Chapter 8 Vectors and Parametric Equations

Transcription

1 Chapter 8 Vectors and Parametrc Eqatons 8- Geometrc Vectors Page 9 Check for Understandng. Sample answer: a a b b Draw a. Then draw b so that ts ntal pont (tp) s on the termnal pont (tal) of a. Draw a dashed lne from the ntal pont of a to the termnal pont of b. The dashed lne s the resltant.. Sample answer: A vector has magntde and drecton. A lne segment has onl length. A vector can be represented b a drected lne segment.. Sample answer: the veloctes of an arplane and a wnd crrent. No, the are oppostes. -. Answers ma var slghtl... cm, 6..9 cm, 7.. cm, cm,. h.9 cos v.9 sn h.66 cm v.8 cm a. b. Use the Pthagorean Theorem. c a b c () () c, c, or abot. m/s Pages 9 9 Eercses..6 cm, 8.. cm, 6.. cm, 7.. cm, 8-. Answers ma var slghtl. 8. r s cm.9 cm. cm r s 9..6 cm 7. cm, 7 cm, 9. s. cm s t t. cm,. s..6 cm, 9.8 cm, 9.8 cm s.9 cm.9 cm, Chapter 8

2 . 6. t r. cm r. cm r t.. cm, r t.9 cm r r 7.. cm, r s. cm s r.9 cm,. t. cm, 8. s r. cm. cm, 8. s. cm,. 8. cm, r s. cm 8. cm 8 s s 8. cm, 8 9. Draw to scale: t s. cm,. Draw to scale: t. cm s r. cm r t 7.7 cm, 7. h.6 cos 8 v.6 sn 8 h.6 cm v. cm. h. cos v. sn h.99 cm v.99 cm. h. cos v. sn h. cm v. cm r r t s.7 cm Chapter 8 6

3 . h. cos v. sn h.8 cm v. cm. c a b c (9.) (.) c 9.68 c or abot.7 m 6. The dfference of the vectors; sample answer: The other dagonal wold be the sm of one of the vectors and the opposte of the other vector, so t wold be the dfference. 7. Yes; sample answer: s. h 7 cos v 7 sn h 6 mph v mph. It s tre when k or when a s the ero vector.. c a b c () () c or abot 7 lb. a 6 6 b 6. cm r 8. 6 N s r s s r N r N a b eqlateral trangle a b lb. The orgn s not n the nteror of the acte angle. d d d or ) ( d or ( ) ( ) ( ) s 6. csc v cos v tan v s cos v c n v s n v sn v c os v cos v n v os v n where n s an nteger 8. 6 N, north of east 9. Sometmes; a b a b a b..7, scl b [, ] scl, for 9. tan 8 9. b b. tan 8 9 b 9.9 cm sn 8 9 h a b a a b. a b..7 h sn 8 9 h.8 cm b a. v. sn h. cos v.8 N h.9 N b. h. cos 78 v. sn 78 h. N v.7 N 7 Chapter 8

4 . v o volme of orgnal bo v n volme of new bo v o o w o h o (w ) w w (w )w w w v n n w n h n (w ) (w ) (w ) (w w )(w ) w 8w w w 8w w 6 w w o o w or 6 h o w or So, the dmensons of the orgnal bo are ft ft 6 ft. g() ( )( ) vertcal: As approaches and, the epresson approaches or. So, and are vertcal asmptotes. horontal: As ncreases postvel or negatvel, the epresson approaches. So, s a horontal asmptote.. Let,, and be consectve odd ntergers. ( ) 8 The correct answer. 8- Algebrac Vectors Pages Check for Understandng. Sample answer: a 8, 6, b 6, 8; eqal vectors have the same magntde and drecton.. Use XY ( ) ( ) and replace the vales for and. (, 6), (, ) XY [ )] ( [6)] ( (8) () 6 or 68. Jacq s correct. The representaton s ncorrect.,, s not eqal to, (),. The correct epresson s j.. MP, () or, MP () () or nts. MP, 6 or, MP () ) ( 6 nts 6. MP (9), or, MP () () nts 7. t v,,, () or, 8. t v,,,,, () or, 9. t 6 v, 6,, 6 8, 8, 6 () or,. t 8 8, 8(), 8() or 8,. 8, 6 86) ( or 8 6j. 7, (7) ) ( 7 7 j. Let T represent the force Terrell eerts. Let W represent the force Mr. Walker eerts. T cos 6 W 6 cos 69.. T sn 6 W 6 sn T 69., 6., W., 6.8 T W 6.6, 96. T W (6. 6) ) ( N Pages Eercses. YZ, 8 or, 6 YZ () 6 or Chapter 8 8

5 . YZ (), 7 or, YZ ) ( 6. YZ (), or, YZ ) ( 7. YZ, or, 7 YZ () ) ( YZ, or, YZ () 8 or 9. YZ (), 9 or, 7 YZ 7 7. YZ 7, 6 or, 6 YZ 6 or. YZ, () or 9, 9 YZ or 9. AB 6, () or, AB ) ( 69 or. a b c 6,, 8 6 (), 8 or,. a b c 6,, 8, 6, 8 (), 6 8 or 8,. a b c 6,, 8 6, 8, 6 6 (8), 6 or, 9 6. a b c 6,, 8, 6, (), 6 or, 7. a b c 6,, 8 6, 6, 6 (6), or, 9 8. a b c 6,, 8 6, 8, 6 6 (8), (6) or, 9. a b 6, 6, or 8, 9. a c, 8 (), 8 or,. a 6b c 66,, 8 6, 8 6, 6 (6), 8 or,. a.b.c.6,., 8.,..8, 9.6. (.8),. 9.6 or 7., 8.. a (b c ) (6,, 8) (, 6, ) ( (), 6 ), or,. a (b c ) b 6,, 8 6, 8, 9 (, 8, 8 (), 9 8 or,. m.n, [, 8,. [ (), 8 (.,. 6., or j 7., ) ( j 8. 6, (6) ) ( 7 6 j 9.., (.) 6. or.. j., () 7 j. 6, (6) ) ( or 6 j. ST (9), or, j. Stdent needs to show that ( v v ) v v ( v v ) ( v v ) v [a, b c, d] e, f a c, b d e, f a c e, b d f a c e, b d f a, b c e, d f a, b [c, d e, f] v ( v v ) a. N F b. sn F F sn N F 9 Chapter 8

6 a. Shore b. sn V k V k sn mph 6a. Snce QR ST, QR ST. So, the are oppostes. 6b. QR and ST have the same magntde, bt opposte drecton. So, the are parallel. Qadrlateral QRST s a parallelogram. 7a. t d r m m/ s V k or s 7b. d rt (. m/s)( s) or m 7c. V B V C.. 6 or abot. m/s 8. cos v ( ) ( ) v cos v v ) V sn v ( v ( ) v sn v 9. PQ 8, (7), PQ () RS 7 8, (7), 7 RS () 7 none. d A B C A B () 7() Srfer V s d 7) ( d or abot. 8. sn sn ( ) sn cos cos sn 6. A sn (k c) A: A 7 A 7 or 7 k: k k 8 c c: 6 k. Let a, b 6, C 6. c 6 ()(6) cos 6. c c P a b c 6 ft s (a b c) s () or 77 k s(s a)(s b)(s c) k 77(7 7 )(7 7 6)(7 7 ) k 7,,766, 79 k 86,7 sq ft. Sample answer:. 6. f() r 9 f() r [, ] scl b [, ] scl ma: (, ), mn: (.67,.8) f() f(), 99,97, 997, 97 7,,,,,, An pper bond s. Alower bond s. as, as Ths statement s tre regardless of the vale of, so t s tre for all real vales of. The correct choce s A. c 6 8 c 8 7 sn (8 8 ) Chapter 8

7 8- Vectors n Three-Dmensonal Space Pages Check for Understandng. Sample answer: sketch a coordnate sstem wth the -aes on the horontal, and the -as pontng p. Then, vector s two nts along the -as, vector j s three nts along the -as, and vector k s for nts along the -as. Draw broken lnes to represent three planes.. Sample answer: To fnd the components of the vector, o wll need the drecton (angle) wth the horontal as. Usng trgonometr, o can obtan the components of the vector.. Sample answer: Nether s correct. The sgn for the j -term mst be the same (), and the coeffcent for the k -term s, so the correct wa to epress the vector as a sm of nt vectors s j.. G(,, 7) OG ) ( RS (), 9, 8 or,, RS ) ( 6 or 9 6. RS, 7, () or 7,, RS 7 ) ( 7 or 9 7. a f g,, 8, 9,, 9,, 9,, 9 9, () or 6,, 8. a g f, 9,,, 8 6, 8,,, 6, 8 (), () or,, 8 9. EF 6 (), 6 (), 6,, j k. EF (), 7, (9),, j k.,,,97, 7 N Pages.. Eercses B(,,) OB ) ( 6 B(7,, ) OB B(,,) OB (). TM,, or,, 8 TM ) ( (8) 8 or 9 6. TM (),, 7 or,, TM () () 7 or 7. TM,, or,, TM ) ( () 8. TM, (), 6 or, 6, TM () 6 () 68 or 7 9. TM (), 8, 6 or, 9, 9 TM 9) ( (9) 7 or 9 Chapter 8

8 . TM, 6, or,, 6 TM ) ( (6). CJ (),, or, 8, ) CJ 8) ( ) ( 76 or 69. 6w 6, 6,,,, 6, 6 6,, 8 8, 6,. v w,,, 6,,,,,, 6, 6,, 8 6, 7,. v w,,, 6,, 9,, 6,, 8,. v w,,, 6,,,, 9,,, 6,, 8 6,, 6..7v. w.7,,., 6,,.,.7.,.,..,.7,. 7. w, 6,,, 8,, (,,,, 8 8. f g h,.,,, 6 6,,,, 6,, ,,, 6, 6 9. LB, 6, 7 or, 8, 8j k. LB (6),, or,, j k. LB 7 9, 7, () or,, 9 j 9k. LB 8, 7, 6 or,, j k. LB 8 (),, () or 7,, 6 7 j 6 k. LB 6 (9),, () or, 7, 7j. G G ( ) ( ) ( ) ( ) ( ) ( ) G G becase ( ) ( ) for all real nmbers and. 6. If m m, m, m, then m (m ) ) (m (m. ) If m m, m, m, then m (m ) m ( ) ) (m. Snce m (m ), m (m ), and m (m ), m m. 7.,, 6,, F O 9,, 9 F O F 9,, 9 or 9,, 9 8. m (,, ) (,, 6 ) (6, 8, 8) (,, ) 9a. OK,, or,, j 9b. TK,, or,,. c b a c,,,, c,, a. 6 O 6 6 b. Fnd dstance between (,, ) and (,, ). d ( ) ( ) ( ) 67 or abot 6 feet c. sn v 6 v sn 6 7 v.. AB ( ) () ( ) or BC ( ) 6 6 or.69 AC ( ) or. No, the dstances between the ponts are not eqal. A and B are nts apart, B and C are.69 nts apart, and A and C are. nts apart..,, (),, 7 Chapter 8

9 . AB, or 8, CD d, d or d, d AB CD 8, d, d D (8, ) sn X. cos X cot X sn X cos X cos X sn X 6. cos v sn X cos X sn X c os X sn X cot X cot X cot X cot X cot X sn v cos v sn v 9 sn v 9 sn v 7. 6 sn v ampltde 6 or 6 perod k rad or rev 8. 6 m n mn rev 6 sec 8 radans per second 9. Yes, becase sbstttng 7 for and for reslts n the neqalt 8 whch s tre. (7) (7) 8. So, A, C, and D are not correct. So, B s not correct. The correct choce s E. 8- Perpendclar Vectors Pages 8 9 Check for Understandng. Sample answer: Vector v w s the negatve of vector w v. j k v w 6 7j k j k j k v w 6 7j k j k j k. a a a a a a a a a a a a a a a a a a a a (a a a a ) (a a a a )j j k (a a a a ) k j k,,. Sample answer: No, becase a vector cannot be perpendclar to tself..,, 7 () (7), no. 8,., 8 8(.) (8) 6 6, es 6., 9, 8,, () 9() 8() 8 6, no j k 7.,,,, j k or,,, es,,,, () () ()(),,,, () () ()() 6 j k j k 8. 6,,,, j 6 k j k or 8,,, es 8,, 6,, 8(6) ()() ()() 8 8 8,,,, 9 8() ()() ()(9) 8 9. Sample answer: Let T(,, ), U(,, ), and V(,, ) TU,, UV,, TU UV j k 8j k or, 8, j k Chapter 8

10 . AB (.6,,.) (,, ).6,,. F,, T AB F j k j k. j k T.8 ( ).8 foot-ponds.6. Pages 9 Eercses..8 6, (6) 8(), es.,, () (), no.,, 6 () ()(6) 6, no. 7,, 7() (), no. 8, (, 8() () 6 6, no 6., 9, 6, 7, (6) 9(7) ()() 6, no 7.,,, 8, () (8) () , no 8.,, 8 6,, (6) () 8() 6 6, es 9. 7,,, 8, 7() ()(8) () 6 9, no. a b, 8,, es b c 8,, 8, no a c,, 9, no j k.,,,, j k j k or,,, es,,,, () () ()(),,,, () () ()() j k.,,,, j k 6j 9k or, 6, 9, es, 6, 9,, ()() (6)() 9() 7 87, 6, 9,, ()() (6)() 9() j k.,,,, j k j k or,,, es,,,, () () (),,,, () () () j k.,,,, j 6k or,, 6, es,, 6,, () () 6() 6,, 6,, () () 6() j k j k.,,,, 8 8j 6k or 8, 8, 6, es 8, 8, 6,, 8() 8() 6() 8 8, 8, 6,, 8() 8() 6() j k Chapter 8

11 j k 6.,, 7,, j k or,,, es,,,, () () () 8 8,, 7,, (7) () () 7. Sample answer: Let v v, v, v and v v, v, v j k j k v (v ) v v v v v v v v v v v v v v v v v v j k j k 8. a (b c ) a a a (b c ) (b c ) (b c ) a (b c ) a (b c ) a (b c ) a (b c ) a [a (b k (b (b c ) a (b c c ) )] [a (b c ) a (b c )]j [a (b c ) a (b c )]k [(a b a c ) (a b a c )] [(a b a c ) (a b a c )]j [(a b a c ) (a b a c )]k [(a b a b ) (a c a c )] [(a b a b ) (a c a c )]j [(a b a b ) (a c a c )]k (a b a b ) (a c a c ) (a b a b )j (a c a c ) j (a b a b )k (a c a c )k [(a b a b ) (a b a b )j (a b a b )k ] [(a c a c ) (a c a c )j (a c a c )k ] a b a a b b a j a b b a k b j k j a b a b a b a b a b a b j k a a a a j a a k c c c c c c ( a b ) ( a c ) 9. Sample answer: Let T(,, ), U(,, ), and V(,, ) TU,, UV,, j k TU UV j k 7j k or, 7,. Sample answer: Let T(,, ), U(,, ), and V(,, ). TU,, UV 8,, j k TU UV 8 j k 8 8 j 6k or,, 6. Sample answer: Let T(,, ), U(,, ), and V(,, ). TU,, UV,, j k TU UV j k j k or,,. The epresson s false. m n and n m have the same magntde bt are opposte n drecton. a. forearm b. T AB F AB. cos ( ),,. sn ( ).(),,. F,, 6 j k AB F.. 6 j k T AB F or abot N-m j k. 8j k A (8) () j k 6 N. m elbow a. o,, 6 c 9, 8, b. o c (9) (8) 6() $, Chapter 8

12 6a. F 6b. W F d cos v W cos W 9 ft-lb 7a. X,, or,, ) Y,, or,, j k X Y j k 8 7j 9k or 8, 7, 9 7b. The cross prodct of two vectors s alwas a vector perpendclar to the two vectors and the plane n whch the le. 8a. v p (q r ) j k q r j k j k or,, p (q r ),,,, () () ()() or nts 8b. () or nts The are the same. 9. Need (kv w ). [k,, ], [k, k, ], k, k, (k ) (k ) k k 9k 9 k 9 9. BA a b a b cos v (a b ) (a b ) a a b b a b a b cos v (a b ) (a b ) a a b b a a b b cos v a a b b a a b b a a b b a b a b cos v a b a b a b a b cos v a b a b a a b b cos v a b a b a b cos v a b a b cos v. AB,, () or,,. D(8, ) E(, ) DE 8, ) or 8, DE (8) ) ( A B or p nts 7 sn f 7 7 cos f 7 7 tan f f. A 6, b, and c 6 a b c bc cos A a 6 ()(6) cos 6 a 8.9 sn s n B n 6 B sn s 8. 9 B 9. or 9 C C 8.9 or 8. tan 7 h cos 7 tan 7 h cos 7. h;. m.7 m ( ) or ( ) 9 So 6 8 The correct choce s B. Chapter 8 6

13 Page. Md-Chapter Q 8-B Graphng Calclator Eploraton: Fndng Cross Prodcts. cm 6 F. cos 6 F. sn 6.6 cm.7 cm..7 cm F 7 cos F 7 sn. mm. mm. CD (9), or, CD ) ( or. CD, 7 7, () or,, CD. r t s 6,, 6, 8, 6 6 8, 6 or, 8 6. r v,, 8, 9,, 9,, 9,, 9 9, () or 6,, 7., 6, () 6() ; es 8.,,,, () ()() () 8 ; no 9.,,,, j k j k j 7k or,, 7, es,, 7,, ()() () (7)(),, 7,, ()() () (7)() 7. Let X(,, ) and Y(7,, 6). XY (7 ) ( ) (6) or abot 6.7 m Page. 9,. 68, 96, 76.,,.,,.,, 7 6.,, 7. 6, 6, v 6 6 ) ( 6 8. v,, v ) ( ) ( 7 9. Sample answer: Insert the followng lnes after the last lne of the gven program. :Dsp LENGTH IS :Dsp ((BZ CY) (CX AZ) (AY BX) ) 8- Applcatons wth Vectors Pages 6 7 Check for Understandng. Sample answer: Pshng an object p the slope reqres less force becase the component of the weght of the object n the drecton of moton s mg sn v. Ths s less than the weght mg of the object, whch s the force that mst be eerted to lft the object straght p.. The tenson ncreases.. Sample answer: Forces are n eqlbrm f the resltant force s O.. Crrent knots. F F (7 cos ) (7 sn )j F F ( os 7 c ) 7 (sn ).9 N tan v 7 7 sn 7 cos v tan 7 sn 7 cos 6. F F j F F.8 N tan v or v tan horontal 8 cos.79 N vertcal 8 sn.7 N 7 Chapter 8

14 8. F ( cos 9 ) ( sn 9 )j or j F ( cos 6 ) ( sn 6 )j or j F F ( ) 7 N tan v or v tan 7 Aforce wth magntde 7 N and drecton 7 8 or wll prodce eqlbrm. 9a. mph 9 lb lb 9b. If v s the angle between the resltant path of the ferr and the lne between the landngs, then sn v or. So v sn, or abot 9.. Pages 7 9 Eercses... N mph. F F 9j F F N tan v 9 or v tan v 6 v ( cos ) ( sn )j or j v v 9 () mph tan v 9 or 8 8 Wnd 7 6 mph v tan Apostve vale for v s abot... v ( cos 6 ) ( sn 6 )j or j v ( cos ) ( sn )j or j v v () 99.9 km/h Snce tan v s ndefned and the vertcal component s postve, v The force mst be at least as great as the component of the weght of the object n the drecton of the ramp. Ths s sn, or abot 7.6 lb. 7. F F ( cos ) ( sn )j F F ( os c ) (sn ) 9.87 N tan v sn cos sn cos v tan 6.7. F (7 cos ) (7 sn )j or j F ( cos ) ( sn )j or j F (6 cos ) (6 sn )j or j tan v sn sn cos cos v tan 7. F F F ( ) ( ) 8.6 lb. F ( cos 6 ) ( sn 6 )j F v tan.6 8. F w sn v. 7 sn v. sn v 7. sn 7 v v 9. F ( cos ) ( sn )j F ( cos ) ( sn )j F F ( os c cos ) n ( s sn ). lb sn sn tan v cos cos or..j ( cos ) ( sn )j or..j F F ) ( 9.8 N Snce tan v s ndefned and the vertcal component s postve, v 9. A force wth magntde 9.8 N and drecton 9 8 or 7 wll prodce eqlbrm. Chapter 8 8

15 . a g sn sn.6 ft/s. F (6 cos ) (6 sn )j F (8 cos (8 sn )j F F (6 s co s 8 co) 6 ( sn 8 ) sn 9.9 N 6 sn 8 sn tan v 6 cos 8 cos v tan 6 sn 8 sn 6 cos 8 cos 6.7 or. west of soth a. lb 6 lb 7 b. F 7 F ( cos 6 ) ( sn 6 )j F F F (7 os c 6 ) cos n ( s 6 ) sn. lb sn 6 sn tan v 7 cos 6 cos v tan sn 6 sn 7 cos 6 cos 8.7 or 8.7 soth of west Snce F F F, the vectors are not n eqlbrm.. W F d [(6 cos ) (6 sn )j ] (6 cos )() (6 sn )(),,69 N-m 6a. Sample answer: The horontal forward force s F cos v. Yo can ncrease the horontal forward force b decreasng the angle v between the handle and the lawn. 6b. Sample answer: Pshng the lawnmower at a lower angle ma case back pan. 7a. tan v 8 or 6 v tan 6 9. soth of east 7b. s 8 8. mph 8. F cos v cos 9.6 N 9. F cos 7. F cos 6. F sn 7. F sn 6. The frst eqaton gves F co s 7. cos 6. F. Sbsttte nto the second eqaton. cos 7. sn 6. cos 6. 7 lb F sn 7. F F 76 lb F co s 7. cos 6. F 76 lb. Sample answer: Method b s better. Let F be the force eerted b the tractor, T be the tenson n the two halves of the rope, and v be the angle between the orgnal lne of the rope and half of the rope after t s plled. At eqlbrm, F T sn v F, or T s n. v So, f v, the force appled to the stmp sng method b s greater than the force eerted b the tractor.. Let T be the tenson n each towlne and sppose the as of the shp s the vertcal drecton. T sn 7 6 T s 6 n 7 9. tons. Let T be the tenson n each wre. The halves of the wre make angles of and wth the horontal. T sn T sn T T T lb. v 9() () () The vectors are not perpendclar snce v.. AB, (), 8, 6,. d v g sn v cos v sn 6 cos 6 9. ft 6. Sample answer: A plot of the data sggests a qadratc fncton. Performng a qadratc regresson and rondng the coeffcents gves b. p. b p.66 b p b..... O. sn 7. cos 7. tan 6. (.,.) (.,.) (.,.6) (.,.) The vertces are at (.,.), (.,.), (.,.) and (.,.6). cost fncton C(p, b) 9p b ( p b) 8p 8b C(.,.) 8(.) 8(.) or 76 C(.,.) 8(.) 8(.) or 8.8 C(.,.) 8(.) 8(.) or C(.,.6) 8(.) 8(.6) or 9.8 The mnmm cost s $76, sng % beef and % pork. p... F (sn 7. cos 7. tan 6. ) 9 Chapter 8

16 8. * *() ( ) [() ()] 6 () 8 The correct choce s A. Vectors and Parametrc Eqatons Pages Check for Understandng. When t, and. When t, 7 and. The graph s a lne throgh (, ) and (7, ).. Sample answer: For ever sngle nt ncrement of t, ncreases nt and ncreases nts. Then, the parametrc eqatons of the lne are t, 6 t.. When t, and, so the lne passes throgh (, ). When t, and, so the lne passes throgh (, ), ts -ntercept. The slope of the lne s or.. (), t, 8, t, 8 t 8t t 8t., t7, 7t t 7t t t t t t 8. t 9. 9t t 9 t t t. 8-6 t 9 9 t 6 a. recever: t t t O defensve plaer:.9t.7t.9t.7t b. t t t When t, the coordnates of the defensve plaer are (.9(),.7()) or (.,.), so the defensve plaer has not et caght the recever. Eercses., 7 t, t 7 t t 7. (), t6,, t6, 6t t 6t t. (6), t, 6, t, 6 t t 6 t t., t7, 7t t 7t t 6., t,, t, t t t 7., () t,, t, t t t t 8. t t t t t 9t. t t. t t. t t The slope s. ( ) 6 t t 6 Chapter 8 6

17 . t t t 7. t t t t 9. t t t t 7. Regardless of the vale of t, s alwas 8, so the parametrc eqatons represent the vertcal lne wth eqaton 8. a., () t, 7, t, 7 b. t 7t t 7t c. t t. t 7t [, ] Tstep [, ] Xscl [, ] Yscl 6. 7 t 7 t t t ( ) 6 8. t 8 8 t t t. [, ] Tstep [, ] Xscl [, ] Yscl. [, ] Tstep [, ] Xscl [, ] Yscl a. t and 7t If t, then and, so the part of the lne to the rght of pont (, ) s obtaned. b. t t t 6. cos t sn t cos t and sn t, so the graph s the segment of the lne wth eqaton from (, ) to (, ). (, ) 6 Chapter 8

18 7a. target drone: ()t t t mssle: t t 7b. t t t t When t, the mssle has a -coordnate of, not, so t does not ntercept the drone. 8a. Ceres: t, t, t Pallas: 7 t, 6 t, t 8b. Addng the eqatons for and for Ceres gves. Sbtractng the eqatons for and for Pallas reslts n. The solton of ths sstem s and. Elmnatng t from the eqatons for and reslts n the sstem 7, whch has solton and. Hence, the paths cross at (,, ). 8c. t v t 7 t v t Ceres s at (,, ) when t bt Pallas s at (,, ) when t. The asterods wll not collde. 9. The lne s parallel to the vector,, 8 or,, 9. The vector eqaton of the lne s,, t,, 9 or,, t,, 9. t t t t 9t 9t. v ( cos ) ( sn )j v ( cos ) ( sn )j v v ( os c ) cos ( sn sn ) 6. km/h sn sn tan v cos cos v tan sn sn cos cos 7 or 7 soth of east.,, () () Snce the nner prodct s not, the vectors are not perpendclar.. Snce A 9, a b, and a b sn A, no solton ests.. A graphng calclator ndcates that there s one real ero and that t s close to. f(), so the ero s eactl... The slope s. [ ()] 6. The lnear veloct of the belt arond the larger plle s ( rpm) 9 n./rev 8 n./mn. The lnear veloct arond the smaller plle mst be the same, so ts anglar veloct s (8 n./mn) rev n. 8 rpm. The correct choce s D. 8-6B Graphng Calclator Eploraton: Modelng wth Parametrc Eqatons Page t 8.(t.8) 8.7t 8.t t.789 t hr or.7 mn. d rt m. The tme for plane to fl mles s 8..7 The tme for plane s Sppose the speed of plane s ncreased b a mph a 8.7 a a mph Modelng Moton Usng Parametrc Eqatons Page Check for Understandng. Sample answer: a rocket lanched at 9 to the horontal; tp-off n basketball. Eqal magntde wth opposte drecton.. The greater the angle of the head of the golf clb, the greater the angle of ntal veloct of the ball.. v v sn v. v v cos v sn cos. ft/s.86 m/s 6. v v cos v v v sn v cos sn 8.6 ft/s.8 ft/s 7. v v cos v v v sn v 7. cos 7. sn 7. m/s.7 m/s Chapter 8 6

19 8 ft h 6 8a. mph mle s ft/s t v cos v t() cos t t v sn v gt h t() sn ()t 6t 8b. Sample graph: Heght (feet) 6 Horontal Dstance (feet) 8c. 6t 6t t 6 t 6 t.8 s 8d. t (.8) 6 ft Pages Eercses 9. v v cos v v v sn v 6 cos 6 6 sn 6. ft/s 6.9 ft/s. v v cos v v v sn v 7 cos.7 7 sn m/s 8.7 m/s. v v cos v v v sn v cos sn ft/s 8.96 ft/s. v v cos v v v sn v 7 cos 8 7 sn 8. ft/s 7.98 ft/s. v v cos v v v sn v 69 cos 7 69 sn 7. d/s. d/s. v v cos v v v sn v 6 cos 9 6 sn 9.9 km/h.98 km/h a. t v cos v t v sn v gt 7t cos 7t sn 6t b. 7t sn 6t t(7 sn 6t) 7 sn 6t 7 sn 6t 7 s n 6 t 7t cos 7 7 s n 6 cos 899. ft or d 6. To fnd the tme the projectle stas n the ar, set and solve for t. t v sn v gt t( v sn v gt) v sn v gt v sn v gt v sn v g t The greater the angle, the greater the tme the projectle stas n the ar. To fnd the horontal dstance covered, sbsttte the epresson for t n the eqaton for. t v cos v v sn v g v g sn v g v cos v As the angle ncreases from to, the horontal dstance ncreases. As the angle ncreases from to 9, the horontal dstance decreases. 7a. when t 7 7 v sn 78 ()7 7 v sn v sn 78 8 v 8 7s n 78 v 8. ft/s 7b. t v cos v d (7)(8.) cos 78 7 d 8. t v cos v t v cos v t v sn v gt v sn v g v v cos v cos v tan v v gcos v The presence of the -term (de to the force of gravt) means that s a qadratc fncton of. Therefore, the path of a projectle s a parabolc arc. 9. To fnd the tme the projectle stas n the ar f the ntal veloct s v, set and solve for t. t v sn v gt t v sn v gt v sn v gt v sn v gt v sn v g t t To fnd the range, sbsttte ths epresson for t n the eqaton for. t v cos v v sn v g v sn v g v cos v 6 Chapter 8

20 If the magntde of the ntal veloct s dobled to v, the range becomes ( v ) sn v g or v sn v g. The projectle wll travel for tmes as far. m h 6 a. 8 km/h km s. m/s t v cos v. t cos t v sn v gt.t sn (9.8)t.t sn.9t The negatve coeffcent n the t-term n the eqaton for ndcates that the arcraft s descendng. The negatve coeffcent n the eqaton for s arbtrar. b..t sn.9t.(.) sn.9 (.). The arcraft has descended abot. m. c.. m. s 69 m/s or 69 m/s km m 6 s h 8 ft h 6 s 68. km/h a. 7 mph m ft/s sn 6t t 8 8 sn 8 sn (6) t t.8 s t v cos v. ft b. 8 8 sn 6t 8 t 6t t 8 sn t t.7 s t v cos v. ft c. From the calclatons n part b, the tme s abot.7s. a. 8 8 sn 8 n s 6) ( b. a. when t.8.8 v sn 8 ()(.8).8 v sn v sn 8 v.7 ft/s b. t v cos v. d a. t v cos v t v sn v gt h t cos t sn 6t b. t cos t cos t sn 6t cos sn 6 cos 6. ft Snce 6., the ball wll clear the fence. c. t sn 6t sn ( n s) (6) t t.68 s t v cos v 8.86 ft. t t t 8 6t 8 6( ) 6 8 6a. mg sn v (9.8) sn. N 6b. mg cos v (9.8) cos 7.9 N 7. cos A A cos A 7 O t Chapter 8 6

21 8. ( ) ()..7 ( ) (). 6. ( ) (.7) (.).7 (.) The correct choce s B. Page stada Hstor of Mathematcs (), stada,(),, ft,, 8,67 m The actal crcmference of Earth s abot,9. mles.. See stdents work. No solton ests.. See stdents work. 8-8 Transformaton Matrces n Three-Dmensonal Space Pages 9 Check for Understandng. Matr T mltples -coordnates b and - and -coordnates b, so t prodces a reflecton over the -plane and ncreases the dmensons two-fold.. CC 8 6, 8 7, or,, The matr s.. VU are the same. T, so the transformatons a. BE,, or,, A(, (), ) A(,, ) C( (),, ) C(,, ) D( (), (), ) D(,, ) F(, (), ) F(,, ) G(,, ) G(,, ) H( (),, ) H(,, ) The matr s. b. () () () () () () () c. G B A H F C E O D () The mage s the reflecton over the -plane. d. The dmensons of the resltng fgre are half the orgnal. 6a. The scale factor of the dlaton s. The translaton ncreases -coordnates b. The matrces are D and T. a-c. Transformaton Orentaton Ste Shape Reflecton es no no Translaton no no no Dlaton no es no 6 Chapter 8

22 6b. Sample answer: If the orgnal prsm has vertces A(,, ) B(,, ), C(,, ), D(,, ), E(,, ), F(,, ), G(,, ), and H(,, ), then the mage has vertces A(,, ), B(,, ), C(,, ), D(,, ), E(,, ), F(,, ), G(,, ), and H(,, ). G H C D C G D F F H E E Pages Eercses 7. FB, 7, or, 6, A(,, (6), ) A(,, C(, 7 (6), ) C(,, ) The matr s AH (), (), or 7,, B(,, ) B(,, ) C(,, ()) C(,, ) D(,, ()) D(,, ) E( 7,, ()) E(,, ) F( 7,, ) F(,, ) G( 7,, ) G(,, ) The matr s 9. CF 6, (), or,, D(,, ()) D(,, ) E(,, ()) E(,, ) 6 The matr s.. () () () () () () () () B A B A D F H A E The reslt s a translaton of nts along the - as and nts along the -as.. () () E G O C B () () () () () () () () () () () () () () The reslt s a translaton of nt along the - as, nts along the -as, and nts along the -as.. () D F G A H () () 8 C B 8 D F A E () () C G B H () () 8 () The reslts s a translaton of nt along the - as, nts along the -as, and nts along the -as. 8 Chapter 8 66

23 . The transformaton does not change the fgre O A 6 B E D H. F The transformaton reslts n reflectons over the and -planes.. C D F E A H B G G G E A F D C B H C The transformaton reslts n reflectons over all three coordnate planes. 6. The matr reslts n a dlaton of scale factor, so the fgre s twce the orgnal se. 7., so the fgre s three tmes the orgnal se and reflected over the -plane , so the fgre s three-forths.7 the orgnal ste and reflected over all three coordnate planes., so the transformaton can 9a. be represented b the matr. 9b. The transformaton wll magnf the - and -dmensons two-fold, and the -dmenson -fold a b c. The reslt s a translaton.6 nts along the -as and 7 nts along the -as.. The matr wold reflect the prsm. over the -plane. The matr.. wold redce ts dmensons b half a. Placng a non ero element n the frst row and thrd colmn wll skew the cbe so that the top s no longer drectl above the bottom. Sample answer: b. Sample graphs: C D G H B A F E C G B H F E D A 67 Chapter 8

24 . The frst transformaton reflects the fgre over all three coordnate planes. The second transformaton stretches the dmensons along the - and -aes and skews t along the -plane. (The frst row of T changes the -coordnate of (,, ) to.). To mltpl the -coordnate b, the frst row of the matr mst be. Snce the -coordnate s mltpled b, the second row s. To convert a -coordnate to, se a thrd row of. The matr s. a. The -coordnates are nchanged, the -coordnates ncrease, and the -coordnates decrease, so the movement s dp-slp b a. La Shawna Jame t 6t 6t 6t 6t 6 6 t t.6 t t 6 7 ft 6b. Snce the stones have the same parametrc eqatons for, the land at the same tme. In part a, t was calclated that the elapsed tme s abot.6 seconds. 7. t t t t A graphng calclator ndcates that there s a solton between and. B Descartes Rle of Sgns, t s the onl solton. When., and when., So the solton to the nearest tenth s... Dvde each sde of the eqatons b,,, and 6, respectvel, so that the left sde s. I. II. III. IV. 8 Onl I and II are eqvalent, so the correct choce s A. Chapter 8 Std Gde and Assessment Page Understandng and Usng the Vocablar. resltant. nt. magntde. cross. nner 6. vector 7. parallel 8. standard 9. drecton. components 6 Sklls and Concepts.. cm,..9 cm,. q p q p. cm,. p. cm,. p q. cm q q. cm p q 8. sec cos coscos. cm 98 p. cm, 98 Chapter 8 68

25 6. p q. cm 8 q p. cm, 8 7. h. cos v. sn h.8 cm v cm 8. h.9 cos v.9 sn h.9 cm v. cm 9. CD 7, or, CD 69 or. CD (), 8 or 6, CD 6 or. CD, 9 () or, CD () 8 or 7. CD (6), or, 8 CD 8) ( 6. v w,,, () or, 6. v w,,, () or,. v w,, 6, 6, 6 6, () or, 7 6. v w,, 6, 6, 6 6, () or, 7. EF 6, (), or,, EF ) ( () 6 8. EF 9, 8, or,, 6 EF () ) (6 9. EF (), (), 7 ) or 6,, 7 EF EF, 7, (8) or 7, 7, EF (7) ) (7 67. w v,,, 7, 8,, (,, 8 (),, (), 7,..v.w., 7,.,,.,.7,.6,.,..6,.7 (.),.,.,.,, 6 () ()6 6 6; no., 6, () 6() 6 8; no.,,,, () () () 8 ; es 6.,, 6,, (6) ()() () 8; no 7.,,,, () () ()() 8 ; no j k 8.,,,, j k 6 j k or 6,, 6,,,, 6() () ()() 6,,,, 6() () ()() ,,,, 9 6j k or 9, 6, 9, 6,,, 9() (6)() () 8 8 9, 6,,, 9() (6)() () 8 8 j k j k 69 Chapter 8

26 .,,,, j k j k 8 9j k or 8, 9, 8, 9,,, (8)() 9()()() 8 8 8, 9,,, (8)() 9() ()() 8. 7,,,, j k j k 9j k or, 9,, 9, 7,, (7) (9) () 7 (8), 9,,, () (9) () (9) 9. Sample answer: Let (,, ), (,, ) and (,, ),, or,, (),, () or 9,,,, 9,, j k 9 j k j 9 k 9 or,,. F F 6j F F 6. N tan v 6 or 6 v tan v j v ( cos 6 ) ( sn 6 )j v v ( s co ) 6 ( 6 ) sn m/s tan v sn 6 cos 6 v tan cos , () t,, t, t t t t 6. (), 9 t7,, 9 t7, 7t 9 t 7t 9 t sn 6 7., t, 6, t, 6 t t 8. t 8t 7 7 8t 6t 9. t t. v v cos v v v sn v cos sn 8.6 ft/s.9 ft/s. v v cos v v v sn v. cos 66. sn 66.7 ft/s.6 ft/s. v v cos v v v sn v 8 cos 8 8 sn 8.89 m/s 8. m/s. CH,, () or 7, 6, A(, (6), ) A(,, ) B(, (6), ) B(,, ) D(,, ) D(,, ) E( (7),, ) E(,, ) F( (7),, ) F(,, ) G( (7), (6), ) G(,, ) The matr for the fgre s. The matr for the translated fgre s. A B H G D C The fgre moves nts along the -as and nts along the -as. E F Chapter 8 7

27 . D C E m The fgre s reflected over the -plane. Page 7 Applcatons and Problem Solvng. AB cos,, sn or,, F,, T AB F ft j k j k j k or,, T ) ( lb-ft 6. t v sn v gt h.(8) sn ()(.). ft 7a. 6 km/h km/h b (6 cos )c (6 sn )j c j b c (6 s co ) 6 ( sn ).7 km/h sn 7b. 6 6 cos sn 6 cos F A O B 6 7. m H G lb 6 8. F 9 F (7 cos ) (7 sn )j or j F F (9 ).6 N 7 tan v or v tan Page 7 Open-Ended Assessment a. Sample answer: X(, ), Y(, ) XY, () or, b. XY () or The magntde of XY onl depends on the dfferences of the coordnates of X and Y, not the actal coordnates. a. Sample answer: P(, ), Q(, ), R(, ), S(, ) PQ, or, RS, or, PQ and RS are parallel becase the have the same drecton. In fact, the are the same vector. b. Sample answer: a 8,, b, 6 a b 8() ()6 or a and b are perpendclar becase ther nner prodct s. Chapter 8 SAT & ACT Preparaton Page 9 SAT and ACT Practce. Recall that the formla for the area of a parallelogram s base tmes heght. Yo know the base s, bt o don t know the heght. Don t be fooled b the segment BD; t s not the heght of the parallelogram. Tr another method to fnd the area. The parallelogram s made p of two trangles. Fnd the area of each trangle. Snce ABCD s a parallelogram, AB DC and AD BC. The two trangles are both rght trangles, and the share a common sde, BD. B SAS, the two trangles are congrent. So o can fnd the area of one trangle and mltpl b. The hpotense of the trangle s and one sde s. Use the Pthagorean Theorem to fnd the other sde. b 9 b 6 b b The heght s. Use the formla for the area of a trangle. A bh A ()() or 6 Snce the parallelogram conssts of two trangles, the area of the parallelogram s 6 or. The correct choce s A. 7 Chapter 8

28 . In order to wrte the eqaton of a crcle, o need to know the coordnates of the center and the length of the rads. The general eqaton for a crcle s ( h) ( k) r, where the center s (h, k) and the rads s r. From the coordnates of ponts A and B, o know the length of the sde s. So the center Q, has coordnates (, ). To calclate the length of the rads, draw the rads OB. Ths creates a - -9 rght trangle. The two legs each have length. The hpotense has length. ( ) ( ) () ( ) () ( ) 8 The correct choce s B.. Wrte the eqaton for the permeter of a rectangle. then replace wth ts vale n terms of. Solve the eqaton for. p p p p p The correct choce s B.. Recall the trangle Ineqalt Theorem: the sm of the lengths of an two sdes of a trangle s greater than the length of the thrd sde. Let represent the length of the thrd sde. 8 8 Snce mst be greater than, cannot be eqal to. The correct choce s A. To check or answer, notce that the other answer choces are greater than and less than, so the are all possble vales for.. Snce the answer choces have fractonal eponents of, start b rewrtng the epresson wth fractonal eponents. Smplf the fractons and se the rles for eponents to combne terms. 9 9 ( ) or The correct choce s E. 6. Ths fgre looks more comple than t s. A semcrcle s jst one half of a crcle. Notce that the answer choces nclde, so don t convert to decmals. Fnd the rads of each sem-crcle. Calclate the area of each sem-crcle. The area of the shaded regon s the area of the large sem-crcle mns the area of the medm sem-crcle pls the area of the small sem-crcle. Large sem-crcle area 9 Medm sem-crcle area Small sem-crcle area Shaded area 9 6 The correct choce s A. 7. The onl vales for whch a ratonal fncton s ndefned are vales whch make the denomnator. Snce f(), the denomnator s onl when or. The correct choce s D. 8. Start b sketchng a dagram of the conter Use or calclator to fnd the area of the whole conter and then sbtract the area of the whte tles n the center. The whte tles cover an area of ( )( ) or (8)(8). ()() (8)(8) 6 Red tles 6 6 The correct choce s B. 9. Frst, fnd the slope of the lne contanng the ponts (, 6) and (, ). m 6 m 9 6 or 8 8 The pont-slope form of the lne s 6 ( ). 6 So the -ntercept of the lne s. The correct choce s B. Chapter 8 7

29 . Wrte an epresson for the sm of the areas of the two trangles. Recall the area of a trangle s one half the base tmes the heght. (AC)(AB) (CE)(ED) From the fgre, o know that ABC and CDE are both sosceles, becase of the angles marked and becase BCD s a lne segment. These two trangles have eqal correspondng angles. Snce the are sosceles trangles, AC AB and CE ED. Use these eqvalent lengths n the epressons for the area sm. (AC)(AB) (CE)(ED) (AC) (CE) [(AC) (CE) ] Usng the Pthagorean Theorem for ACE, o know that (AC) (CE) (AE) or. So the sm of the two areas s (). Yo can grd the answer ether as. or as. 7 Chapter 8