MAC2313_Test1_Spring2008_Solutions

MAC2313_Test1_Spring2008_Solutions - Page 1 of5 MAC 2313...

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Unformatted text preview: Page 1 of5 MAC 2313 Name Test 1 Date: January 24, 2008 Clearly show all work for full credit. 1. (4 points) Find an equation of the sphere with center (6,5,-2) that passes through the point (4,5,-2). r9: (47*!)1 (5:93 t 02:02)? 2 q (Mr + (w); + ma)”: H 2. (4 points) Find the center and radius of the sphere given by x2 + y2 + 22 = 4x — 2y. ”Kam- ‘ix +va+9v til =0 Cac- 933% (v HY“ 22‘: "lt/ (finder " ((2, fil) O.) Raoul; : (‘3 jg; Page 2 of 5 3. (4 points each) Given a = 3i — 2k and b = i — j + k , compute each of the following: M K . N h - a, _, -7 (a) 2a—3b 2 (@1174 ‘11?) A (EX-33) r313): 3C7+ 3.} “'- '7k (b) lal >— ~ Ci+o+LI :: J73 (C) M) '2 3m + 0(«0 .9210): I (d) axb L J k .42 K]: “5 <~QJ”SJ“3> (e) the angle 6’ between a and b > \ r O _ 5V L .. ——L—-—-—-‘ :1» Q = 80-8 403(91 ‘“ 1'2? x (f?! ““ (JE)LJ‘3’> (t) the scalar projection of 3 onto b «a 8, v) 53‘ ‘0 ,— J... O a = “.7“ ~— ”‘ Pt: I I If? (g) the vector projection of a onto b 1/ > . f7 cQIkP~7a _ ’1. r: < "’— v Praia: r-——.%—I-tl>’ Iii) 32323 i? l L h th d' to sin f -: 3i () e 1recrnco eso a Cu__g(o<\_ $3 6.0; C 63 = 0 C05 (20 1‘ :% 4 (4 points) Find the volume of the parallelepiped determined by vectors a= <1,1,—,=1>b <1, -,,11> andc=<-,11,1>. 2:7 =21 LJkiJ '~)11*11~I fixedi to“: J! [114‘ (”UK 8:) <11'l>v<§lao>--L1I V: I LII LI 5'3 ‘5 , , . chc, : {'o?)';)0> Page 3 of5 5. (8 points) A pilot is steering a plane in the direction of N3 O°E at a speed of 200 mph. The wind is blowing at a speed of 10 mph from the direction N75°E. Find the true course and ground speed of the plane. ”7: PL.“ vafbr : <aw cone-o“) , 20c. siutcc>°)> .___. < ioo, mfg» «.1 4 (00) 573.31» Cf) v.34 vac/(w :— (JOCosU‘iS‘),105i¥1(‘i€$’°)‘7 res Ilia/wk - 03+ q], C3=< 610.34, {70.52) R taizei: "3"? => 6:: $910 0' 1 ‘10»8: arm lift: (540.3qafl70‘692 2-4— 143.“: TM CawfiSa': ~91an True Speed 1 [€3,043 mink 6. (2 points each) Describe, in words, the intersection of the graph of x2 + 4 y2 — 222 = —8 with each of the following planes, and also state the equation of each intersection. (a) the yz-plane (70:0) )Lypcrimxek’: L‘VQ‘ QED—T‘g (b) the xz—plane (4/16) I Q ’3- .— kyper’iaetea' ’X - 92' ~ 8) (C) the xy-plane (2:0) Ito ihiflsaebh S‘nce» X9+qfior~3 Las no SchgftbA (d) the planezz—Z the, Pm“: (O) O} ‘Q\ Snag, XD+HYQ“A(”Q)D=‘S( =57 XQrLlfif‘l: O :2) 'x:O:'~1 (e) the planez= 3 eflf’osgi 12+ 7112-2(3332 ~8 :3 XQ+LIYQ :: IO Page 4 of5 7. (4 points) For r(t) =t2 i+ t 11 j+ ln(t k, find the domain of r and compute limr(t) 3:; £50 11*“ Domain: P05 61139 Peta/2&1 nméers excepé [1“: W564, { M [a Ida—.20 . :1 £1 _ .L SW, A z: :1, L; ,9, N. @1111» 12 , 1:9! to” 4i 6-?! £113., [57 7—366134‘Ja30> 8. (4 points) Write the rectangular equation 2 — x2 + y2 — y using spherical coordinates. “Z: (‘2 --’y Z: €EOS(¢) F: €S¢‘n(¢)y "ftrsierQl 1‘ (355161951316) :2 ECU; (Cb) ":2 €9s‘_na(®) “‘" F$iin(¢\$ir\(e§ 9. (4 points) Change the point with spherical coordinate (22 2\/—,—— 37f 2,”) to cylindrical coordinates. Sm“), (Z): 1%: (fie. {Dairié I: On (2‘4 Xy-/>2€&4:¢, ) .i, Z: O 5.11% F: P 1 (mag-49133:?) I" 10. (2 points each) For the line 1n 3- -space through the points:(6, 1,-3) and Q”,(2 4 ,,5) determine 31—) (a) avector equation of the hne l7 rm< —‘-I 3 52> (Agra? P: (e, 8‘33 F3651: {(9,150 + t <-q,3,8> (b) parametric equations for the line ’X: é*"{6 NY: (it—36 Z: “3+31i: (c) symmetric equations for the line Page 5 of5 ll. (5 points) Find an equation of the plane in R3 that passes through the point (-2,8,10) and is perpendicular to the line x =1+t , y 2 2t , z = 4 —— 3t. 7?: if: <i)91*3‘> 2 (W) 7+ 90,40 ‘ECz-I’Oh : o I 12. (5 points) Determine whether the lines L1 and L2 given below are parallel, skew, or intersecting. If they are intersecting, find the point of intersection. L1: x=l+2t,y=2t,z=4—3t ‘07: 4;,Q)-3\/ L2: x=6t,y=l+6t,z=5—9t V3: <G,é,”“i> 1):: 33? 33> L. // La 13. (5 points) Find an equation of the plane in R3 that passes through the point (1 ,—l,1) and contains the line with symmetric equations x = 2 y = 32 . P: (0,0,0) [We (2: (‘e)'3,a\ m Page m at [at L&é [2: (23") i» a.) ~$i , :3:— PE? '2 < 93;» x7”: PR: < 1,4, D W: Cfxvzz 5,"‘i)”4> U859 P ‘1 (0)0’03 : wal‘ixj "q? :0 l4. (5 points) Find the point at which the line with parametric equations x=1+2t, y =4t, z =2~3t intersects theplane x+2y—z+l =0. M+Q~pe+l =0 :2» (It-atlraCHQ-(av3t3tlT-O :> :35 :0 .—~.:> £20 H l a (xi—ER (l ...
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