareness of errors in measurement and of giving results in an appropriate number of significant digits. Use estimation and approximations to check the accuracy of results Convert values between decimal notation, scientific notation and engineering notation Use the laws of indices Factorise algebraic expressions using common factors TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013 Page 11 of 670

6 1.1 Rational and Irrational numbers Any number that can be expressed in the form b a, where a and b are integers and b , is said to be rational. For example, the numbers,, are rational Any number that cannot expressed in the form b a,is said to be irrational. Examples of irrational numbers include π (the ratio of the diameter of a circle to its circumference), 1 3, 2, Surds Note that 4 = 2 is a rational number since 2 can be written as 1 2. Consider the term 3 ( = _ ), which cannot be expressed in the form b a. This term is irrational and is called a surd. Any number written with a radical ( ) sign, that is also irrational, is called a surd. Other examples include 5, 3 7, 1 + 2, 7-5. Example 1-1: Determine whether the following numbers are rational or irrational. a) 0.9 = 9 rational b) 16 = 4 rational c) 5 = irrational Page 12 of 670 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013

7 d) 3 1 = irrational e) 3 27 = 3 rational f) 3 8 = 2 rational 1.3 Manipulating Surds There are some basic rules that must be followed when working with surds: ab = a x b a b = b Example 1-2: Simplify leaving your answer as a surd (where applicable) (a) a 8 = 4 2 = 4 2 = 2 2 (b) 2 27 = = = = 6 3 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013 Page 13 of 670

8 (c) 4 9 = 4 9 = 2 3 (d) (e) (f) 18 2 = = = = = = 8 = 3 3 = = = = 6 = 2 Page 14 of 670 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013

9 Activity 1 1. Use a calculator to evaluate the following (to two decimal places) a) b) c) x d) 3.14 x e) f) Simplify the following, leaving your answer as a surd where applicable a) 2 3 b) c) d) 2 e) f) Simplify, leaving your answer as a surd where applicable a) 32 b) c) 72 2 d) x 40 8 e) f) 2 6 x TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013 Page 15 of 670

10 1.4 Indices Base : Scientific and Engineering Notation Indices: Base Numbers such as, 0, 00 or 1, 0.1, 0.01 etc, can be expressed in index form by noting the following relationship: 1 = 1 followed by one zero. 2 = 0 1 followed by two zeros. 3 = 00 1 followed by three zeros. n = followed by n zeros. We could state further that: 0 = 1 1 followed by no zeros. 1 = preceded by one zero. 2 = preceded by two zeros. 3 = preceded by three zeros. When multiplying or dividing numbers expressed with indices some basic rules apply: Rule 1 a b = a+b Rule 2 a b = a b Page 16 of 670 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013

11 Example 1-3: Simplify the following using the rules above: (a) 2 3 = 2+3 = 5 = 0,000 (d) 2 2 = 2 2 = 0 = 1 (b) (c) 2 3 = 2 3 = 1 = = 2 4 = 2 = 1 0 = (e) (f) 4 4 = = = = = 00 = = = TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013 Page 17 of 670

12 1.5 Scientific and Engineering Notation Scientific Notation Scientific notation (sometimes called standard form) requires numbers to be expressed n in the form a where 1 a <. (a must be bigger than or equal to 1, but less than ). Example 1-4: Express the following in scientific notation. a) c) = 4.46 b) 8900 = = d) 38.6 = = = 7.2 The decimal point is placed, generally between the first two digits to create a number between 1 and, and adjustments made to the index of. Compare the index with the number of places the decimal point has moved. Example 1-5: Express the following in scientific notation. (a) = 4.6 (b) = (c) = (d) = In the examples: Large numbers have positive indices. Small numbers have negative indices. Page 18 of 670 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013

13 1.5.2 Engineering Notation n Engineering notation also requires numbers to be written in the form a, but where 1 a < 00 and the index of is a multiple of three (,,, etc). Example 1-6: Write the following in engineering notation. (a) = 446 (b) = (c) (e) (g) = 12.5 (d) 3 = (f) = (h) = = = 721 Activity 2 1. Use the index laws to simplify, leaving answers in index form. (a) 6 4 (b) 5 (c) (d) (e) (f) 2 3 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013 Page 19 of 670

14 Section 2 Algebra TABLE OF CONTENTS 2 ALGEBRA Basics of Algebra Axioms of Equality Algebraic Representation Factors Fundamental Operations in Algebra Rules of Algebra Associative Law Distributive Law Working with Algebraic Expressions Parentheses Associating Words and Mathematical Symbols Order of Operations Terms of an Expression Factors of a Term Expressions Exponents The Base and Power of Exponents Simplifying Expressions Self Test Answers to Self Test Answers to Exercises Solution of Simple Equations Some Basic Facts About Equations Root of an Equation Equation Types Identities False Statements TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013 Page 31 of 670

15 Provide Computational solutions to Power Engineering Problems Conditional Equation Roots of Equations Equivalent Equations Axioms Inverse Operations Equations with Literal Terms on Both Sides Removing Parentheses in Equations Fractional Equations Handling Binomial Denominators Polynomials Working with Formulas Self Test Answers to Self Test Answers to Exercises... 1 Page 32 of 670 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013

16 2 ALGEBRA Objectives After completion of the material in this section you will be able to Describe the general format and representation of an algebraic equation; Define literal and numerical factors; Describe and apply the fundamental rules of algebra which include: the commutative law, the associative law and the distributive law; Express a simple algebraic expression in plain words and also write an algebraic expression from a written statement; Know and apply the order of mathematical operations; Recognize and arrange the factors and terms of an algebraic expression and understand the concept and application of key words such as coefficient, exponent and base. About this Section This section is primarily intended to serve as an introduction to algebra and to show you how it relates to what you already know about arithmetic. Mature aged students,, particularly those who studied algebra some time ago will benefit from doing a review before continuing with the more complex mathematics encountered later in this module. To use the learning material to its greatest benefit, you must work through enough exercises to assure mastery of the concepts. You can measure your mastery by taking the self test at the end of this section. Most students will find the earlier material quite familiar however the focus quickly changes to applied concepts and problem solving techniques that are used in the electrotechnology areas. When you first start this section, or any other section, scan through the pages reading the headings and main points. If it seems familiar to you, do a few exercises in each topic and at least one or two problems from each question of the self test. If you feel you can do this easily, move on to the next section. 2.1 Basics of Algebra There is nothing mysterious about algebra. It is simply another handy mathematical tool-like arithmetic. The beauty of algebra is that it will provide you with some quick and easy ways to solve problems that may have stumped you before. It will also provide you with a new insight into the world of numbers. Fifty or one hundred years ago studying algebra was not as important as it is now. Today, a knowledge of algebra is essential in such diverse fields as engineering, economics, insurance, architecture, statistics, and all of the physical sciences. TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013 Page 33 of 670

17 Provide Computational solutions to Power Engineering Problems Although algebra is really not a difficult subject, it does require that you be willing to learn the meaning of some new words. One of the most important aspects of algebra is the language it uses. To learn its language you must be willing to listen carefully when we are discussing any points that may be new to you. You will quickly discover that algebra is really just a logical extension of arithmetic. Most of the problems you can solve with algebra are based on operations you learned to perform in arithmetic: addition (x), subtraction (-), multiplication (x), and division ( ). The symbol shown after each wording is the one commonly used in arithmetic to indicate (in mathematical language) the operation to be performed. Exercise 2-1: Test your memory. Write the name used to indicate the results of each of these arithmetical operations. The result of addition is called The result of subtraction is called The result of multiplication is called The result of division is called When you have attempted every question, turn to the back of this section to check your answers. If you missed any of these, review them again before you go on. These same operations are performed in algebra-with one major difference. In algebra letters are frequently used to represent numbers. Why? Simply because we don t always know the numerical value of certain quantities or terms at the outset of a problem. Since we wish to identify the quantity in some way, we use a letter of the alphabet to represent it until its value can be determined. In fact, finding the values of unknown quantities is one of the things we must do frequently in algebra. Exercise 2-2: Identify the statement below which you think best represents the main point we have been discussing above. Arithmetic uses letters in place of numbers. In algebra, letters represent numbers we know beforehand. Algebra differs from arithmetic in its frequent use of letters to represent numbers Axioms of Equality It is this use of letters to represent unknown numbers that makes it possible in algebra to translate long word statements into brief mathematical expressions. We are going to look into the matter of how to do this very shortly. But before we do so it is important that we review certain axioms of equality, because these are fundamental to our work from this point on. Page 34 of 670 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013

18 Let s start with the wording axiom itself, a word you may be familiar with from arithmetic (or geometry, if you have studied it). If you can t remember, the definition of an axiom is a basic assumption accepted as true without proof. Now we you re going to pick up the pace. Don t worry if you don t instantly grasp every concept. And remember, if something isn t clear the first time around, it will pay to spend another few minutes rereading the entire frame. In both geometry and arithmetic we make use of the following axioms. (a) If equals are added to equals, the sums are equal. Example: 4 = 6 2 ; therefore, adding 2 to each side yields: = (6 2) + 2 (b) If equals are subtracted from equals, the differences are equal. Example: 6 = 4 + 2; therefore, subtracting 2 from each side gives: 6 2 = (4 + 2) 2 (c) If equals are multiplied by equals, the products are equal. Example: Consider the fraction: 14 = 7. Therefore, multiplying both sides by 2 results in the 2 following: = 2 7 (d) If equals are divided by equals, the quotients are equal (provided the denominator is not zero). Example: 7 2 = 14; therefore, dividing both sides by 2 gives, (7 2) 2 = 14 2 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013 Page 35 of 670

19 Provide Computational solutions to Power Engineering Problems You should be familiar with these axioms already. You must apply them strictly in manipulating algebraic expressions. Many of the difficulties and mistakes of math students are due to their failure to recognize and apply one or more of these axioms. We will be referring to them from time to time and checking up to make sure you haven t forgotten them Algebraic Representation How does the use of letters, numbers, and mathematical symbols make possible the translation of long word statements into short mathematical sentences or expressions? Let s see. Example: The sum of five times a numbered and two times the same number is equal to seven times the number. How can we represent this situation most simply? Solution: If let n represent the number we are talking about, we can say that same thing with this short algebraic sentence: 5n + 2n = 7n Example: Three times a number subtracted from eight times of the same number equals five times the number. Solution: 8n 3n = 5n Equations An algebraic equation like 8n 3n = 5n is called an equation because it represents two things that are equal to one another. In this example the two quantities that are equal are (8n 3n) and 5n. In this equation we have used an algebraic convention that we should discuss for a moment. In arithmetic we indicate multiplication by use of the times sign (x). In algebra there are other ways of expressing the idea of multiplication. Suppose, as in the previous answer, we wish to express the idea of eight times a number. We can do this in any of the following ways: 8 n or 8 n or 8(n)or 8n Both the dot and the parentheses are acceptable, but omission of the multiplication sign, as in 8n above, is preferred. The times sign is seldom used in algebra since it could be mistaken for the letter x. For two numbers, parentheses are often preferred over a raised dot, which might be confused with a decimal or a full stop, even though the decimal and full stop are identified with a dot placed down low. Page 36 of 670 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013

20 Exercise 2-3: Rewrite the following algebraic representations without using the times sign. (a) 6 w (b) r s t (c) 5 a b (d) 4 a Factors We can use any symbol to represent a number. Traditionally, we use only the letters of the alphabet in addition to the number symbols, called numerals. A letter that represents a number is termed a literal number. Literal in algebra means letter. As you may recall from arithmetic, when two numbers are multiplied together, the two numbers themselves are called factors. Thus when multiplying in algebra we refer either to numerical factors (digits) or literal factors (letters). Definitions We will discuss the meaning and use of the words term, factor, and expression in detail later on; we define them for the present as follows: Term: a term is either a single number or the product of one or more numerical and/or literal factors. For example, 8, 32, and yk are terms. Factor: a factor is any one of the individual letters or numbers in a term. Thus, 5, c 2, and d are factors of the term 5c 2 d. Expression: An expression (meaning algebraic expression) consists of one or more terms connected by plus or minus signs. Examples: In the algebraic term: 7xyz, x, y, and z ae the literal factors and 7 is the numerical factor. In the term jk 3tm, j, k, t and m are the literal factors and 3 is the numerical factor. In the term 12z(9ak), z, a, and k are the literal factors; 12 and 9 are the numerical factors. TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013 Page 37 of 670

21 Provide Computational solutions to Power Engineering Problems Exercise 2-4: See if you can identify the literal factors in the following terms. (a) 7x (b) 4abc (c) 3s(5rt) (d) xy 8ab We ve been following a lot of new terms that you and you re probably asking yourself: Do I need to know all this stuff? Our answer, surprisingly, is No! Then why do we make such a big deal out of words like literal number, numerical factor, term, and expression? We have a couple of reasons. First, the ideas and concepts in algebra - just as in real life - need names. So we make up names for them. Second, a name sometimes saves us from having to write out a whole sentence. But don t worry, if you don t immediately recognise the name of a particular term, you can almost always figure out what it means from the context. Now let us consider the symbols that algebra borrows from arithmetic to indicate the fundamental operations of addition, subtraction, multiplication, and division. We have already discussed the multiplication symbol (x).just like the multiplication symbol, the division symbol ( ) is rarely used in algebra. More often we used the colon(:) or the fraction bar (-) and sometimes we use the slash (/). Thus, for x y we would write: x: y or x or x/y y All three mean divided x by y. The addition symbol and subtraction symbol are exactly the same in algebra as they are in arithmetic. Exercise 2-5: Translate the following verbal statements into algebraic expressions using the information given above. (a) x divided by y (b) 4 times a times c (c) the sum of y and cd, divided by seven plus x (d) 8rs minus 2b, all divided by six times x times y Page 38 of 670 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013

22 If your answer for questions (c) or (d) was wrong, then you possibly did not take the commas into consideration. For instance, the correct answer to question (c) was y+cd 7+x, and not y + cd 7+x. One special situation you must watch out for is the case where a letter represents zero. We know from arithmetic that adding or subtracting zero to or from another number does not change the value of an expression. You will also recall that multiplying any number by zero (or zero by another number) gives zero as a result. What happens when you divide any number by zero? The answer is that a number can be divided by zero an infinite number of times. So practically, division by zero is meaningless; it is an undefined operation. These meaningless expressions are easy to recognize when you see a zero in the denominator (as bottom half) of a fraction. But when the denominator contains a letter (either one or more), you must be very careful to make sure that this letter does not represent zero, or that some value assigned to the letter does not cause the value of the denominator to become zero. In fact, mathematicians would have a fit if they read these previous statements. Dividing by zero, or into zero, is a serious subject. However, for now, we will ignore the deeper aspects of this assumption. Example: These are meaningless expressions: 5 0 ; x 0 In the expression 3, if x = 0 the division would be impossible as 3 could be divided x by zero an infinite number of times. Exercise 2-6: Indicate which values of the letters in the denominators would result in an impossible division. For example: (a) 2 d (b) xy 5d (c) 25 c 2 (d) 3 ab Note: Division of a number into zero is not meaningless. The fraction actually equals zero. Example: a ; where a = 0, then 0 = 0 as dividing zero eight times is still zero. 8 8 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013 Page 39 of 670

23 Provide Computational solutions to Power Engineering Problems Before we go on, test yourself to see that you remember these important axioms of equality. Exercise 2-7: Complete the statement below. (a) If equals are added to, the sums are equal. (b) If equals are by equals, the products are equal. (c) Expressions that are equal to the same quantity are. (d) If equals are subtracted from equals, but differences are (e) If equals are divided by equals the are equal (provided the is not zero). If you missed any of these, you can refresh your memory by rereading frame Fundamental Operations in Algebra In arithmetic we use all of the fundamental operations (adding, subtracting, dividing, multiplying) as they relate to numerical values. And now we need to extend and apply these operations to literal values (letters) used to represent numbers. Such literal values can be used to represent either fixed values (constants) or variable numbers (variables). In the paragraphs that follow we will discuss some specific laws, or rules, that you will use extensively (and nearly automatically once you are thoroughly familiar with them) throughout your study of algebra and thereafter, no matter what branch of mathematics you study. You are already familiar with some of these rules from arithmetic, although probably not by name. For example, you doubtless would accept without argument the fact that: Or that: = = 2 3 Remember: Get used to seeing and using the dot instead of the times sign for multiplication. Instead of numerals in the above examples we could have used letters. Thus, a + b = b + a or a b = b a (More simply, ab = ba) Page 40 of 670 TAFE NSW (TES, Industry Skills Unit Meadowbank) 2013

Math 90 Lecture Notes Chapter 1

Math 90 Lecture Notes Chapter 1 Section 1.1: Introduction to Algebra This textbook stresses Problem Solving! Solving problems is one of the main goals of mathematics. Think of mathematics as a language,