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Mathematics 20 Module 1 Lesson 2 Mathematics 20 Irrational Numbers and Square Root Radicals 51 Lesson 2 Mathematics 20 52 Lesson 2 Irrational Numbers and Square Root Radicals Introduction In this lesson a new set of numbers called irrational numbers is added to the family of natural numbers, whole numbers, integers, and rational numbers. The irrational numbers together with the rational numbers form the real number system whose graph is the solid number line with no gaps. There are many varieties of irrational numbers, but the particular kind studied in this and the next lesson is the irrational number obtained by taking the square root of a whole number. The classes of numbers which form the real number system belong to a yet larger system of numbers called the complex number system. Whole Number Math Triangular numbers are those which can be represented by triangles as shown by the examples. Which two consecutive triangular numbers have a difference of 10? What is the difference of the differences between consecutive numbers? Mathematics 20 53 Lesson 2 Square numbers are those which can be represented by a square. Which two consecutive square numbers have a difference of 13? What is the difference of the differences between consecutive numbers? Mathematics 20 54 Lesson 2 Objectives After completing this lesson, you will be able to • define and illustrate by example the term absolute value, and simplify expressions that contain the absolute value of a number as well as the absolute value of a variable. • define and illustrate by example an irrational number. • solve problems which use the terms square root radical, radicand, principal square root, and perfect square. • express square root radicals in simplest mixed radical form. • work with a variety of given formulas that contain a squared term or a square root radical. Mathematics 20 55 Lesson 2 Mathematics 20 56 Lesson 2 2.1 Absolute Value The Number Line The point on the number line representing the number zero is chosen arbitrarily and this point is called the origin. The point for the number 1 is chosen a certain distance to the right of the origin and all other points representing integers are equally spaced, as shown on the diagram. The points which represent rational numbers are located so that the number represents the distance between the point and the origin. For example, if the scale is such that the point 1 is 1 cm from the origin then the point for the rational number 2.5 is 2.5 cm from the origin. Very often units such as cm are not stated. In such a case the point 1 is said to be one unit away from the origin, 2 is two units away from the origin, etc. There are exactly two points that are a non-zero distance from the origin. For example the points -10 and 10 are both ten units from the origin. Only one point, the origin, is 0 units away from the origin. Absolute Value The absolute value of a number is the distance between the origin and the point representing the number. Distance is always non negative. Two vertical bars are used to symbolize absolute value. • 3 = 3 The absolute value of 3 is 3. • 3 = 3 The absolute value of 3 is 3. • 7 The absolute value of 7 • 0 = 0 The absolute value of 0 is 0. 1 1 = 7 2 2 1 1 is 7 2 2 What two numbers have absolute value equal to 7? This is the same as asking for the solution to |x| = 7 . Mathematics 20 57 Lesson 2 The two values for x are 7 and 7 since each point representing these numbers is 7 units away from the origin. Symbols can be used for number systems. • • • • • I N W Q - Integers ..., 2, 1, 0, 1, 2, ... Natural numbers 1, 2, 3, 4, ... Whole numbers 0, 1, 2, 3, 4 , ... Rational numbers Real Numbers Example 1 Draw the graph of all the integers on the number line whose absolute values are greater than or equal to 2. Solution: Find all the integers which are a distance greater than or equal to 2 from the origin. This is the same as solving the inequality |x| • 2, x I. All the points to the left of 2 , including 2 , together with all the points to the right of 2, including 2, have absolute value greater than or equal to 2. The dark dots and the dark arrows show the graph of x 2 . Example 2 Draw the graph on the number line of all the integers whose absolute value is less than 3. Solution: This is the same as solving the inequality |x| < 3 , • x I. All the points between 3 and 3, not including 3 and 3, have absolute values less than 3. Mathematics 20 58 Lesson 2 Example 3 Draw the graph of all the rational numbers whose absolute value is less than 3. Solution: This is the same as solving |x| < 3 , • • x Q. The graph is a solid line between 3 and 3 . Dots cannot be used here since rational numbers are close together. The empty circles or open dots on 3 and 3 show that 3 and 3 are not included in the solution set. Absolute value can be used to find the distance between two points on the number line. If a and b are two points on a number line, the distance between these two points is |a b| or |b a| . • • The rule is to subtract the numbers in any order and take the absolute value. The absolute value ensures that the distance is a positive number. Example 4 Find the distance between 8 and 11 . Solution: 8 (11) = 8 11 = 19 = 19 Take the absolute value of the difference. or (11) 8 = 19 = 19 Both ways result in the same answer. The usual rules of order of operations are applied when working with absolute values. Just as for brackets, the expression inside the absolute value sign is to be evaluated first. Mathematics 20 59 Lesson 2 Example 5 Evaluate 5 3(4 ) 8 + 6 2 . Solution: 5 3(4 ) 8 + 6 2 Write the original expression. Evaluate inside the signs first. = 17 2 2 Take the absolute values. Simplify. = = 17 2 2 32 Example 6 Solve for y in |x| + |y| = 8, if x = 3 . Solution: y = 8 x Isolate |y| . Substitute x = 3 . Evaluate the right side. y = 8 3 y = 8 3 y = 5 y = 5 and y = 5 There are two points on the number line which are 5 units away from the origin. Exercise 2.1 1. Simplify each absolute value expression. a. 23 b. 204 c. 9 2 d. 16 + 8 e. 4 + 4 f. 4 + 3 g. 20 3 22 Mathematics 20 60 Lesson 2 2. Find the distance between the given points using |a - b| . a. b. c. d. 3. 4. 5. 19 and 25 36 and 17 99 and 99 376 and 540 Draw the graph of the solution to each equation or inequality on the number line. Assume only integer solutions. a. |x| = 0 b. |x| 2 c. |x| 3 > 2 d. |x| < 5 e. |x| + 2 6 Evaluate each expression. a. 9 4 b. 9 8 2 c. 2 1 4 + 3 7 2 7 3 d. 9 e. 8 2 6 3 3 4 3 Solve for the variable in each case. a. |x| = 5 b. |x| 3 = 1 c. 2x = 4 d. 2x + 3x 4x = 7 Mathematics 20 First collect similar terms. 61 Lesson 2 2.2 Irrational Numbers and Square Root Radicals Rational Numbers The natural numbers, whole numbers, and integers are each part of the larger set of numbers called the rational numbers. Every rational number can be written in the fraction form a , where a and b b are integers, and b is not equal to zero. a Q = a ,b I ; b 0 b “such that” Every rational number can also be written as either a repeating decimal or a terminating decimal (the zeros repeat). • • 2 = 0 .181818 ... 11 1 = 0 .25000 ... 4 = 0.25 The digits 1 and 8 repeat. Only the zeros repeat. This is a terminating decimal. Irrational Numbers There are also decimal numbers that do not repeat or terminate. An example of this is the number 2.0100100010000100000.... This number has a non repeating decimal since the number of zero digits increases by one after each 1 digit. A number like this cannot be written in the form of a fraction a and is, therefore, not a b rational number. Any number whose decimal is non repeating and non terminating is called an irrational number. Mathematics 20 62 Lesson 2 The rational numbers and the irrational numbers together form the set of real numbers and the letter is used to represent this set. The number line is the graph of the set of real numbers. Each real number corresponds to some point on the line and, conversely, each point on the line corresponds to a real number. From now on, the number line will be called the real number line. • The graphs of sets of real numbers appear as heavily shaded lines, with solid (or darkened) heads when indicating continuation in the direction of an inequality. • The graphs of sets of integers are shaded dots spaced one unit apart. Example 1 Draw the graphs of the following sets. a) x| x > 2, x b) {x | x > 2 , x I } Solution: a) x| x > • 2, x The empty circle indicates that 2 is not included in the set but the graph includes every real number larger than 2. Mathematics 20 63 Lesson 2 b) {x | x > 2, x I} • The solid dots start at 3 because 2 is not included and there are no integers between 2 and 3. The solid arrow head indicates an indefinite continuation of the solution set, that is, all integers greater than 2. Square Root Radicals You know how to find the square of a number. The square of 2 is 2 2 = 4 . The square of 2 is 2 2 = 4 . Both integers squared have a value of 4. The reverse procedure is to find the square root of a number. • • • One square root of 4 is 2. The other square root is 2 . Similarly, one square root of 9 is 3 and another is 3 . In general: a is the square root of b if a a = b , or a 2 = b . Some numbers like 2, 3, and 5 do not have integer square roots but have irrational roots. It is often not necessary to know the decimal form of the square root so the symbol a is used to denote the positive square root of the number a. The sign is called the radical sign, and the number or expression under the radical sign is called the radicand. a is called a radical or the square root of a. Mathematics 20 64 Lesson 2 For example, the expression 10 is a radical which means the positive square root of 10. Different calculators evaluate square roots differently. If it does not follow the procedure that is used in this course, check your manual. Example 1 Use the square root function on your calculator to find 2. Solution: clear 2 enter display: 1.4142136... Therefore, 2 = 1.4142136 .... Because of space limitations, some calculators report only the first 7 decimals. This gives only an approximation to the infinite decimal. The entire decimal, however, is non repeating and the number is irrational. Mathematics 20 65 Lesson 2 Activity 2.2 The 5 = 2.236068 . Square each decimal approximation using one more decimal place each time. Value 2.2 2 = 2.23 2 = 2.236 2 = 2.23606 2 2.236068 2 = = Notice that the squares approach the value 5 as more decimal places are included. Principal Square Root The number 9 has two square roots. The two square roots are 3 and 3 because: 3 3= 9 • 3 3 = 9 • • 3 is the positive square root. 3 is the negative square root. • This is written in radical form as: 3= 9 3 = 9 positive square root negative square root There is only one square root of zero. If a a = 0 , a must equal 0. 0= 0 Mathematics 20 66 Lesson 2 The number 9 has no real number square roots. It is impossible to find two equal numbers that when multiplied together have a negative product. Negative numbers have no square root. • 9 is undefined. Examples: • • • • The principal square root of 49 is 49 = 7 . The negative square root of 64 is 64 = 8 . All the square roots of 16 are 16 , 16 , or 4, 4 . All the square roots of 15.21 are 15 .21 , 15 .21 , or 3.9, 3.9 . The above examples illustrate the following general properties of square root radicals. Properties of Square Root Radicals • The radicand can be only zero or positive. • Each positive number c has two square roots, • The positive root is called the principal root and usually has no sign attached to it. c, c . Notice that the calculator gives only the principal root. Sometimes an equation has a squared term in it. 2 x = 25 In this case, x can be equal to 25 or 25 . Examples: • • The solution to x 2 = 100 is x = 100 , x = 100 , or x = 10 . The solution to x 2 = 42 .25 is x = 42.25 , or x = 6.5 . Questions with square root equations are more common in real world situations. These types of questions will be explored more in section 2.4. Mathematics 20 67 Lesson 2 The solution to the equation x 2 = c , for c > 0 , is x = c and x = c . The following examples will show you some of the different terminology that is used when defining radicals. It is important to read the question to determine the root that is being used. • The number which has 15 and 15 as square roots is 15 or 15 . The number is 225. • The number which has 2 6 as a principal root is • The number which has x as a root is x • • The solution to x = 5 is x = 5 2 = 25 . 2 If x = 2 .6 , then x = 2 .6 = 6 .76 . 2 6 2 2 = 6. =x . Exercise 2.2 1. Which numbers are rational and which are irrational? a. b. c. d. e. 2. 3. 0.24567 49 3 1.1511511151 1115 ... 2 .153525000 ... Draw the graph of each set on the real number line. a. b. c. d. x < 2 , x {x | -1 x < 2, x I} x| x < 3.5 , x {x | x < 3.5, x I} a. b. c. d. Evaluate 169 , 196 , 225 . Find the square roots of 37.4544. Find the principal root of 625. Find the solutions to y 2 = 28 .09 . e. x| 1 Find the solutions to x 2 = 16 . Mathematics 20 2 68 Lesson 2 4. a. b. c. d. e. f. g. What number has 7 as a root? What number has 16 as a root? If x = 5 , then what is x? Find the solution to x = 3.75 . Find the solution to x = 10 . 2 Evaluate 3 . Evaluate 36 4 . 2 2 2.3 Expressing Radicals in Simplest Form Numbers such as 1, 4, 9, 16, can be written as a product of two equal whole numbers. 1 = 12 , 4 = 2 2 , 9 = 3 2 , 16 = 4 2 Any positive integer which can be written as a product of two equal integers is called a perfect square. Activity 2.3 A partial list of perfect squares is given. Continue the sequence by supplying at least 5 more perfect squares. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, . . . Perfect squares will be used in simplifying radicals to make the radicand as small as possible. Perfect squares are useful because their square roots are integers. • 4 22 2 • 9 32 3 In general: 2 x = x, for x > 0 , x . Mathematics 20 69 Lesson 2 Rules for multiplication and division of radicals are required for simplifying. These are similar to rules of exponents. a b = ab a a for a, b 0 , b 0 . = b b A radical is in simplest form if the radicand is made to be the smallest whole number possible. This is done by removing perfect square factors from the radicand. The examples illustrate how this simplification is done. The object is to factor the radicand into as many perfect squares as possible. Example 1 Simplify 8. Solution: Write the original expression. Factor the radicand. Use the product rule. Simplify the radical 4 . = = = = 8 4 2 4 2 2 2 2 2 Example 2 Simplify 2450 . Solution: The number 2450 can be factored many different ways. One way is 2450 = 50 49 = 2 25 49 . Two of the factors, 25 and 49, are perfect squares and can therefore be simplified. The factor 2 cannot be factored further. Write the original expression. Factor the radicand. Use the product rule. Simplify the radicals. = = = = Mathematics 20 70 2450 49 25 2 49 25 2 7 5 2 35 2 Lesson 2 Example 3 Simplify 900 . 2 Solution: 900 Write the original expression. Use the quotient rule. = Simplify. Factor the radicand. Use the product rule. Simplify the radical 225 . = = = = = 2 900 2 450 225 2 225 2 15 2 15 2 Example 4 Simplify 3 3 3 7 7 13 13 2 . Solution: By grouping pairs of equal factors, perfect squares are formed. Write the original expression. 3 3 3 7 7 13 13 2 2 2 2 3 3 7 13 2 Group equal factors. = Use the product rule. = Simplify each radical. Regroup whole numbers and radicals. Multiply whole numbers together. Multiply radicals together. = 273 6 Mathematics 20 2 2 2 3 3 7 13 = 3 3 7 13 2 = 3 7 13 3 2 71 2 Lesson 2 An entire radical consists only of the radical sign and the radicand. Some examples of entire radicals are: • 2 12 300 A mixed radical is the product of an entire radical with a real number not in radical form (a coefficient). Some examples of mixed radicals are: • 2 12 3 300 5 2 • The coefficients are 2, 3 , and 5. These examples show that simplifying entire radicals to mixed radicals with lowest possible radicand involves factoring into perfect squares. The reverse process of changing mixed radicals into entire radicals requires the squaring of numbers. Any number can be changed to entire radical form by making the square of the number into the radicand. Example 5 Change 2 and 7 to entire radical form. Solution: Square each of the numbers. 2= 2 2 = 4 7= 2 7 = 49 Example 6 Change the mixed radical 2 5 into an entire radical. Solution: Write the original expression. Change the whole number into radical form. Multiply the radicals. 2 5 = 4 5 = 4 5 = 20 The coefficient of a mixed radical may be a negative number. If this is the case, the negative sign remains outside the radical sign when you change the mixed radical into entire radical form. Mathematics 20 72 Lesson 2 Example 7 Change 6 7 into an entire radical. Solution: Write the original expression. Change 6 to radical form. Multiply the radicals. 6 7 = 62 7 = 36 7 = 36 7 = 252 Exercise 2.3 1. Write each as a square or as a product of squares. a. b. c. d. e. f. 2. 100 324 81 400 3 3 3 3 5 5 72 8 Express each entire radical in simplest form. a. b. c. d. e. f. g. h. i. j. k. l. m. 2 2 3 3 5 9 7 5 5 2 2 2 2 49 13 13 2 200 121 18 54 90 20 28 360 432 234 Mathematics 20 73 Lesson 2 3. Write each mixed radical in simplest form. a. b. c. d. e. f. g. h. i. j. 4. 25 25 9 64 2 81 3 16 3 25 5 9 6 50 2 3 72 8 63 2 18 3 3 80 4 9 32 2 Express each mixed radical as an entire radical. a. b. c. d. e. f. g. 2 2 8 3 3 5 6 5 50 2 10 7 4 50 5 Mathematics 20 74 Lesson 2 2.4 Applications of Radicals The Pythagorean Theorem for right triangles requires the use of radicals. For any right triangle, the sum of the squares of the lengths of the legs, a and b, equals the square of the length of the hypotenuse, c. leg 2 leg 2 hyp 2 a 2 b2 c2 The principal square root of both sides of the equation gives a formula for the length of the hypotenuse c. 2 2 2 c = a + b 2 2 2 c = a + b c= 2 2 a + b The right side cannot be simplified further. Example 1 For the given triangle, express the length of the hypotenuse as the simplest possible mixed radical. Solution: Use c = 2 2 a + b , where c = x, a = 4 , b = 6 . Write the formula. c= 2 2 a + b Substitute the known values. x= ( 4 )2 + ( 6 )2 Evaluate. x = 16 + 36 x = 52 x = 4 13 x = 2 13 Since 13 cannot be factored further, this is the simplest mixed radical. Mathematics 20 75 Lesson 2 This answer is in radical (or exact) form. A decimal approximation can be found using a calculator. x 42 62 clear ( 4 yx 2 + 6 yx 2 ) enter display: 7.21110255093 7.21 Example 2 The area of a square with sides of length x is given by the formula A x 2 . What must the length of each side be if the area is 500 square units? Solution: It is convenient to isolate the variable x first. This is done by taking the principal square root of both sides. Note that the length and area are always positive, so only the principal root is considered. Write the formula. A = x2 Take the square root of both sides. A= x x= A 2 x = 500 = 100 5 x = 100 5 x 10 5 units Substitute A = 500 . This answer is in radical (exact) form. A decimal approximation can be found using your calculator. clear 500 enter display: 22.360679775 22.36 Mathematics 20 76 Lesson 2 Exercises 2.4 1. a. Solve for x. x 12 3 b. Solve for d. 2d t g c. Solve for w. 2 gE V w 2. Express the length of the hypotenuse in exact form. 3. Express the length of the side of the triangle in both exact and approximate decimal form. Mathematics 20 77 Lesson 2 4. The formula for the distance one can see to the horizon is given by d 2 13 h , where d is the distance to the horizon in kilometres, and h is the height in metres above ground level a person is located. a. b. 5. Isolate the variable d. Calculate d if a person is observing from the top of a 400 m building. Isolate r in the formula for the area of a circle A = r 2 . Mathematics 20 78 Lesson 2 Conclusion In the introduction you were given a Whole Number Math problem. The solution to this problem is: • Two consecutive triangular numbers that have a difference of 10 are 45 and 55. The difference of the differences is 1. • Two consecutive square numbers that have a difference of 13 are 36 and 49. The difference of the differences is 2. Summary The following is a list of concepts that you have learned in this lesson: • The absolute value of a number can be thought of as the distance between the origin and the point representing the number on the number line. • Equations and inequalities with absolute values have the following solutions: | x| < c | x| = c | x| > c • The radicand can only be zero or a positive number. • Each positive number c has two square roots, • The positive root is called the principal root and usually has no sign attached to it. • The calculator gives only the principal root. • The solution to the equation x 2 = c , for c > 0 , is x = • The solution to the equation Mathematics 20 c, c . c , and x = c . x = a, is x = a 2 . 79 Lesson 2 Answers to Exercises Exercise 2.1 1. a. b. c. d. e. f. g. 23 204 7 8 0 7 5 2. a. b. c. d. 6 53 198 164 3. a. b. c. d. e. 4. 5. Mathematics 20 a. b. c. 36 144 2 3 10 2 4 8 d. e. 9 3 4 3 3 3 a. b. c. d. x x x x 27 88 = = = = 5 2 2 7 80 Lesson 2 Exercise 2.2 1. 2. a. rational b. rational c. irrational d. irrational e. rational a. b. 101 c. d. Exercise 2.3 Mathematics 20 3. a. b. c. d. e. 13, ±14, 15 ±6.12 25 ±5.3 ±2 4. a. b. c. d. e. f. g. 7 256 25 14.0625 100 3 144 1. a. b. c. d. e. f. 10 2 18 2 9 20 2 2 2 2 3 3 5 2 2 2 2 3 4 2 81 Lesson 2 2. a. b. c. d. e. f. g. h. i. j. k. l. m. 15 3 60 7 91 2 10 2 11 3 2 3 6 3 10 2 5 2 7 6 10 12 3 3 26 3. a. b. d. e. f. g. 25 5 125 72 29 3 34 2 1 60 3 36 2 18 2 8 9 7 8 3 7 24 7 h. i. j. 2 2 3 5 18 2 a. b. c. d. e. f. 4 2 8 64 3 192 45 180 5000 100 7 700 16 50 32 25 c. 4. g. Mathematics 20 82 Lesson 2 Exercise 2.4 1. x 12 3 a. x 144 3 x 432 2d g 2d t2 g 2 t g 2d t b. d t2g 2 2 gE w 2 gE V2 w 2 V w 2 gE 2 gE w V2 V c. 2. x 6 2 14 2 x 232 x 2 58 3. x 15 2 3 2 x 225 9 x 216 14 .7 x 6 6 4. a. b. d 13 h d 13 400 d 20 13 5. Mathematics 20 r2 A r A 83 Lesson 2 Mathematics 20 84 Lesson 2 Mathematics 20 Module 1 Assignment 2 Mathematics 20 85 Lesson 2 Mathematics 20 86 Lesson 2 Optional insert: Assignment #2 frontal sheet here. Mathematics 20 87 Lesson 2 Mathematics 20 88 Lesson 2 Assignment 2 Values (40) A. Multiple Choice: Select the best answer for each of the following and place a () beside it. 1. The absolute value of 16 is ***. ____ ____ ____ ____ 2. 5. only 2 only 4 2 ±2 a. b. c. d. 0 10 10 25 If 2 x 3 y = 6 , then x is equal to ***. ____ a. 6+ 3y ____ b. ____ c. ____ d. 6 + 3y 3 6+ y 2 6+ 3y 2 4 5 expressed as an entire radical is ***. ____ ____ ____ ____ Mathematics 20 a. b. c. d. The value of 5 5 is ***. ____ ____ ____ ____ 4. 16 ±4 ±16 4 If the absolute value of a number is 2, the number is ***. ____ ____ ____ ____ 3. a. b. c. d. a. b. c. d. 20 9 80 10 89 Lesson 2 6. 12 2 expressed as an entire radical is ***. ____ ____ ____ ____ 7. 10. 5 12 10 30 30 10 10 3 a. b. c. d. 18 10 6 5 3 20 2 45 The graph of y > 2 , y I is ***. ____ a. ____ b. ____ c. ____ d. The one rational number is ***. ____ ____ ____ ____ Mathematics 20 a. b. c. d. 180 expressed as a mixed radical in simplest form is ***. ____ ____ ____ ____ 9. 24 288 48 144 300 written as a mixed radical in simplest form is ***. ____ ____ ____ ____ 8. a. b. c. d. a. b. c. d. 2 3 4 5 90 Lesson 2 11. The graph of |x| < 3 , x is ***. ____ a. 3 0 3 ____ b. ____ c. ____ d. 3 12. a. b. c. d. 10 5 2 50 5 2 a. b. c. d. x = 72 is ***. 5184 144 6 2 6 2 The expression 3 2 2 14 ____ ____ ____ ____ Mathematics 20 2 71 71 71 71 The complete solution to ____ ____ ____ ____ 15. a. b. c. d. The complete solution to 2 x 2 = 100 is ***. ____ ____ ____ ____ 14. 3 The principal square root of 71 is ***. ____ ____ ____ ____ 13. 0 a. b. c. d. 2 in simplest form is equal to ***. 7 6 8 12 2 12 2 12 91 Lesson 2 16. The mixed radical 8 3 in entire radical form is ***. ____ ____ ____ ____ 17. 1352 676 26 2 13 2 a. b. c. d. 20 5 10 50 10 5 The diagonal of a square is ____ ____ ____ ____ 20. a. b. c. d. The length of the diagonal of a rectangle with sides 15 and 5 is ***. ____ ____ ____ ____ 19. 192 24 72 192 The mixed radical 2 338 in simplest radical form is ***. ____ ____ ____ ____ 18. a. b. c. d. a. b. c. d. If T = 2 128 m 16 m 64 m 8m l , then l is equal to ***. g a. gT2 4 2 ____ b. ____ c. ____ d. g T2 4 2 gT2 2 T g ____ 128 m . Each side is ***. 2 Mathematics 20 2 92 Lesson 2 Part B can be answered in the space provided. You also have the option to do the remaining questions in this assignment on separate lined paper. If you choose this option, please complete the entire question on the separate paper. Evaluation of your solution to each problem will be based on the following. (40) B. • A correct mathematical method for solving the problem is shown. • The final answer is accurate and a check of the answer is shown where asked for by the question. • The solution is written in a style that is clear, logical, well organized, uses proper terms, and states a conclusion. 1. Evaluate 3 5 27 36 + 4 4 . 2. Evaluate x y x 2 y2 , if x = 3 and y = 2 . 3. Draw the graph of a. |x| < 2.5 , x b. |x| 1.6 , x I (4 each) Mathematics 20 93 Lesson 2 4. Find the distance between 3 and 15 using a b . 5. Find the solution to 2 a. x = 6 .25 . b. x = 3.56 . 6. Simplify 5 49 2 25 . 25 7. Mathematics 20 Express 1575 as a mixed radical in simplest form. 94 Lesson 2 17 75 as an entire radical. 5 8. Express 9. The distance in metres a free falling object travels is given by the 1 equation d gt 2 , where t is in seconds and g which is the 2 acceleration due to gravity is 9.8. a. b. 10. Isolate the variable t. How long does it take an object to fall 1000 m? The period of a pendulum in seconds is given by the formula l , where l is the length of the pendulum in metres and g is T = 2 g the acceleration due to gravity. Calculate to two decimal places the acceleration due to gravity if it takes a 10 m long pendulum 6.35 seconds to complete one cycle; i.e., the period is 6.35 s. Mathematics 20 95 Lesson 2 Answer Part C on separate lined paper. Please include any tables or graphs that you are required to do with the assignment. (20) C. 1. 2. a. Find all twelve integer ordered pairs that satisfy the equation |x| + |y| = 3 . Organize the ordered pairs into a table of values. b. Plot the ordered pairs on the attached grid and connect the points. c. What geometric figure is formed? This example shows how to locate the irrational numbers 2 and 3 on the number line by use of a compass and a ruler. You are asked to locate the irrational numbers 5 , 6 , and 7 by the same method. Example: On the real number line draw a right triangle with sides of length 1. By Pythagoras Theorem the hypotenuse is of length 2 . With a compass centred at the origin and radius the length of the hypotenuse, draw an arc intersecting the real line. Label the point of intersection 2. Next, draw a right triangle with base length 2 , and height 1. By Pythagoras Theorem the hypotenuse is 3 . With a compass centred at the origin and radius the length of the hypotenuse draw an arc intersecting the real line at the point labelled 3 . 100 Mathematics 20 96 Lesson 2 y x Mathematics 20 97 Lesson 2