SHW6-R 8@. (a) 4. (a) With the notations in the figure, With the notations in the figure, AG BH Consider G. ΑG tan G tan 50 tan 50 Consider CHG. GH tan H GH tan 70 tan 50 tan 70 GH tan 50 The speed of the aeroplane tan 70 tan 50 0 s 84 /s (cor. to sig. fig.) (b) Consider CHG. cosh CH cos 70 tan 50 CH CH tan 50 cos 70 Consider BCH. BH tan BCH CH tan 50 cos 70 BCH. (cor. to sig.fig.) The angle of elevation of B fro C is.. (b) 5 )80 54 5 Consider AOX..5 cos OA.5 OA cos 54 4.5 (cor. to sig.fig.) With the notations in the figure, VA VO OA (Pyth. theore) VO.0 Area of base ABCDE 5 area of AOB.5 cos54.5 5 5 sin 54 cos54 4.09 Volue of the right pyraid fored area of base ABCDE VO 4.09.0 6 (cor. to sig.fig.) M
7. (a) (i) Consider ADF. DF sin DAF AD DF sin 0 4 DF HK DF Consider AHK. HK sin HAK AH sin 6 AH AH sin 6 4.56 Consider BHK. HK sin HBK BH sin 0 BH BH sin 0 5.8476 4.56 (cor. to sig. fig.) 5.85 (cor. to sig. fig.) (ii) Consider ABH. By the cosine forula, we have AH BH AB cos AHB ( AH)( BH) 4.56 5.8476 5 (4.56)(5.8476) AHB 55.7767 55.8 (cor.to sig.fig.) (b) Tie required for the blue toy car to reach H 4.56 0. /s 5. s (cor. to sig. fig.) Tie required for the green toy car to reach H 5.8476 0.4 /s 4.6 s (cor. to sig. fig.) The two toy cars will not reach H at the sae tie. 8. (a) (i) BD BD AD OB BD AB 0 0 00 00 In VOB, VB OB VB 00 4 0.7 (Pyth. theore) (property of square) VO 8 (Pyth. theore) (cor. to sig. fig.) (ii) Consider VAB. VA VB 4 By the cosine forula, we have VB AB VA cos VBA ( VB)( AB) (b) ( 4) 0 ( 4) ( 4)(0) VBA 6.0765 6. AN sin VBA AB AN 0 sin 6.0765 8.857 8.84 (cor. to sig. fig.) (cor. to sig. fig.) A (c) VAB VCB CN VB and CN = AN The angle between the planes VAB and VBC is ANC. BD 00 Consider N. By the cosine forula, we have AN CN cos ANC ( AN)( CN) 8.857 8.857 ( 00) (8.857)(8.857) ANC 06 (cor. to sig.fig.) The angle between the planes VAB and VBC is 06.
6. (a) Consider Figure (a). AE CE (Pyth. theore) AE (5 5) 6 8 ABD ~ E (AAA) AD AB AE (corr. sides, ~ s) AD 5 8 0 AD 4 (b) (i) The angle between the line and the horizontal table is CAE. Consider DAE in Figure (b). By the cosine forula, we have EA AD DE ( AD)( DE) cos ADE EA 4 4 (4)(4) cos 40.76 Consider CAE in Figure (b). CE tan CAE EA 6.76 CAE 65.4854 65.5 (cor.to sig.fig.) The angle between the line and the horizontal table is 65.5. (ii) Consider CAE in Figure (b). CE sin CAE 6 sin 65.4854 6 sin 65.4854 Consider B in Figure (b). By the cosine forula, we have AB BC cos CBA ( AB)( BC) 5 5 6 sin 65.4854 (5)(5) CBA 8.55 Area of ABC AB BC sin CBA 5 5 sin8.55.4 (cor. to sig.fig.) M+ 9. (a) Let M be the projection of Q on CD. AFP CHQ (RHS) FP HQ (b) (i) PE QE QM ED 6 Consider PEQ. PE QE QE PQ ( ) 9 QE CM CD MD 9 6 Consider CMQ. CQ QM CQ 6 6 7.445 CM (Pyth. theore) 9 (Pyth. theore) 7.4 (cor. to sig.fig.) Let X and Y be the id-points of PQ and respectively. Let Z be the projection of Y on the plane EFGH. The angle between the planes QP and D is XYD, i.e. XYD. Consider BCD. BD BC BD CD 6 6 7 YD BD 7 7 Consider EXQ. (Pyth. theore)
EQ EX EX XQ 9 XZ EZ EX YD EX 7 (Pyth. theore) Consider XYZ. YZ tan YXZ XZ 6 7 YXZ 65.4 (cor. to sig.fig.) XYD YXZ (alt. s, YD// ZE) 65.4 (cor.to sig. fig.) M+ (ii) XY is a line of greatest slope of the inclined plane QP. The angle between the planes QP and D, i.e. is greater than the angle between the line QC and the plane D. To s clai is agreed.
SHW6-P 4@. (a) (i) Consider ABC. (b) (i) By the cosine forula, we have BC AB cos B ( )( BC) 5 5 0 (5)(5) B 5.0 5. (cor.to sig.fig.) BCX B 5. (ii) In BCX, BCX BXC CBX 80 ( su of ) CBX 80 5.0 70 56.8699 By the sine forula, we have CX BC sin CBX sin BXC 5 sin 56.8699 CX sin70.676.4 (cor. to sig.fig.) Let A be the projection of A on the horizontal plane. AX CX (5.676).64 BX BC sin BCX sin BXC 5 sin 5.005 BX sin 70.7707 A AX sin 70 0.90840 X AX cos 70.97850047 B X XB 6.748674 Consider A BC. C B BC ( B)( BC) cos CBX C 5.95454 5. (cor. to sig. fig.) Consider AA C. C A (Pyth. theore) 8.7857476 8.7 (cor. to sig. fig.) M+ (ii) The tetrahedron ABXC has a axiu height when AXB is perpendicular to the horizontal plane. i.e. The tetrahedron ABXC has a axiu volue when AXB is perpendicular to the horizontal plane. When increases fro 5 to 8.7, the volue increases; when increases fro 8.7 to 5, the volue decreases. +A. (a) (i) Consider CPQ. By the cosine forula, we have PQ CP CQ ( CP)( CQ)cos PCQ PQ 6 5 (6)(5) cos80 9.94 9.9 (cor. to sig. fig.) (ii) Consider P. cos P CP cos 5 6 6cos 5 4.5009 AP sin P CP AP sin 5 6 AP 6sin 5 6.769 Consider BCQ. BC cos BCQ CQ BC cos55 5 BC 5cos55 8.606 BQ sin BCQ CQ BQ sin 55 5 BQ 5sin 55.87 Let N be a point on BQ such that PN BQ. NQ BQ AP (.87 6.769) 5.554 Consider NPQ. NP NQ NP PQ (Pyth. theore) 9.94 5.554 9.60 AB NP 9.60 Consider ABC. By the cosine forula, we have BC AB cos B ( )( BC) B 09.850 4.5009 8.606 9.60 (4.5009)(8.606) 09 (cor. to sig. fig.) M+
(b) D B 80 (adj. s on st. line) D 80 09.850 70.650 Consider D. By the cosine forula, we have AD DC ( )( DC)cos D AD 4.5009 8.606 (4.5009)(8.606) cos70.650 4.94 Consider ADP. DP AP DP AD (Pyth. theore) 6.769 4.94 5.7 (cor. to sig. fig.) DP is shorter than CP. Sion s clai is disagreed. +A. (a) Let M be the id-point of FD. AF = AD and EF = ED with DM = FM AM DF and EM DF (prop. of isos. ) DF is the line of intersection of the planes AFD and EFD with AM DF and EM DF. The angle between the planes AFD and EFD is AME. Consider EFM. EM FM EF (Pyth. theore) (b) (i) Consider AEM. AE tan AME EM AE tan 60 64 AE 64 tan 60.0454.0 (cor. to sig. fig.) Distance between XY and BC EM 64.0454 tan 64 7.9 (cor. to sig.fig.) M+ M+ (ii) XF is not a line of greatest slope on the plane XFDY. The angle between the planes XFDY and EFD ust be greater than that between the line XF and the plane EFD. +A EM 0 64
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