Chapter 6 Area and Volume
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- Lorin Bradford
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1 Capte 6 Aea and Volume Execise 6. Q. (i) Aea of paallelogam ( ax)( x) Aea of ectangle ax ( x + ax)( x) x x ( + a) a x a Faction x ( + a) + a (ii) Aea of paallelogam Aea of ectangle 5 ax (( + 5 x)( x ax) ) 0ax 6x + 8ax ax x ax 6x 0 x ( a 8) 0 a 8 0 a 8 Q. (i) Aea of dak paallelogams x x 6x + x x x x 5x x (ii) Aea of ligte section Aea of ectangle 0x 0x 5x. x 0x 5x 0x 5x x (iii) Dak to Ligt 0x : 5x : Q. Heigt ( x 5) cm Base x cm Aea 5cm ( x)( x 5) 5 x 5x 0 x 5x 0 0 ( x )( x + 8) 0 x o x 8 Since x > 0 x cm base cm eigt 5 8cm. 7
2 Q. a + ( 9 a) a a + a 68 a 98a a 9a ( a 0)( a 9) 0 a 0cm o a 9cm a cm 9 a Q5. x + ( x + ) x + x x + Diagonal + 5 Steel cable 5 + ( 0. 5) m (x + )m 50 cm x m 50 cm Q6. Let A + B+ C 80 and A > B > C C B A + B + C 80 6 A + A + A 80 7 and B A 7 A + 9A + 6A 80 C A A A A 6 05 B ( 05) C ( 05) 0 Q7. A D a B E B b C a A 8
3 (i) A paallelogam (ii) Aea of paallelogam ( a + b) AD + BC (iii) Te tapezium te aea of paallelogam ( a + b). a + b fomula fo te aea of a tapezium. Q8. Tee times widt exceeds twice lengt by cm. Let x widt and y lengt x y + x y... A Fou times its lengt is cm moe tan its peimete. y ( x + y) + y x + y + x y... B A: x y B: x y A B: x 5cm ( 5) y y y cm Q9. p A A. p + q p + q q q p A A. p + p q + q q q p A q + p q + A p p > q >. p p p > p q > p p q > p 9
4 Compae A and A, i.e. Since p q > p p q + q p + q > A > A Compae A and A, i.e. p > q p > q p + q p q + q and p q + p p q + q and p q + p p q + q > A > A A > A > A 0
5 Q0. Note: Te skip as a constant dept it will be alf-full wen te two aeas on te side ae equal. Aea of te top, A Aea of te bottom, A Total aea cm 8 0 A: t x + 0 y 60 b x + 0. y 0 + x ( 8 y) xy + 0y 0 xy 0 0y 0 0y x y 8 cm 0 + x and B: 8 y 60 ( ) 60 0 y + 8x xy y y y y 0 y y 0 y y 0 y + 0 y 0 y y + 6y 96 0 a, b 6, c 96 y 0 cm x alf-full 0 cm y ± ( ) y. 65 cm 8 ± 0 0 0(. 65 x cm )
6 + Q. (i) a a a a + a a a Aea. a. a. a a a 7 a a a 7 a cm a cm. (ii) cm 8 A ( 0. 75) cm A absinc sin cm 5.75 Q. aea of ectangle ( 8. 5) m m aea of tiangle. 8 (. 8) sin m 5 m 8. m
7 Q. (i) (ii) 60 EOD 60 6 ODE OED equal adii ODE + OED 0 ODE 60 (iii) Aea EOD a b sinq 5 5 sin cm Aea of exagon ABCDEFA 6 Aea EOD ( ) cm 6. 95cm Q. Pentagon squae exagon (i) a q 5 60 b 60 Pentagon: intenal angle Base angles a ( ) 5 60 Hexagon: intenal angle base angles 60 q 60 0 squae: b
8 (ii) Aea: squae 6 cm exagon 6 sin60 cm pentagon: tan Aea 8.. tan Total aea. 8 cm cm cm cm 67.5 Q5. 5 cm 6 + Aea 0 cm Q6. B cm 6 cm 5 cm 5 cm cm A D C (i) Aea ABD : Aea CBD AD. : DC. AD : DC (ii) ABCD is a tapezium AB is paallel to DC. Aea of ADC Aea BDC (same base and eigt) Aea c + Aea b Aea d + Aea b Aea c Aea d c d same pependicula
9 (iii) Aea of tapezium a + b + c + d c DX b But a XB d ab cd ab c A Aea c Aea a X Aea d B c ab Aea of tapezium a + b + c + d a + b + c D Aea b C a + b + ab. Execise 6. Q. (i) Peimete cuved (ii) Aea p. 0 p cm p. 60. p cm + Total peimete cm 9.7 cm Q. (i) Aea p. 7 cm cm (ii) Peimete 60 + p. 8 cm l cm l + 60 Q. (a) Aea p. (b) Aea pr p p R (c) Aea Aea of squae of side x + a Aea of cicle of adius a Aea x + a pa 5
10 (d) Aea Aea of ectangle + Aea of cicle + a ab p pa ab + (e) Aea a a ( a) x x a a x a (f) Aea ( a)( a) sin60. a a x a q Q. P + q + q P + q q R S Q5. RTS 0. adians ROS RTS 0. 8 adians RS 8. 5 ( 0. 8) O 6. 8 cm T Q6. l + C l B l (i) Aea of squae BCDE l 6 A
11 (ii) Aea of cicle Aea of squae Aea of saded section p ( p ) Q7. 80 p cm p Aea p. p. (. 7) cm. Because te adius depends on p, i.e. an iational numbe tat cannot be expessed as a faction. Q8. (i) Let cicle adius. Aea of oute squae l + l Aea of inne squae ( l)( l) l Aea : Aea inne oute : : (ii) sin 0 R R.sin 0 R R Aea : Aea cicumcicle incicle pr pr : p pr : R 0 : : 7
12 Q9. Aea of Tapezium wee R. Given p R 6 p p p p p R R 7 6 R p p Aea tapezium p p Q0. (i) p p AOB p 60 (ii) OAB ABO OAB 60 ABO A AB. Cicumfeence AB + BO + secto p (iii) Aea Aea + A tiangle secto p B 60 O 60 p +. sin. + p + p 8
13 (iv) Aea non-saded Aeasemi-cicle + p p p p p + p Q. Aeasaded Aeasecto Aea tiangle q sinq ( q sin q). p At q : p p Aea mino segment sin q p B C Aea majo segment p Aea mino segment p p p p + p + p + p p Ratio: Aea majo segment:aea mino segment + : p + p : p + : p 9
14 0 Q. Radius of disc 5 Aea of 5 discs 5 ( p5 ) 5p Heigt of fame + Aea of fame cm 0 cm Aea of emaining space 56. 5p 5. 7 cm Q. Aea secto q : Aea peimete + q 8 ( ) q 8 ( + q) q ( ) cm o 8 cm Q. (i) and (ii) m I 8 m II m III 5 m m 8 m 0
15 (iii) Aea I Aea II Aea III ( p ) ( p ) ( ) p 8 8p 9p p Aea TOTAL m Execise 6. Q. (i) (a). Aea ( 0. 8 ) m 0. 5 m (b) m Aea of Tiangle sin 60. Aea of ectangles Total aea m 8. 5 m (c) 0 mm Aea of Tiangle 0 0 sin Total aea mm
16 (d) cm Aea of squae cm Aea of paallelogam ( 00) + ( 80) Aea Total 560 cm (e) 0 6 cm Aea Aea Aea tiangle squae TOTAL 6 cm ( 8) + (f) cm 0 7 Aea Aea Aea tiangle ectangle squae 7 7 sin Aea TOTAL 9 + ( 0 ) cm
17 (ii) Volume( a) (. )( 0. 8)(. 0). 08. m Volume( b) Aea eigt. 5 base. m. m Volume ( c) Aea base eigt l Heigt of pyamid l l mm 800 Volume Volume ( d) 75. mm Aea base 80 0 eigt 0 mm 0 0 mm base 0 0 Cos0 0 Cos cm Volume( e) Aea base eigt l l 0 cm
18 Volume cm + Volume f Volume Volume cuboid pism ( 0 7 7) + ( base) cm Q. (i) Aea ( a) p p 0 57 mm Aea + b pl p ( 0)( 0) + p p 0 57 mm Aea + c p p p + p mm Aea Aea + d p p p p ( 5) mm ( e) ( ) + ( ) + ( ) p p p p p mm
19 Aea ( f ) + p p + p + p + p p mm + Aea Aea g p pl mm p. ( 0) p mm 805 mm Aea ( l) Note: l mm Aea i Note: ( 0) mm (i) 0 mm mm mm mm (i) f, d, b a, e, g,, c, i. 5
20 (ii) Volume( a) p. p mm 0 Volume( b) p p mm 0 Volume ( c) p p.. 9 mm Volume( d) p p. ( 5) 7069 mm Volume( e) p Volume f. p mm ( p ). p mm 6
21 Vol( g) p. p mm 6 Vol ( 6) 056 mm (ii) Vol ( i) f, d, a, b, e, g,, c, i + 0 mm 6 0 Q. (i) Vol Aea Dept (ii). 9 m tapezium Volume pick-up a a9. 60 Weigt pick-up. tonnes 00 kg a( 00) 0 a90 Weigt pick-up option is bette value. (iii) (iv) Volume a + a + tan q skip.. w a + tan q.. w q 5 Volume a.9 a +. tan 5. a a.5 a + tan q m bottom. 5 m. 5 m top tan m. 7 m tanθ q a a w 7
22 Q. (i) x +. 5 cm ½ x ½ (ii) Tiangula pism (iii) Volume Aea Lengt tiangle ( ) 8 cm (iv) Let eigt cm. a Let widt cm. Aea of Tapezium cm Sum of lengts of paallel sides a + b a + b b cm a + b 8 Let a 5, b Base cm, Top 5 cm, Heigt cm, Widt cm Q5. (i) Volume of spee Volume of cube p. p cm cm Volume taken away cm 8
23 (ii) Volume of spee enclosing cube p. p cm 5 5 cm Q6. 8 lites pe minute 8000 cm pe minute (i) sotest time smallest tank and fastest dain-ate. 0 m 0.95 m.5 m.5 m. m.5 m volume m 7875 cm Dain-ate 8. 9l pe minute 890 cm pe minute 7875 Time min min s min (ii) Longest time biggest time and slowest dain-ate.0 m.0 m.5 m.5 m. m. m volume m 6 cm Dain-ate 7. 5 lites pe minute 7500 cm pe minute 6 Time s 6 minutes 9
24 Q7. (i) Volume Aea base eigt ( 0 5) 700 cm (ii) l 0 5 cm 0 cm 5 Note: cm l 5 7½ cm 5 + : l + ( 7 ). 866 l Aea cm 50
25 Q8. Note: adius of cicle 8 cm Volume volume + volume ( cuboid) cuboid cylinde ( 8 ) p. 8.. cm cm cm 8 cm 8 cm cm Suface Aea p( 8 )( ) cm + p( 8) 0 cm (A quate-cicle emoved) Q9. x, y, z lengts p, a numbes. (a) px + py + p z p x + y + z x lengt lengt aea y lengt lengt aea z lengt lengt aea p x + y + z aea (b) ax + py lengt + lengt lengt. (c) axz numbe lengt lengt. aea. (d) a. p. y numbe numbe lengt lengt. (e) axy + paz numbe lengt lengt + numbe numbe lengt. inconsistent. (f) ax + xy numbe lengt + lengt lengt. inconsistent. (g) a x y z numbe lengt lengt lengt volume. 5
26 () x y + y z + z x lengt lengt + lengt lengt + lengt lengt volume Q0. (i) A. x z (ii) lengt. lengt lengt, consistent. V x Ay lengt (iii) V xy + z (iv) ( lengt) ( lengt)., inconsistent. lengt + lengt lengt lengt lengt, inconsistent. ( lengt) + lengt A x + y + z ( lengt) ( lengt) + ( lengt) + ( lengt) ( lengt), consistent. ( lengt) ( lengt) ( lengt + lengt + lengt) ( lengt). lengt ( lengt), (v) V A x + y + z (vi) V A + y x lengt ( lengt) + lengt lengt consistent. lengt lengt, inconsistent. + (vii) x y + z lengt lengt + lengt lengt, consistent. Q. (i) Vp ( a a) a If a 6 cm and 7 cm, V cm 5
27 (ii) If V 00 cm, a 5 cm cm l l cm l 5 5 cm l.5 l l + (. 5). 5 A cm
28 (iii) Vp. a. If a m, and m, and V Vp, V. m. m Note: l m 6 m m + side l m m p m back bottom side l 5 m p m + m m Aea A back + A bottom + Asides m m m m. m 5 m + m + m m l m Q. (i) (a) V (b) V box spee ( ) ( ) ( ) 7 cm p. ( 7) 7 cm (c) V unoccupied cm cm % 06 unoccupied % 8% 5
29 (ii) V cylinde p. ( 7) ( ) 5 cm V unoccupied cm % unoccupied 55.. % less space is unoccupied in cylinde. Q cm 7 cm V lage cone p cm p. 7.( 0. 5) 7. 5p cm V small cone p.. (. 5). 5p Vstoppe 7. 5p. 5p 58p 96. cm 6 7 cm 6 55
30 Execise 6. Q. Aea y + + ( + + ) y n y y... y n ( ) 0 mm Scale 000 : 000 mm: mm ( 000)( 000) mm 6 0 mm : mm : mm Aea 0 0 ( mm) ( Tue) 6 0 m m ( 000) ( mm ) 0. ectaes Q. Aea y + + ( + + ) y n y y... y n cm (i) (ii) % eo. 6% 7. Aea [ ( ) ] cm Q. (i) Aea y + + ( + + ) y n y y... y n + + ( ) sq.units (ii) Aea [ ( ) ] 7.75 sq.units 56
31 Q. y y x; at x 0. 5, y at x, y at x. 5, y. 5. at x, y x A y + + ( + + ) y n y y... y n ( ). 8 sq.units Q5. A y + + ( + + ) y n y y... y n 0. 5 [ ( ) ] sq.units 0. 5 A [ ( ) ]. 65 sq.units A : A :. 65 :. Bot aeas equal A Let x maximum (. ) sq.units x. 75 Let maximum ( + ) sq.units Fo equal aeas:. 5< x <
32 Q6. Aea y + + ( + + ) y n y y ( +. ) [ ] 0. 8 cm cm 0 km cm 00 km Aea km Revision Execise (Coe) Q. Aea A + C Aea B ( same base and same pependicula eigt) ( + ) Aea A + C Aea P C ( same base and same pependicula eigt) Aea P Aea A Aea B Aea C Aea P Aea C P : Q C : A+B C : P+B C : C+C C : 5C : 5 Midpoint C C C C C Q A B P Midpoint C Q. (a) 8 sufaces Aea ( 0 5) + ( 0 0) p p cm Volume ( 0 5 0) + ( p. 5. 5) cm 58
33 (b) sufaces Aea ( p (. 5) ) + ( p. 5 0) + p 6. 5 mm p. 6 p (c) Volume ( p. 5 0) + p ( 6) mm Aea of back Aea of bottom Aea of teads Aea of ises Aea of sides , 500 cm Volume cm Q. (i) Let COB q l q l ac lengt 6. 5q adius q. 8 adians (ii) (iii) Aea.. q cm Aea of cicle p ( 5) Aea of mino secto (subtacting) Aea of majo secto Aea of mino:aea of majo : :. 9 59
34 Q. Volume Aea of tapezium widt m m ( 00 cm), 000, 000 cm lites cm lite Capacity l 50, 000l 50, 000 Time minutes ous ous Q5. (i) (ii) A q. x.. x. 0 x x 5 cm A q p p p 5 p. x. q 5 x 60 x p (iii) x. 96 cm A q x 0 x. 5 x adians 60
35 Q6. Cuved suface aea q l. q But s lq and s p lq p p q l Cuved suface aea l. pl p l Q7. (i) Aea of ectangle sq.units Aea of tiangle ( 5. ) 5. 6 sq.units Aea unde gap sq.units (ii) Aea y + + ( + + ) y n y y... y n ( ) 0. 7 sq.units Q8. (i) Peimete m m (ii) Aea ( )m ( 0 ) + ( ) c b a m m m a + 0m b c 0 m + 8 m + 6
36 Q9. Lage adius 8 cm Small adius cm (i) Peimete peimete of lage cicle + peimete of small cicle + peimete of small cicle ( ) + ( ) p 8 p 8p + 8p 6p cm (ii) Aea aea of lage cicle aea of small cicle + aea of small cicle aea of lage cicle p. 8 p cm Q0. Cost a A A suface aea p cost + p p0 00p a + a8, p Q. Volume Volume of cube 8 + Volume of cylinde p p p + Volume of cone p p 8 + p +. 7 cm. 7 cm 6
37 Revision Execise (Advanced) Q. (i) ac lengt q (ii) peimete q + Aea q q +.. q q A A 5 65 A 65 5 Te maximum value of A occus wen 5 0, i. e. 5. (iii) CEB q q 65 q adians (iv) Let 5 A cm is te maximum aea. Note: if A if 6 A 65 ( 6 5) 6 6
38 Q. Hole of adius cm diamete cm. seet of widt and lengt m 00 cm can ave 50 oles cut along lengt and along widt Metod A: numbe of oles oles Aea of seet cm 0, 000 cm Aea of ole p p. p cm Aea of 500 oles p cm Waste 0, 000 7, cm % waste % 0, % 6
39 Metod B cm st nd ow as 50 oles ow as 9 oles d ow as 50 oles, etc. cm ow needs cm of metal Numbe of ows cm cm cm cm ows need + cm of metal cm ows need + cm of metal n ows need + n cm of metal + ( n ) 00 cm, wee n numbe of ows ( n ) 98 n ows 57 complete ows 9 ows of 50 oles 50 8 ows of 9 oles 7 8 oles Aea of 8 oles p cm Waste 0, cm. 00 % waste 0, 000. % Q. (i) Aea of PQO sinq sinq (ii) Aea of segment Aea of secto Aea of tiangle q sinq ( q sinq) 65
40 (iii) Aea of PQN ( q sin q) Also, aea of PQN sin ( p q) sin( p q) ( q sin q) sin p cosq cos psinq q sinq ( ) 0.cosq sin q q sin q sin q q sin q sin q q q sinq 0 Q. (a) suface aea sufaces oute cuved inne cuved p p( 5)( 00) 9000p p p 8000p end ings pr p p( 5) p( 0) 850p cm Total aea p cm Volume pr p ( R ) ( ) p p cm 58 cm 66
41 (b) suface aea 6 sufaces oute cuved ( pr ) p. ( 0). 5 50p mm inne cuved ( p ) p end ings pr p R p p mm flat aeas 5 5 ( ) ( ) 09 mm 50 mm 75p mm p Total aea p Volume pr p p ( R ) p. 5 ( 0 5 ) mm 767 mm (c) suface aea sufaces cm emispee ( p ) (. p. ) p pl cone p.. ( 8. ) 7. 76p Total aea cm cm p 8 cm l l cm Volume p. p c m + 67
42 Q5. (i) Aea of secto q p. p. 89 cm (ii) Aea of segment aea of secto aea of tiangle. )( ) 89 (. 57 cm sin p Aea bisected aea of tiangle.sin (. ). 895 Aea of segment sin (. 895) 5 m Q6. (i) sin a 5 a sin adians a. 8 adians x α m (ii) l q m x m (iii) (iv) Sotest distance x m Aeasecto q ( ).(. 8) 78. Aea tiangle sinq ( ).sin(. 8) 9. Aea segment m 68
43 Q7. aea of secto q aea of tiangle sinq aea of segment q sinq ( q sinq) (i) Aea of majo segment p aea of mino segment p ( q sin q). sin p q q + Wen q,. sin p + q in adians (ii) sinq sinq sin. 5 cm suface aea p p (. 5) cm.0 cm θ adian Q8. (i) A B acceleating nealy unifom/constant B C acceleating( not unifom) C D beginning to slow down E F aving stopped, begins to move off again. (ii) speed time distance (iii) [ ] + n + ( n ) Aea distance y y y y y [ ( ) ] 7 km min 7 km min min km km 69
44 a Q9. cos0 a cos0 a Base of tetaedon a a 8a a a Volume of cylinde p a p... a pa a a Q x x 0x x 6x 780 x 0 x y 0 + y 78 y 0 y ( 78) + 78y 6y 87 y cos a a Aea of lage tiangle ( 0)( 0) sin ( ) 0 78 sin Aea of small tiangle Aea of tapezium mm 70 α 0
45 Revision Execise (Extended-Response Questions) Q. (i) (a) l q (b) A q (ii) Lengt of wie + + q + q q Aea q (iii) Aea (iv) A.5 + q, p Maximum point (,) max m 0.5 (v) 5 A q q q adians 7
46 Q. (i) Goup one aeas ae te same. Aea of semi-cicle ( p ) x p px 0. 9x 8 Aea of equilateal tiangle q 60 A ( x)( x) sin 60 x. x 0. x Tei claim is false. Goup two Diffeence in aeas ( ) x 0. 00x 0. 00x 00 % diffeence 0. 9x 0. 8% Tei claim is tue. Goup tee 0. 0x 00 % diffeence 0. x 9. % Tei claim is also tue. 7
47 (ii) Aea of semi-cicle p p x 8 Aea of isosceles tiangle ( x) tan q ( x)( x) x tan q. x p x tan q 8 p tan q q θ x x tan θ x x x tan θ (iii) Volume of emispee p p x. 09x Volume of cone p x tan ( ) p. x. 0. 8x Te volumes ae not te same. ( ) Q. (i) Aea of base 0 0 (ii) Cube cm (iii) Volume cm 7
48 (iv) Volume Maximum volume (v), 5, (vi) is not possible since te base is ( 0 ) ( 0 6) 6 Q. (i) Aea y + + ( + + ) y n y y... y n ( ). 5[ 0] 50 cm (ii) p p. 0 A % eo % (iii) a 6. 6 b 8. 7 c a b c a b c (iv) Aea + + a + b + c + + c + b + a 0 0 ( a + b + c + ) 8 ( a + b + c + ) 7
49 ( ) (v) Aea (. ) + (. ) + (. ) + (vi) (. ) 5 cm A. 95( 5) 7. 8 cm 0 cm A. 95( 0) 9. 5 cm 5 cm A. 95( 5) (vii) Using A p 6. 8 cm : p ( 5 ) at 5, A 9. 7 cm at p ( 0 ) 0, A cm p ( 5 ) at 5, A 5. cm % eo. 8% % % 5. As can be seen, tee is a consistent eo of.8% wit tis fomula. We also note tat te fomula eading is always.8% less tan te tue eading as would seem easonable as te tapezia all lie below te eal cuve. Q5. (i)(a) lengt: + l + + l + l 0 l 5 (b) widt: + + w + + w 0 eigt: (c) eigt: eigt 75
50 (ii) Capacity : Volume l w (iii) Squae base : l w cm ( ) (iv) V ( 5) ( 5) 500 cm (v) is a oot ( solution) 5 0 is a facto is te second facto a, b 0, c 50 0 ± ± 00 0 ± 0 0 ± o , as >, te lengt of cadboad.. 9 cm would also give a capacity of 500 cm. 76
51 (vi) Capacity (cm ) (v) (iv) (cm) (vii) Capacity inceased by 0% Volume cm Tis is not possible since no value of could poduce a value of 550 cm. 77
52
FORMULAE. 8. a 2 + b 2 + c 2 ab bc ca = 1 2 [(a b)2 + (b c) 2 + (c a) 2 ] 10. (a b) 3 = a 3 b 3 3ab (a b) = a 3 3a 2 b + 3ab 2 b 3
![FORMULAE. 8. a 2 + b 2 + c 2 ab bc ca = 1 2 [(a b)2 + (b c) 2 + (c a) 2 ] 10. (a b) 3 = a 3 b 3 3ab (a b) = a 3 3a 2 b + 3ab 2 b 3](/thumbs/78/77696958.jpg "FORMULAE. 8. a 2 + b 2 + c 2 ab bc ca = 1 2 [(a b)2 + (b c) 2 + (c a) 2 ] 10. (a b) 3 = a 3 b 3 3ab (a b) = a 3 3a 2 b + 3ab 2 b 3")
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