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Square Roots and Irrational Numbers

Square Roots and Irrational Numbers. PRE-ALGEBRA LESSON 11-1. A blue cube is 3 times as tall as a red cube. How many red cubes can fit into the blue cube?. 27. 11-1. Square Roots and Irrational Numbers. PRE-ALGEBRA LESSON 11-1. (For help, go to Lesson 4-2.).

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Square Roots and Irrational Numbers

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  1. Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11-1 A blue cube is 3 times as tall as a red cube. How many red cubes can fit into the blue cube? 27 11-1

  2. Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11-1 (For help, go to Lesson 4-2.) Write the numbers in each list without exponents. 1. 12, 22, 32, . . ., 1222. 102, 202, 302, . . ., 1202 Check Skills You’ll Need 11-1

  3. Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11-1 Solutions 1. 12, 22, 32, . . . , 122 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 2. 102, 202, 302, . . . , 1202 100, 400, 900, 1,600, 2,500, 3,600, 4,900, 6,400, 8,100, 10,000, 12,100, 14,400 11-1

  4. a. 144 144 = 12 b.  – 81 – 81 = – 9 Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11-1 Simplify each square root. Quick Check 11-1

  5. Use the formula. d = 1.5h Replace h with 20. d = 1.5(20) Multiply. d = < 25 < 30 30 36 Find perfect squares close to 30. 25 = 5 Find the square root of the closest perfect square. Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11-1 You can use the formula d = 1.5h to estimate the distance d, in miles, to a horizon line when your eyes are h feet above the ground. Estimate the distance to the horizon seen by a lifeguard whose eyes are 20 feet above the ground. Quick Check The lifeguard can see about 5 miles to the horizon. 11-1

  6. a.  49 c.  3 e. – 15 Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11-1 Identify each number as rational or irrational. Explain. rational, because 49 is a perfect square b. 0.16 rational, because it is a terminating decimal irrational, because 3 is not a perfect square d. 0.3333 . . . rational, because it is a repeating decimal irrational, because 15 is not a perfect square f. 12.69 rational, because it is a terminating decimal g. 0.1234567 . . . Quick Check irrational, because it neither terminates nor repeats 11-1

  7. Square Roots and Irrational Numbers PRE-ALGEBRA LESSON 11-1 Simplify each square root or estimate to the nearest integer. 1. – 100 2. 57 Identify each number as rational or irrational. 3. 48 4. 0.0125 5. The formula d = 1.5h , where h equals the height, in feet, of the viewer’s eyes, estimates the distance d, in miles, to the horizon from the viewer. Find the distance to the horizon for a person whose eyes are 6 ft above the ground. –10 8 irrational rational 3 mi 11-1

  8. The Pythagorean Theorem PRE-ALGEBRA LESSON 11-2 The Jones’ Organic Farm has 18 tomato plants and 30 string bean plants. Farmer Jones wants every row to contain at least two tomato plants and two bean plants. There should be as many rows as possible, and all the rows must be the same. How should Farmer Jones plant the rows? 6 rows, with each row containing 5 bean plants and 3 tomato plants 11-2

  9. The Pythagorean Theorem PRE-ALGEBRA LESSON 11-2 (For help, go to Lesson 4-2.) Simplify. 1. 42 + 622. 52 + 82 3. 72 + 924. 92 + 32 Check Skills You’ll Need 11-2

  10. The Pythagorean Theorem PRE-ALGEBRA LESSON 11-2 Solutions 1. 42 + 622. 52 + 82 16 + 36 = 52 25 + 64 = 89 3. 72 + 924. 92 + 32 49 + 81 = 130 81 + 9 = 90 11-2

  11. c2 = a2 + b2 Use the Pythagorean Theorem. c2 = 282 + 212 Replace a with 28, and b with 21. Simplify. c2 = 1,225 Find the positive square root of each side. c = 1,225 = 35 The Pythagorean Theorem PRE-ALGEBRA LESSON 11-2 Find c, the length of the hypotenuse. The length of the hypotenuse is 35 cm. Quick Check 11-2

  12. a2 + b2 =c2 Use the Pythagorean Theorem. 72 + x2 =142 Replace a with 7, b with x, and c with 14. 49 + x2 =196 Simplify. x2 =147 Subtract 49 from each side. x=147 Find the positive square root of each side. The Pythagorean Theorem PRE-ALGEBRA LESSON 11-2 Find the value of x in the triangle. Round to the nearest tenth. 11-2

  13. Then use one of the two methods below to approximate . 147 Method 1 Use a calculator. A calculator value for is 12.124356. Round to the nearest tenth. Method 2 Use a table of square roots. Use the table on page 778. Find the number closest to 147 in the N2 column. Then find the corresponding value in the N column. It is a little over 12. x 12.1  x 12.1  147 Estimate the nearest tenth. The Pythagorean Theorem PRE-ALGEBRA LESSON 11-2 (continued) Quick Check The value of x is about 12.1 in. 11-2

  14. c2 = a2 + b2 Use the Pythagorean Theorem. c2 = 102 + 102 Replace a with 10 (half the span), and b with 10. c2 = 100 + 100 Square 10. c2 = 200 Add. c = 200 Find the positive square root. c 14.1 Round to the nearest tenth. The Pythagorean Theorem PRE-ALGEBRA LESSON 11-2 The carpentry terms span, rise, and rafter length are illustrated in the diagram. A carpenter wants to make a roof that has a span of 20 ft and a rise of 10 ft. What should the rafter length be? Quick Check The rafter length should be about 14.1 ft. 11-2

  15. a2 + b2 = c2 Write the equation to check. Replace a and b with the shorter lengths and c with the longest length. 102 + 242262 Simplify. 100 + 576 676 676 = 676 The Pythagorean Theorem PRE-ALGEBRA LESSON 11-2 Is a triangle with sides 10 cm, 24 cm, and 26 cm a right triangle? The triangle is a right triangle. Quick Check 11-2

  16. The Pythagorean Theorem PRE-ALGEBRA LESSON 11-2 Find the missing length. Round to the nearest tenth. 1.a = 7, b = 8, c = 2.a = 9, c = 17, b = 3. Is a triangle with sides 6.9 ft, 9.2 ft, and 11.5 ft a right triangle? Explain. 4. What is the rise of a roof if the span is 30 ft and the rafter length is 16 ft? Refer to the diagram on page 586. 10.6 14.4 yes; 6.92 + 9.22 = 11.52 about 5.6 ft 11-2

  17. Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11-3 Find the number halfway between 0.784 and 0.76. 0.772 11-3

  18. Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11-3 (For help, go to Lesson 1-10.) Write the coordinates of each point. 1.A2.D3.G4.J Check Skills You’ll Need 11-3

  19. Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11-3 Solutions 1.A (–3, 4) 2. D (0, 3) 3.G (–4, –2) 4. J (3, –1) 11-3

  20. d = (x2 – x1)2 + (y2 – y1)2 Use the Distance Formula. d = (8 – 3)2 + (3 – (–2 ))2 Replace (x2, y2) with (8, 3) and (x1, y1) with (3, –2). 52 + 52 d = Simplify. 50 d = d 7.1 Find the exact distance. Round to the nearest tenth. Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11-3 Find the distance between T(3, –2) and V(8, 3). The distance between T and V is about 7.1 units. Quick Check 11-3

  21. WX= (–2 – (–3))2 + (–1 – 2)2 = 1 + 9 = 10 Replace (x2, y2) with (–2, –1) and (x1, y1) with (–3, 2). XY= (4 – (–2))2 + (0 – (–1)2 Replace (x2, y2) with (4, 0) and (x1, y1) with (–2, –1). 37 Simplify. = 36 + 1 = Simplify. Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11-3 Find the perimeter of WXYZ. The points are W (–3, 2), X (–2, –1), Y (4, 0), Z (1, 5). Use the Distance Formula to find the side lengths. 11-3

  22. YZ= (1 – 4)2 + (5 – 0)2 Replace (x2, y2) with (1, 5) and (x1, y1) with (4, 0). = 9 + 25 = 34 Simplify. Replace (x2, y2) with (–3, 2) and (x1, y1) with (1, 5). ZW= (–3 – 1)2 + (2 – 5)2 = 16 + 9 = 25 = 5 Simplify. Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11-3 (continued) 11-3

  23. perimeter = + + + 520.1 10 37 34 Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11-3 (continued) The perimeter is about 20.1 units. Quick Check 11-3

  24. 1 2 1 2 1 2 1 2 4 + 9 2 y1 + y2 2 x1 + x2 2 , Use the Midpoint Formula. –3 + 2 2 = , Replace (x1, y1) with (4, –3) and (x2, y2) with (9, 2). = , Simplify the numerators. = 6 , – Write the fractions in simplest form. 13 2 –1 2 The coordinates of the midpoint of TV are 6 , – . Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11-3 Find the midpoint of TV. Quick Check 11-3

  25. 9.1; (–2 , – ) 13.6; (1 , 2) 1 2 1 2 1 2 Distance and Midpoint Formulas PRE-ALGEBRA LESSON 11-3 Find the length (to the nearest tenth) and midpoint of each segment with the given endpoints. 1. A(–2, –5) and B(–3, 4) 2.D(–4, 6) and E(7, –2) 3. Find the perimeter of ABC, with coordinates A(–3, 0), B(0, 4), and C(3, 0). 16 11-3

  26. , 5 15 , , = = = = 15 24 24 8 15 5 24 15 8 24 5 8 8 5 Problem Solving Strategy: Write a Proportion PRE-ALGEBRA LESSON 11-4 Use these numbers to write as many proportions as you can: 5, 8, 15, 24 11-4

  27. 1 3 h 5 1 4 8 x 2 7 a 12 20 25 c 35 Problem Solving Strategy: Write a Proportion PRE-ALGEBRA LESSON 11-4 (For help, go to Lesson 6-2.) Solve each proportion. 1. = 2. = 3. =4. = Check Skills You’ll Need 11-4

  28. 1 3 h 5 1 4 8 x 2 7 Problem Solving Strategy: Write a Proportion PRE-ALGEBRA LESSON 11-4 Solutions 1. = 2. = 3 • a = 1 • 12 25 • h = 5 • 20 3a = 12 25h = 100 a = 4 h = 4 3. = 4. = 1 • x = 4 • 8 7 • c = 2 • 35 x = 32 7c = 70 c = 10 a 12 20 25 c 35 11-4

  29. Since the triangles are similar, and you know three lengths, writing and solving a proportion is a good strategy to use. It is helpful to draw the triangles as separate figures. Problem Solving Strategy: Write a Proportion PRE-ALGEBRA LESSON 11-4 At a given time of day, a building of unknown height casts a shadow that is 24 feet long. At the same time of day, a post that is 8 feet tall casts a shadow that is 4 feet long. What is the height x of the building? 11-4

  30. 8 x = Write a proportion. 4 24 Problem Solving Strategy: Write a Proportion PRE-ALGEBRA LESSON 11-4 (continued) Write a proportion using the legs of the similar right triangles. 4x = 24(8) Write cross products. 4x = 192 Simplify. x = 48 Divide each side by 4. Quick Check The height of the building is 48 ft. 11-4

  31. Problem Solving Strategy: Write a Proportion PRE-ALGEBRA LESSON 11-4 Write a proportion and solve. 1. On the blueprints for a rectangular floor, the width of the floor is 6 in. The diagonal distance across the floor is 10 in. If the width of the actual floor is 32 ft, what is the actual diagonal distance across the floor? 2. A right triangle with side lengths 3 cm, 4 cm, and 5 cm is similar to a right triangle with a 20-cm hypotenuse. Find the perimeter of the larger triangle. 3. A 6-ft-tall man standing near a geyser has a shadow 4.5 ft long. The geyser has a shadow 15 ft long. What is the height of the geyser? about 53 ft 48 cm 20 ft 11-4

  32. Special Right Triangles PRE-ALGEBRA LESSON 11-5 One angle measure of a right triangle is 75 degrees. What is the measurement, in degrees, of the other acute angle of the triangle? 15 degrees 11-5

  33. Special Right Triangles PRE-ALGEBRA LESSON 11-5 (For help, go to Lesson 11-2.) Find the missing side of each right triangle. 1. legs: 6 m and 8 m 2. leg: 9 m; hypotenuse: 15 m 3. legs: 27 m and 36 m 4. leg: 48 m; hypotenuse: 60 m Check Skills You’ll Need 11-5

  34. Special Right Triangles PRE-ALGEBRA LESSON 11-5 Solutions 1.c2 = a2 + b22.a2 + b2 = c2 c2 = 62 + 82 92 + b2 = 152 c2 = 100 81 + b2 = 225 c = 100 = 10 m b2 = 144 b = 144 = 12 m 3.c2 = a2 + b24.a2 + b2 = c2 c2 = 272 + 362 482 + b2 = 602 c2 = 2025 2304 + b2 = 3600 c = 2025 = 45 m b2 = 1296 b = 1296 = 36 m 11-5

  35. hypotenuse = leg • 2 Use the 45°-45°-90° relationship. y = 10 • 2 The length of the leg is 10. 14.1 Use a calculator. Special Right Triangles PRE-ALGEBRA LESSON 11-5 Find the length of the hypotenuse in the triangle. The length of the hypotenuse is about 14.1 cm. Quick Check 11-5

  36. hypotenuse = leg • 2 Use the 45°-45°-90° relationship. y = 20 • 2 The length of the leg is 20. 28.3 Use a calculator. Special Right Triangles PRE-ALGEBRA LESSON 11-5 Patrice folds square napkins diagonally to put on a table. The side length of each napkin is 20 in. How long is the diagonal? The diagonal length is about 28.3 in. Quick Check 11-5

  37. hypotenuse = 2 • shorter leg 14 = 2 • bThe length of the hypotenuse is 14. = Divide each side by 2. 7 = bSimplify. longer leg = shorter leg • 3 a = 7 • 3 The length of the shorter leg is 7. a 12.1 Use a calculator. 14 2 2b 2 Special Right Triangles PRE-ALGEBRA LESSON 11-5 Find the missing lengths in the triangle. The length of the shorter leg is 7 ft. The length of the longer leg is about 12.1 ft. Quick Check 11-5

  38. 3 3 in. Special Right Triangles PRE-ALGEBRA LESSON 11-5 Find each missing length. 1. Find the length of the legs of a 45°-45°-90° triangle with a hypotenuse of 4 2 cm. 2. Find the length of the longer leg of a 30°-60°-90° triangle with a hypotenuse of 6 in. 3. Kit folds a bandana diagonally before tying it around her head. The side length of the bandana is 16 in. About how long is the diagonal? 4 cm about 22.6 in. 11-5

  39. Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11-6 A piece of rope 68 in. long is to be cut into two pieces. How long will each piece be if one piece is cut three times longer than the other piece? 17 in. and 51 in. 11-6

  40. Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11-6 (For help, go to Lesson 6-3.) Solve each problem. 1. A 6-ft man casts an 8-ft shadow while a nearby flagpole casts a 20-ft shadow. How tall is the flagpole? 2. When a 12-ft tall building casts a 22-ft shadow, how long is the shadow of a nearby 14-ft tree? Check Skills You’ll Need 11-6

  41. 120 8 308 12 6 8 2 3 12x 12 x 20 12 22 14 x 8x 8 Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11-6 Solutions 1. = 2. = 6 • 20 = 8 • x 22 • 14 = 12 • x 120 = 8x 308 = 12x = = x = 15 ft x = 25 ft 11-6

  42. 3 4 3 5 4 5 opposite hypotenuse sin A = = = adjacent hypotenuse cos A = = = opposite adjacent tan A = = = 12 16 12 20 16 20 Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11-6 Find the sine, cosine, and tangent of A. Quick Check 11-6

  43. sin 18° 0.3090 Scientific calculator: Enter 18 and press the key labeled SIN, COS, or TAN. cos 18° 0.9511 Table: Find 18° in the first column. Look across to find the appropriate ratio. tan 18° 0.3249 Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11-6 Find the trigonometric ratios of 18° using a scientific calculator or the table on page 779. Round to four decimal places. Quick Check 11-6

  44. sin A = Use the sine ratio. opposite hypotenuse sin 40° = Substitute 40° for the angle, 10 for the height, and w for the hypotenuse. 10 sin 40° w = Divide each side by sin 40°. w 15.6 Use a calculator. 10 w Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11-6 The diagram shows a doorstop in the shape of a wedge. What is the length of the hypotenuse of the doorstop? You know the angle and the side opposite the angle. You want to find w, the length of the hypotenuse. w(sin 40°) = 10 Multiply each side by w. Quick Check The hypotenuse is about 15.6 cm long. 11-6

  45. 12 13 5 13 12 5 , , sin 72° 0.9511; cos 72° 0.3090; tan 72° 3.0777 Sine, Cosine, and Tangent Ratios PRE-ALGEBRA LESSON 11-6 Solve. 1. In ABC, AB = 5, AC = 12, and BC = 13. If A is a right angle, find the sine, cosine, and tangent of B. 2. One angle of a right triangle is 35°, and the adjacent leg is 15. a. What is the length of the opposite leg? b. What is the length of the hypotenuse? 3. Find the sine, cosine, and tangent of 72° using a calculator or a table. about 10.5 about 18.3 11-6

  46. Angles of Elevation and Depression PRE-ALGEBRA LESSON 11-7 An airplane flies at an average speed of 275 miles per hour. How far does the airplane fly in 150 minutes? 687.5 miles 11-7

  47. Angles of Elevation and Depression PRE-ALGEBRA LESSON 11-7 (For help, go to Lesson 2-3.) Find each trigonometric ratio. 1. sin 45° 2. cos 32° 3. tan 18° 4. sin 68° 5. cos 88° 6. tan 84° Check Skills You’ll Need 11-7

  48. Angles of Elevation and Depression PRE-ALGEBRA LESSON 11-7 Solutions 1. sin 45° 0.7071 2. cos 32° 0.8480 3. tan 18° 0.3249 4. sin 68° 0.9272 5. cos 88° 0.0349 6. tan 84° 9.5144 11-7

  49. Draw a picture. opposite hypotenuse sin A = Choose an appropriate trigonometric ratio. sin 52° = Substitute. 24 hSimplify. h 30 Angles of Elevation and Depression PRE-ALGEBRA LESSON 11-7 Janine is flying a kite. She lets out 30 yd of string and anchors it to the ground. She determines that the angle of elevation of the kite is 52°. What is the height h of the kite from the ground? 30(sin 52°) = hMultiply each side by 30. Quick Check The kite is about 24 yd from the ground. 11-7

  50. Draw a picture. opposite adjacent Choose an appropriate trigonometric ratio. tan A = Substitute 61 for the angle measure and 30 for the adjacent side. tan 61° = 54 hUse a calculator or a table. h 30 Angles of Elevation and Depression PRE-ALGEBRA LESSON 11-7 Quick Check Greg wants to find the height of a tree. From his position 30 ft from the base of the tree, he sees the top of the tree at an angle of elevation of 61°. Greg’s eyes are 6 ft from the ground. How tall is the tree, to the nearest foot? 30(tan 61°) = hMultiply each side by 30. 54 + 6 = 60 Add 6 to account for the height of Greg’s eyes from the ground. The tree is about 60 ft tall. 11-7

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