1 Secton 8.1 Non-rght Trangles: Law of Snes and Cosnes 519 Secton 8.1 Exercses Solve for the unknown sdes and angles of the trangles shown Assume A s opposte sde a, B s opposte sde b, and C s opposte sde c. Solve each trangle for the unknown sdes and angles f possble. If there s more than one possble soluton, gve both. 9. A43, C 69, b A35, C 73, b A119, a6, b C 113, b10, c B50, a105, b B67, a49, b A43.1, a184., b A36.6, a186., b 4.
2 50 Chapter 8 Solve for the unknown sdes and angles of the trangles shown Assume A s opposte sde a, B s opposte sde b, and C s opposte sde c. Solve each trangle for the unknown sdes and angles f possble. If there s more than one possble soluton, gve both. 1. C 41., a.49, b B58.7, a10.6, c A10, b6, c 7 4. C 115, a18, b 3 5. Fnd the area of a trangle wth sdes length 18, 1, and 3 6. Fnd the area of a trangle wth sdes length 0, 6, To fnd the dstance across a small lake, a surveyor has taken the measurements shown. Fnd the dstance across the lake. 800 ft 900 ft To fnd the dstance between two ctes, a satellte calculates the dstances and angle shown (not to scale). Fnd the dstance between the ctes. 370 km km
3 Secton 8.1 Non-rght Trangles: Law of Snes and Cosnes To determne how far a boat s from shore, two radar statons 500 feet apart determne the angles out to the boat, as shown. Fnd the dstance of the boat from the staton A, and the dstance of the boat from shore. A B 30. The path of a satellte orbtng the earth causes t to pass drectly over two trackng statons A and B, whch are 69 m apart. When the satellte s on one sde of the two statons, the angles of elevaton at A and B are measured to be 86. and 83.9, respectvely. How far s the satellte from staton A and how hgh s the satellte above the ground? A 83.9 B A communcatons tower s located at the top of a steep hll, as shown. The angle of nclnaton of the hll s 67. A guy wre s to be attached to the top of the tower and to the ground, 165 m downhll from the base of the tower. The angle formed by the guy wre s 16. Fnd the length of the cable requred for the guy wre m 3. The roof of a house s at a 0 angle. An 8 foot solar panel s to be mounted on the roof, and should be angled 38 for optmal results. How long does the vertcal support holdng up the back of the panel need to be? 0 8 ft A 17 foot tower s located on a hll that s nclned 38 to the horzontal. A guy wre s to be attached to the top of the tower and anchored at a pont 64 feet downhll from the base of the tower. Fnd the length of wre needed ft 17 ft
4 5 Chapter A 113 foot tower s located on a hll that s nclned 34 to the horzontal. A guy wre s to be attached to the top of the tower and anchored at a pont 98 feet uphll from the base of the tower. Fnd the length of wre needed. 113 ft ft 35. A plot s flyng over a straght hghway. He determnes the angles of depresson to two mleposts, 6.6 km apart, to be 37 and 44, as shown n the fgure. Fnd the dstance of the plane from pont A, and the elevaton of the plane A B 36. A plot s flyng over a straght hghway. He determnes the angles of depresson to two mleposts, 4.3 km apart, to be 3 and 56, as shown n the fgure. Fnd the dstance of the plane from pont A, and the elevaton of the plane A B 37. To estmate the heght of a buldng, two students fnd the angle of elevaton from a pont (at ground level) down the street from the buldng to the top of the buldng s 39. From a pont that s 300 feet closer to the buldng, the angle of elevaton (at ground level) to the top of the buldng s 50. If we assume that the street s level, use ths nformaton to estmate the heght of the buldng. 38. To estmate the heght of a buldng, two students fnd the angle of elevaton from a pont (at ground level) down the street from the buldng to the top of the buldng s 35. From a pont that s 300 feet closer to the buldng, the angle of elevaton (at ground level) to the top of the buldng s 53. If we assume that the street s level, use ths nformaton to estmate the heght of the buldng. 39. A plot fles n a straght path for 1 hour 30 mn. She then makes a course correcton, headng 10 degrees to the rght of her orgnal course, and fles hours n the new drecton. If she mantans a constant speed of 680 mles per hour, how far s she from her startng poston?
5 Secton 8.1 Non-rght Trangles: Law of Snes and Cosnes Two planes leave the same arport at the same tme. One fles at 0 degrees east of north at 500 mles per hour. The second fles at 30 east of south at 600 mles per hour. How far apart are the planes after hours? 41. The four sequental sdes of a quadrlateral have lengths 4.5 cm, 7.9 cm, 9.4 cm, and 1.9 cm. The angle between the two smallest sdes s 117. What s the area of ths quadrlateral? 4. The four sequental sdes of a quadrlateral have lengths 5.7 cm, 7. cm, 9.4 cm, and 1.8 cm. The angle between the two smallest sdes s 106. What s the area of ths quadrlateral? 43. Three crcles wth rad 6, 7, and 8 respectvely, all touch as shown. Fnd the shaded area bounded by the three crcles. 44. A rectangle s nscrbed n a crcle of radus 10 cm as shown. Fnd the shaded area, nsde the crcle but outsde the rectangle. 55
6 54 Chapter 8 Secton 8. Exercses Convert the Polar coordnate to a Cartesan coordnate ,. 6, 3. 4, , , , , 8. 5, 9. 3, 6 10., (3, ) 1. (7,1) Convert the Cartesan coordnate to a Polar coordnate 13. (4,) 14. (8, 8) 15. ( 4, 6) 16. ( 5,1) 17. (3, 5) 18. (6, 5) , ( 4, 7) Convert the Cartesan equaton to a Polar equaton 1. x 3. y 4 3. y 4x 4. y x 4 5. x y 4y 6. x y 3x 7. x y x 8. x y 3y Convert the Polar equaton to a Cartesan equaton r 3sn r 4cos r sn 4 7cos 3. r cos 6 3sn 33. r sec 34. r 3csc 35. r rcos 36. r 4sec csc
7 Secton 8. Polar Coordnates 55 Match each equaton wth one of the graphs shown. r cos r sn r 4 3cos 40. r 3 4cos r 4. r sn A B C D E F Match each equaton wth one of the graphs shown. 43. r log 44. r cos 45. r cos 46. r sn cos 47. r 1 sn3 48. r 1 sn A B C D E F
8 56 Chapter 8 Sketch a graph of the polar equaton r 3cos r sn 51. r 3sn 5. r 4sn4 53. r 5sn3 54. r 4sn5 55. r 3cos 56. r 4cos4 57. r cos 58. r 3 3sn 59. r 1 3sn 60. r 4cos r 6. r 63. r 3 sec, a conchods r, a ltuus r sn tan, a cssod 66. r 1 sn, a hppopede 1 Ths curve was the nspraton for the artwork featured on the cover of ths book.
9 Secton 8.3 Polar Form of Complex Numbers 57 Secton 8.3 Exercses Smplfy each expresson to a sngle complex number Smplfy each expresson to a sngle complex number 3 (5 3 ) (6 ) (3 ) (4 ) 1. 5 (3 ) (5) (4 ) ( 3 ) (4 ) Rewrte each complex number from polar form nto a b form 9. 3e e 31. e 8e e e 7 4 Rewrte each complex number nto polar re form
10 58 Chapter 8 Compute each of the followng, leavng the result n polar e e e 4e re 53. form 6e 3e e 55. 6e 4 e e e e 3 Compute each of the followng, smplfyng the result nto a b form Solve each of the followng equatons for all complex solutons z 66. z z z 1
11 Secton 8.4 Vectors 59 Secton 8.4 Exercses Wrte the vector shown n component form 1.. Gven the vectors shown, sketch u v, u v, and u Wrte each vector below as a combnaton of the vectors u and v from queston # From the gven magntude and drecton n standard poston, wrte the vector n component form. 7. Magntude: 6, Drecton: Magntude: 10, Drecton: Magntude: 8, Drecton: Magntude: 7, Drecton: 305 Fnd the magntude and drecton of the vector 11. 0, , , , 7 15., , , , , , 9 Usng the vectors gven, compute u v, u v, and u 3v 1. u, 3, v 1, 5. u 3,4, v, 1
12 530 Chapter 8 3. A woman leaves home and walks 3 mles west, then mles southwest. How far from home s she, and what drecton must she walk to head drectly home? 4. A boat leaves the marna and sals 6 mles north, then mles northeast. How far from the marna s the boat, and what drecton must t sal to head drectly back to the marna? 5. A person starts walkng from home and walks 4 mles East, mles Southeast, 5 mles South, 4 mles Southwest, and mles East. How far total have they walked? If they walked straght home, how far would they have to walk? 6. A person starts walkng from home and walks 4 mles East, 7 mles Southeast, 6 mles South, 5 mles Southwest, and 3 mles East. How far total have they walked? If they walked straght home, how far would they have to walk? 7. Three forces act on an object: F, 5, F 0,1, F 4, 7 force on the object Fnd the net 8. Three forces act on an object: F,5, F 8,3, F 0, 7 on the object Fnd the net force 9. A person starts walkng from home and walks 3 mles at 0 North of West, then 5 mles at 10 West of South, then 4 mles at 15 North of East. If they walked straght home, how far would they have to walk, and n what drecton? 30. A person starts walkng from home and walks 6 mles at 40 North of East, then mles at 15 East of South, then 5 mles at 30 South of West. If they walked straght home, how far would they have to walk, and n what drecton? 31. An arplane s headng north at an arspeed of 600 km/hr, but there s a wnd blowng from the southwest at 80 km/hr. How many degrees off course wll the plane end up flyng, and what s the plane s speed relatve to the ground? 3. An arplane s headng north at an arspeed of 500 km/hr, but there s a wnd blowng from the northwest at 50 km/hr. How many degrees off course wll the plane end up flyng, and what s the plane s speed relatve to the ground? 33. An arplane needs to head due north, but there s a wnd blowng from the southwest at 60 km/hr. The plane fles wth an arspeed of 550 km/hr. To end up flyng due north, the plot wll need to fly the plane how many degrees west of north?
13 Secton 8.4 Vectors An arplane needs to head due north, but there s a wnd blowng from the northwest at 80 km/hr. The plane fles wth an arspeed of 500 km/hr. To end up flyng due north, the plot wll need to fly the plane how many degrees west of north? 35. As part of a vdeo game, the pont (5, 7) s rotated counterclockwse about the orgn through an angle of 35 degrees. Fnd the new coordnates of ths pont. 36. As part of a vdeo game, the pont (7, 3) s rotated counterclockwse about the orgn through an angle of 40 degrees. Fnd the new coordnates of ths pont. 37. Two chldren are throwng a ball back-and-forth straght across the back seat of a car. The ball s beng thrown 10 mph relatve to the car, and the car s travellng 5 mph down the road. If one chld doesn't catch the ball and t fles out the wndow, n what drecton does the ball fly (gnorng wnd resstance)? 38. Two chldren are throwng a ball back-and-forth straght across the back seat of a car. The ball s beng thrown 8 mph relatve to the car, and the car s travellng 45 mph down the road. If one chld doesn't catch the ball and t fles out the wndow, n what drecton does the ball fly (gnorng wnd resstance)?
14 53 Chapter 8 Secton 8.5 Exercses Match each of the equatons wth one of the graphs below. xt t x t t1 1.. y t t 1 y t t 4. x t y t sn( t) 4cos( t) 5. x t t y t 3 t x t y t x t y t 4sn cos t t t 3t A B C D E F From each par of graphs n the x-t and y-t planes shown, sketch a graph n the x-y plane
15 Secton 8.5 Parametrc Equatons 533 From each graph n the x-y plane shown, sketch a graph of the parameter functons n the x-t and y-t planes Sketch the parametrc equaton for t x t 1t y t t t 3 t x t y t Elmnate the parameter t to rewrte the parametrc equaton as a Cartesan equaton x t 5 t xt 6 3t y t 8t y t 10 t x t t1 y t 3 t x t y t e t 15t y t t 3 x t t t x t y t x t y t e e t 6t 4cos 5sn t t x t 3t1 16. y t t xt 4log t 18. y t 3t 4 xt t t 0. y t t 5 xt t. 10 y t t xt 3sn t 4. y t 6cos t
16 534 Chapter 8 Parameterze (wrte a parametrc equaton for) each Cartesan equaton y x 3x 3 y x sn x xy3logy y 8. x y y y 9. x y x y Parameterze the graphs shown Parameterze the lne from ( 1,5) to (,3) so that the lne s at ( 1,5) at t = 0, and at (, 3) at t = Parameterze the lne from (4,1) to (6, ) so that the lne s at (4,1) at t = 0, and at (6, ) at t = 1.
17 Secton 8.5 Parametrc Equatons 535 The graphs below are created by parameterc equatons of the form Fnd the values of a, b, c, and d to acheve each graph. cos sn x t a bt y t c dt An object s thrown n the ar wth vertcal velocty 0 ft/s and horzontal velocty 15 ft/s. The object s heght can be descrbed by the equaton y t 16t 0t, whle the object moves horzontally wth constant velocty 15 ft/s. Wrte parametrc equatons for the object s poston, then elmnate tme to wrte heght as a functon of horzontal poston. 4. A skateboarder rdng at a constant 9 ft/s throws a ball n the ar, the heght of whch can be descrbed by the equaton y t 16t 10t 5. Wrte parametrc equatons for the ball s poston, then elmnate tme to wrte heght as a functon of horzontal poston.
18 536 Chapter A carnval rde has a large rotatng arm wth dameter 40 feet centered 35 feet off the ground. At each end of the large arm are two smaller rotatng arms wth dameter 16 feet each. The larger arm rotates once every 5 seconds, whle the smaller arms rotate once every seconds. If you board the rde when the pont P s closest to the ground, wrte a parametrc equaton for your poston over tme. P 44. A hypocyclod s a shape s the shape generated by trackng a fxed pont on a small crcle as t rolls around the nsde of a larger crcle. If the smaller crcle has radus 1 and the large crcle has radus 6, fnd parametrc equatons for the poston of the pont P as the smaller wheel rolls n the drecton ndcated. P
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Student Name : MOUNT SAINT JOSEPH MILPERRA MOUNT SAINT JOSEPH I YEAR 8 MATHEMATICS 17th November 2006 70 MINUTES (plus 5 mnutes readng tme)... :... INSTRUCTIONS: > WRITE IN BLUE OR BLACK PEN ONLY > ALL
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Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts
CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The
An Algorthm to Solve the Inverse Knematcs Problem of a Robotc Manpulator Based on Rotaton Vectors Mohamad Z. Al-az*, Mazn Z. Othman**, and Baker B. Al-Bahr* *AL-Nahran Unversty, Computer Eng. Dep., Baghdad,
PHYS 050 Sprng 06 Name: Test 3 Aprl 7, 06 INSTRUCTIONS: a) No books or notes are permtted. b) You may use a calculator. c) You must solve all problems begnnng wth the equatons on the Inormaton Sheet provded
Chapter 9 (a) The magntude of the magnetc feld due to the current n the wre, at a pont a dstance r from the wre, s gven by r Wth r = ft = 6 m, we have c4 T m AhbAg 6 33 T 33 T m b g (b) Ths s about one-sxth
(ISO/IEC - 700-00 Certfed) WINTER 07 EXAMINATION Model wer ject Code: Important Instructons to Eamners: ) The answers should be eamned by key words and not as word-to-word as gven n the model answer scheme.
PHYS 1443 Secton 004 Lecture #1 Thursday, Oct., 014 Work-Knetc Energy Theorem Work under rcton Potental Energy and the Conservatve Force Gravtatonal Potental Energy Elastc Potental Energy Conservaton o
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi IN THIS CHAPTER, you wll learn how to solve problems
Chapter 3 Student Lecture Notes 3-1 Chapter 3 Descrbng Data Usng Numercal Measures Fall 2006 Fundamentals of Busness Statstcs 1 Chapter Goals To establsh the usefulness of summary measures of data. The
PHYS 1101 Practce problem set 1, Chapter 3: 1,, 4, 57, 61, 83 Chapter 33: 7, 1, 3, 38, 44, 49, 76 3.1. Vsualze: Please reer to Fgure Ex3.1. Solve: Because B s n the same drecton as the ntegraton path s
Modelng curves Types of Curves Graphs: y = ax+b, y = sn(x) Implct ax + by + c = 0, x 2 +y 2 =r 2 Parametrc: x = ax + bxt x = cos t y = ay + byt y = snt Parametrc are the most common mplct are also used,
A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular