2 Further pplitions of re n volume 2A Are of prts of the irle 2B Are of omposite shpes 2C Simpson s rule 2D Surfe re of yliners n spheres 2E Volume of omposite solis 2F Error in mesurement Syllus referene Mesurement 5 Further pplitions of re n volume This hpter will over fi ning re, surfe re n volume of simple n omplex shpes. Simpson s rule will e introue s metho of pproximting the re of irregulr fi gures.
ARE YOU READY? Digitl o SkillSHEET 2.1 o-1303 Are of irle Digitl o SkillSHEET 2.2 o-1304 Ares of squres, retngles n tringles plus Digitl o SkillSHEET 2.6 o-1312 Volume of ues n retngulr Try the questions elow. If you hve iffiulty with ny of them, extr help n e otine y ompleting the mthing SkillSHEET. Either lik on the SkillSHEET ion next to the question on the ebookplus or sk your teher for opy. Are of irle 1 Fin the re of irle with: rius 4 m rius 19.6 m Ares of squres, retngles n tringles imeter 9 m 2 Fin the re of eh of the following. 10.9 m 4.5 m 3.7 m imeter 19.7 m. 7.6 m 13.8 m Volume of ues n retngulr prisms (3, 3); Volume of tringulr prisms (3) plus Digitl o SkillSHEET 2.7 o-1313 Volume of tringulr prisms 3 Fin the volume of: 18 m 24 m 11 m 9 m 8 m 26 m 6 m Volume of yliners (4); Volume of sphere (4) Digitl o SkillSHEET 2.8 o-1314 Volume of yliners 4 Fin the volume of: 19 m 8 m 12 m Digitl o SkillSHEET 2.9 o-1315 Volume of sphere Volume of pyrmi 5 Fin the volume of: 10 m Digitl o SkillSHEET 2.10 o-1316 Volume of pyrmi Digitl o SkillSHEET 2.11 o-1317 Error in liner mesurement Error in liner mesurement 7 m 6 For eh of the following liner mesurements, stte the limits etween whih the true limits tully lie. 15 m (mesure orret to the nerest entimetre) 8.3 m (mesure orret to 1 eiml ple) 4800 km (mesure orret to the nerest 100 km) 40 Mths Quest Generl Mths HSC Course
2A Are of prts of the irle From previous work you shoul know tht the re of irle n e lulte using the formul: A = πr 2 WORKED EXAMPLE 1 Clulte the re of irle with rius of 7.2 m. Give your nswer orret to 2 eiml ples. THINK WRITE 1 Write the formul. A = π r 2 2 Sustitute for the rius. = π (7.2) 2 3 Clulte the re. = 162.86 m 2 A setor is the prt of irle etween two rii s shown on the right. To lulte the re of setor we fin the frtion of the irle forme y the setor. For exmple, semi irle is hlf of irle n so the re of semiirle is hlf the re of full irle. A qurnt is qurter of irle n so the re is qurter tht of full irle. For other setors the re is lulte y using the ngle etween the rii s frtion of 360 n then multiplying y the re of the full irle. This n e written using the formul: θ A= π r2 360 where θ is the ngle etween the two rii. WORKED EXAMPLE 2 Clulte the re of the setor rwn on the right. Give your nswer orret to 1 eiml ple. 80 5 m THINK 1 Write the formul. 2 Sustitute for θ n r. WRITE A = θ 360 πr 2 = 80 π 360 π 52 3 Clulte the re. = 17.5 m 2 Chpter 2 Further pplitions of re n volume 41
An nnulus is the re etween two irles tht hve the sme entre (i.e. onentri irles). The re of n nnulus is foun y sutrting the re of the smller irle from the re of the lrger irle. This trnsltes to the formul A = π( π R 2 r 2 ), where R is the rius of the outer irle n r is the rius of the inner irle. WORKED EXAMPLE 3 Clulte the re of the nnulus on the right. Give your nswer orret to 1 eiml ple. 5.7 m Tutoril int-2411 Worke exmple 3 3.2 m THINK WRITE 1 Write the formul. A = π( π R 2 r 2 ) 2 Sustitute R = 5.7 n r = 3.2. = π(5.7 2 3.2 2 ) 3 Clulte. = 69.9 m 2 An ellipse is n ovl shpe n therefore oes not hve onstnt rius. The gretest istne from the entre of the ellipse to the irumferene is lle the semi-mjor xis,, while the smllest istne is lle the semi-minor xis,, s shown in the figure on the right. The re of n ellipse is lulte using the formul, foun on the formul sheet: A = ππ semi-mjor xis () semi-minor xis () WORKED EXAMPLE 4 Clulte the re of the ellipse rwn on the right. Give your nswer orret to 2 eiml ples. 6.6 m 4.2 m THINK WRITE 1 Write the formul. A = ππ 2 Sustitute for the vlues of n. = π 6.6 4.2 3 Clulte the re. = 87.08 m 2 42 Mths Quest Generl Mths HSC Course
REMEMBER 1. The re of irle is foun using the formul A = πrπ r 2. 2. The re of setor n e foun y multiplying the re of full irle y the frtion of the irle given y the ngle in the setor. You n use the formul A = θ 360 πrπ 2. 3. An nnulus is the re etween two onentri irles. The re is foun y using the formul A = π( π R 2 r 2 ), where R is the rius of the outer irle n r is the rius of the inner irle. 4. An ellipse is n ovl shpe. The re is lulte using the formul A = ππ, where is the length of the semi-mjor xis n is the length of the semi-minor xis. EXERCISE 2A Digitl o SkillSHEET 2.1 o-1303 Are of irle Are of prts of the irle 1 WE 1 Clulte the re of the irle rwn on the right, orret to 1 eiml ple. 6.4 m 2 Clulte the re of eh of the irles rwn elow, orret to 2 eiml ples. 33 mm 9 m 7.4 m 26.5 m e f 3.84 m 6.02 m Chpter 2 Further pplitions of re n volume 43
3 Clulte the re of irle tht hs imeter of 15 m. Give your nswer orret to 1 eiml ple. 4 WE2 Clulte the re of the setor rwn on the right. Give your nswer orret to 1 eiml ple. 7.2 m 5 Clulte the re of eh of the setors rwn elow. Give eh nswer orret to 2 eiml ples. 60 5.2 m 135 74 mm 23 m 150 9.2 mm 20 e f 240 19.5 m 72 39 mm 6 Clulte, orret to 1 eiml ple, the re of semiirle with imeter of 45.9 m. 7 WE3 Clulte the re of the nnulus shown t right, orret to 1 eiml ple. 12 m 6 m 8 Clulte the re of eh nnulus rwn elow, orret to 3 signifint figures. 20 m 9.7 m 77 mm 18 m 4.2 m 13 mm 44 Mths Quest Generl Mths HSC Course
9 A irulr gren of imeter 5 m is to hve onrete li roun it. The onrete is to e 1 m wie. Wht is the rius of the gren? Wht is the rius of the onrete irle? Clulte the re of the onrete, orret to 1 eiml ple. 10 WE4 Clulte the re of the ellipse rwn on the right, orret to 1 eiml ple. 10 m 6 m 11 Clulte the re of eh of the ellipses rwn elow. Give eh nswer orret to the nerest whole numer. 56 mm 34 mm 13.6 m 7.2 m 21 m 14 m 12 MC The re of irle with imeter of 4.8 m is losest to: A 15 m 2 B 18 m 2 C 36 m 2 D 72 m 2 13 MC Whih of the following lultions will give the re of the setor shown on the right? A 1 8 π 42 B 1 8 π 82 45 C 1 4 π 42 D 1 4 π 82 14 MC The re of the ellipse rwn on the right is losest to: A 32 400 m 2 B 324 m 2 C 5900 m 2 D 59 m 2 1.2 m 8 m 86 m Further evelopment 15 A irulr re is pegge out n hs imeter of 10 m. Clulte the re of this irle, orret to 1 eiml ple. A gren is to e ug whih is 3 m wie roun the re tht hs een pegge out. Clulte the re of the gren to e ug. Give your nswer orret to 1 eiml ple. In the gren setor with n ngle of 75 t the entre is to e use to plnt roses. Clulte the re of the rose gren, orret to 1 eiml ple. 16 A irle hs imeter of 20 m. Clulte the re of this irle, orret to 2 eiml ples. An ellipse is rwn suh tht the rius of the irle forms the semi-mjor xis. The semiminor xis is to hve length equl to hlf the rius of the irle. Clulte the length of the semi-minor xis. Clulte the re of the ellipse, orret to 2 eiml ples. Chpter 2 Further pplitions of re n volume 45
2B 17 The re in front of uiling is retngulr in shpe, mesuring 50 metres y 15 metres. At night seurity light sns the re. The seurity light, if positione vertilly ginst the wll t groun level, illumintes n re of the wll tht is setor of irle of rius 15 m n hs n ngle of 60 t the entre. Wht is the re of the retngulr frontge? Wht is the re tht is illuminte t ny one time y the seurity light? Give your nswer orret to 1 eiml ple. Wht perentge of the frontge is illuminte t ny one time? Give your nswer orret to 1 eiml ple. 18 A irulr isln is in the entre of irulr lke suh tht the surfe of the wter in the lke forms n nnulus. The rius of the lke is 10 m greter thn the rius of the isln. Given tht the isln hs rius of 20 m, fin the re of the surfe of the wter. Give your nswer orret to the nerest squre metre. Tori lims tht the surfe re of the lke will remin the sme regrless of the two rii s long s the ifferene of 10 metres remins unhnge. Is Tori orret? Use lultions to justify your response. 19 An ellipse hs n re of 50 m 2 n the length of one semi-xis is 5 m. Fin, orret to 1 eiml ple, the length of the other semi-xis. Whih of the two xes is the semi-mjor xis n wht is its length? Wht is the length of the semi-minor xis? 20 An r of length 1 m stns on irle of rius 1 m. Fin the size of the ngle sutene t the entre orret to the nerest egree. 21 An ellipse hs semi-mjor n semi-minor xes of 6 m n 4 m respetively. Clulte the re of the ellipse orret to 1 eiml ple. Frnk lims tht the ellipse will hve the sme re s irle of rius 5 m. Is Frnk orret? Fin the rius of the irle tht hs the sme re s the ellipse. (Answer orret to 1 eiml ple.) Are of omposite shpes A omposite shpe is shpe tht is me up of two or more regulr shpes. The re of omposite shpe is foun y splitting the re into two or more regulr shpes n lulting the re of eh seprtely efore ing them together. In mny ses it will e neessry to lulte the length of missing sie efore lulting the re. There will sometimes e more thn one wy to split the omposite shpe. WORKED EXAMPLE 5 Fin the re of the figure t right. 6 m 18 m 10 m THINK 1 Copy the igrm n ivie the shpe into two retngles. WRITE 6 m 18 m A 1 A 2 8 m 12 m 10 m 12 m 46 Mths Quest Generl Mths HSC Course
2 Clulte the length of the missing sie in retngle 1. (Write this on the igrm.) 18 10 = 8 m 3 Clulte the re of retngle 1. A 1 = 6 8 = 48 m 2 4 Clulte the re of retngle 2. A 2 = 10 12 = 120 m 2 5 A together the two res. Are = 48 + 120 = 168 m 2 Composite res tht involve tringles my require you to lso mke lultion using Pythgors theorem. WORKED EXAMPLE 6 Fin the re of the figure on the right. 13 m Tutoril int-2412 Worke exmple 6 10 m 24 m THINK WRITE 1 Drw the tringle t the top n ut the isoseles tringle in hlf. 13 m 12 m 2 Clulte the perpeniulr height using Pythgors theorem. 2 = 2 2 = 13 2 12 2 = 169 144 = 25 = 25 = 5 m 3 Clulte the re of the tringle. A = 1 2 24 5 = 60 m 2 4 Clulte the re of the retngle. A = 24 10 = 240 m 2 5 A the two res together. Are = 60 + 240 = 300 m 2 Composite res n lso e lulte y using sutrtion rther thn ition. In these ses we lulte the lrger re n sutrt the smller re in the sme wy s we i with nnuluses in the previous setion. Chpter 2 Further pplitions of re n volume 47
WORKED EXAMPLE 7 Fin the she re in the figure on the right. 6 m 20 m 30 m THINK WRITE 1 Clulte the re of the retngle. A = 30 20 = 600 m 2 2 Clulte the re of the irle. A = π 6 2 = 113.1 m 2 3 Sutrt the res. Are = 600 113.1 = 486.9 m 2 REMEMBER 1. 2. 3. To fin the re of ny omposite figure, ivie the shpe into smller regulr shpes n lulte eh re seprtely. You my hve to use Pythgors theorem to fin missing piees of informtion. Chek if the est wy to solve the question is y ing two res or y sutrting one re from the other to fin the remining re. EXERCISE 2B Digitl o SkillSHEET 2.2 o-1304 Ares of squres, retngles n tringles Are of omposite shpes 1 WE5 Copy the figure on the right into your workook n lulte its re y iviing it into two retngles. 2 Fin the re of eh of the figures elow. Where neessry, give your nswer orret to 1 eiml ple. 7 m 5 m 5 m 16 m 22 m 18 m 25 m 12 m 18 m 4 m 40 m 20 m 19 m 11 m 6 m 48 Mths Quest Generl Mths HSC Course
e f 4 m 8 m 12 m 8 m 4 m 16 m 4 m Digitl o SkillSHEET 2.3 o-1305 Using Pythgors theorem 3 Look t the tringle on the right. Use Pythgors theorem to fin the perpeniulr height of the tringle. Clulte the re of the tringle. 4 Below is n isoseles tringle. 8 m 10 m 17 m 6 m 15 m Digitl o Exel spresheet o-1306 Pythgors 12 m Use Pythgors theorem to fin the perpeniulr height of the tringle, orret to 1 eiml ple. Clulte the re of the tringle. 5 Clulte the re of eh of the tringles elow. Where neessry, give your nswer orret to 1 eiml ple. 25 m 26 m 48 m Digitl o GC progrm Csio o-1307 Mensurtion 24 m 6 WE6 Fin the re of eh of the omposite figures rwn elow. 13 m 17 m 25 mm 124 mm 52 mm 15 mm 48 mm 12 m 13 m 54 mm 30 m 7 MC The re of the omposite figure on the right is losest to: A 139 m 2 B 257 m 2 C 314 m 2 D 414 m 2 10 m 8 MC The re of the figure rwn on the right is: A 36 m 2 B 54 m 2 C 72 m 2 D 144 m 2 12 m 6 m Chpter 2 Further pplitions of re n volume 49
9 A lok of ln is in the shpe of squre with n equilterl tringle on top. Eh sie of the lok of ln is 50 m. Drw igrm of the lok of ln. Fin the perimeter of the lok of ln. Fin the re of the lok of ln. 10 WE7 In eh of the following, fin the re of the she region. Where neessry, give your nswer orret to 1 eiml ple. 12 m 10 m 4 m 9 m 8 m 16 m 3 m 9 m e f 7.1 m 10 m 112 mm 36 mm 40 mm 1.9 m 7.4 m 3.1 m 95 mm 11 An thletis trk onsists of retngle with two semiirulr ens. The imensions re shown in the igrm elow. 70 m 82 m 90 m The trk is to hve syntheti running surfe li. Clulte the re whih is to e li with the running surfe, orret to the nerest squre metre. 12 A gren is to hve onrete pth li roun it. The gren is retngulr in shpe n mesures 40 m y 25 m. The pth roun it is to e 1 m wie. Drw igrm of the gren n the pth. Clulte the re of the gren. Clulte the re of the onrete tht nees to e li. If the ost of lying onrete is $17.50 per m 2, lulte the ost of lying the pth. Further evelopment 13 Fin the re of eh of the following figures. 10 m 25 m 24 m 4 m 30 m 25 m 20 m 40 m 20 m 50 Mths Quest Generl Mths HSC Course
e f 12 m 76 mm 6 m 12 m 40 m 32 mm 15 m 14 Convert the following res to the units given in rkets. 20 000 mm 2 (m 2 ) 3500 000 m 2 (m 2 ) 0.005 m 2 (m 2 ) 0.035 m 2 (mm 2 ) e 13 400 m 2 (km 2 ) f 375 000 m2 (hetres) g 2 750 000 000 mm 2 (m 2 ) h 0.043 km 2 (m 2 ) 15 Fin the re of the regulr hexgon shown in the igrm elow. 2.40 m 4.16 m 16 A utting le for rft knife hs the imensions shown in the igrm. Wht is the re of steel in the le (to the nerest mm 2 )? 32 mm 20 mm 5 mm 38 mm 17 Emm left her horse tie to riling in pok while she htte to frien. The horse is tie to one en of the riling s shown elow. 2 m Riling 2 m Fin the re tht the horse hs ess to orret to 1 eiml ple. The horse is now tie to the entre of the riling s shown elow. r = 1 m r = 2 m r = 1 m Fin the re tht the horse now hs ess to. Chpter 2 Further pplitions of re n volume 51
18 Fin the res of the following figures (to 1 eiml ple). 14 m 16 m 12 m 25 m 4.5 m 19 m 30 m 2 m 14 m 36 mm 24 mm 95 mm 80 mm e 24 m 20 m 16 m 32 m 20 m f 5.5 m 6.5 m 11 m 17 m 22 m 42 m 3.5 m 2C Simpson s rule Simpson s rule is metho use to pproximte the re of n irregulr figure. Simpson s rule pproxi mtes n re y tking stright ounry n iviing the re into two strips. The height of eh strip (h) is mesure. Three mesurements re then tken perpeniulr to the stright ounry, s shown in the figure on the right. The formul for Simpson s rule is: f h m 10 m h l A h 3 ( f + 4 m + l ) where h = istne etween suessive mesurements f = first mesurement m = mile mesurement = lst mesurement. l WORKED EXAMPLE 8 Use Simpson s rule to pproximte the re shown on the right. 30 m 10 m THINK WRITE 90 m 1 Clulte h. h = 90 2 = 45 2 Write own the vlues of f, m n l. f f = 10, m = 30, l = 0 52 Mths Quest Generl Mths HSC Course
3 Write the formul. A h 3 ( f + 4 m + l ) 4 Sustitute. A 45 (10 + 4 30 + 0) 3 5 Clulte. = 15 130 1950 m 2 Coul Simpson s rule e use to estimte the res of these irregulr shpes from nture? Simpson s rule n e use to pproximte the re of n irregulr shpe without stright ege. This is one y onstruting line s in the igrm in the worke exmple elow n pproximting the re of eh setion seprtely. WORKED EXAMPLE 9 Use Simpson s rule to fin n pproximtion for the re shown on the right. 30 m 30 m 30 m 17 m 10 m THINK WRITE 1 Write own the vlue of h. h = 30 2 For the top re, write own the vlues of f, m n l. f = 0, m = 30, l = 10 3 Write the formul. A h 3 ( f + 4 m + l ) 4 Sustitute. A 30 (0 + 4 30 + 10) 3 5 Clulte the top re. 10 130 1300 m 2 6 For the ottom re, write own the vlues of f, m n l. f = 0, m = 17, l = 0 Chpter 2 Further pplitions of re n volume 53
7 Write own the formul. A h 3 ( f + 4 m + l ) 8 Sustitute. A 30 3 (0 + 4 17 + 0) 9 Clulte the ottom re. 10 68 680 m 2 10 A the two res together. Are 1300 + 680 1980 m 2 Simpson s rule pproximtes n re, it oes not give n ext mesurement. To otin etter pproximtion, Simpson s rule n e pplie severl times to the re. This is one y splitting the re in hlf n pplying Simpson s rule seprtely to eh hlf. WORKED EXAMPLE 10 Use two pplitions of Simpson s rule to pproximte the re on the right. 32 m 31 m 24 m 29 m 30 m Tutoril int-2413 Worke exmple 10 105 m THINK 1 Clulte h y iviing 105 y 4. (We re using 4 su-intervls.) 2 Apply Simpson s rule to the left hlf. Write the vlues of f, m n l. WRITE h = 105 4 = 26.25 f f = 32, m = 31, l = 24 3 Write the formul. A h 3 ( f + 4 m + l ) 4 Sustitute. A 5 Clulte the pproximte re of the left hlf. 8.75 180 1575 m 2 6 Apply Simpson s rule to the right hlf. Write the vlues of f, m n l. f 26. 25 (32 + 4 31 + 24) 3 f = 24, m = 29, l = 30 7 Write the formul. A h 3 ( f + 4 m + l ) 26. 25 8 Sustitute. A (24 + 4 29 + 30) 3 9 Clulte the pproximte re of the right hlf. 8.75 170 1487.5 m 2 10 A the res together. Are 1575 + 1487.5 3062.5 m 2 54 Mths Quest Generl Mths HSC Course
REMEMBER 1. Simpson s rule is metho of pproximting irregulr res. 2. The Simpson s rule formul is A h 3 ( f + 4 m + l ), where h is the istne etween suessive mesurements, f is the first mesurement, m is the mile mesurement n l is the lst mesurement. 3. A etter pproximtion of n re n e foun y using Simpson s rule severl times. EXERCISE 2C Simpson s rule Digitl o SkillSHEET 2.4 o-1308 Sustitution into formuls 1 WE8 The igrm on the right is of prt of reek. Stte the vlue of h. Stte the vlue of f, m n l. Use Simpson s rule to pproximte the re of this setion of the reek. 40 m 9 m 18 m 60 m 2 Use Simpson s rule to pproximte eh of the res elow. 10 m 35 m 12 m 6 m 28 m 72 m 12 m 0 m 16 m 54 m 40 m 48 m 3 WE9 The irregulr re on the right hs een ivie into two res lelle A 1 (upper re) n A 2 (lower re). Use Simpson s rule to fin n pproximtion for A l. Use Simpson s rule to fin n pproximtion for A 2. Wht is the pproximte totl re of the figure? 7 m 30 m A 1 11 m 19 m 80 m 5 m 18 m A 2 Chpter 2 Further pplitions of re n volume 55
4 Use Simpson s rule to fin n pproximtion for eh of the res elow. 22 m 11 m 17 m 45 m 45 m 12 m 14 m 6 m 0 16 m 27 m 23 m 16 m 10 m 12 m 21 m 31 m 27 m 27 m 40 m 21 m 5 MC Consier the figure rwn on the right. Simpson s rule gives n pproximte re of: A 1200 m 2 B 2400 m 2 C 3495 m 2 D 6990 m 2 6 MC If we pply Simpson s rule twie, how mny mesurements from the trverse line nee to e tken? A 4 B 5 C 7 D 9 7 WE 10 Use Simpson s rule twie to pproximte the re on the right. 8 Use Simpson s rule twie to pproximte eh of the res rwn elow. 50 m 22 m 45 m 10 m 51 m 90 m 36 m 18 m 18 m 18 m 18 m 25 m 7 m 54 m 63 m 45 m 102 m 87 m 42 m 71 m 20 m 44 m 50 m 60 m 60 m 60 m 60 m 11 m 33 m 22 m 21 m 21 m 21 m 21 m 10 m 10 m 10 m 10 m 9 The figure on the right is of ross-setion of wterwy. Use Simpson s rule one to fin n pproximte re of this setion of ln. Use Simpson s rule twie to otin etter pproximtion for the ross-setion. 15 m 32 m 36 m 27 m 56 Mths Quest Generl Mths HSC Course
10 Apply Simpson s rule four times to pproximte the re on the right. Digitl o WorkSHEET 2.1 o-1309 20 m 30 m 35 m 36 m 38 m 41 m 45 m 30 m 24 m 9 m 9 m 9 m 9 m 9 m 9 m 9 m 9 m Further evelopment 11 Explin why Simpson s rule n not e use to fin the re of the figure elow. 18 m 21 m 12 m 15 m 10 m 10 m 10 m 12 The figure elow shows the entrne to ve. 7 m 1.8 m 1.9 m 6 m 6 m Fin the re of the entrne. The ve is 25 metres long n pproximtely hs the sme ross-setion for its entire epth. Approximte the volume of the ve. 13 The figure elow shows the ross setion of river. 15 m 15 m 8 m Fin the re of the ross-setion. 14 The figure elow shows semiirle of rius 12 m. 12 m Fin the re of the semiirle orret to the nerest m 2. Joe fins the re of the semiirle using Simpson s rule, tking h = 12 m, the mile vlue s 12 m, while the first n lst vlues re oth zero. Wht nswer oes Joe get? Fin the perentge error in using Simpson s rule to fin the re of this semiirle. Chpter 2 Further pplitions of re n volume 57
15 A otnist nees to estimte the numer of trees in ertin heln re. It is known tht there re pproximtely 32 trees in every 100 squre metres. A igrm of the re is rwn elow. 20 m 60 m 35 m 17 m Complete the estimte of the numer of trees to the nerest 10 trees. 16 The figure elow is top view of n in groun swimming pool. The pool is symmetril n hs length of 16 metres. At its two wiest points the with of the pool is 10 metres n t the nrrow point in the mile it is 5 metres wie. The pool is 2.5 metres eep t ll points. Use Simpson s rule to estimte the volume of the pool. 2D Surfe re of yliners n spheres From erlier work you shoul rememer tht surfe re is the re of ll surfes of 3-imensionl shpe. Consier lose yliner with rius (r) n perpeniulr height (h). The surfe of the yliner onsists of two irles n retngle. Are of top = π r 2 Are of ottom = π r 2 The retngulr sie of the yliner will hve length equl to the irumferene of the irle (2π π r) n with equl to the height (h) of the yliner. Are of sie = 2π π rh r h The surfe re of the lose yliner n e lulte using the formul: SA = 2π π r 2 + 2π π rh 58 Mths Quest Generl Mths HSC Course
WORKED EXAMPLE 11 Clulte the surfe re of the lose yliner rwn on the right. Give your nswer orret to 1 eiml ple. 10 m 9 m THINK WRITE 1 Write the formul. SA = 2π π r 2 + 2π π rh 2 Sustitute the vlues of r n h. = 2 π 9 2 + 2 π 9 10 3 Clulte the surfe re. = 1074.4 m 2 For yliners, efore lulting the surfe re you nee to onsier whether the yliner is open or lose. In the se of n open yliner there is no top n so the formul nees to e written s: SA = π r 2 + 2π π rh Note: On the formul sheet in the exm only the formul for the lose yliner is provie. You will nee to hek the question n pt the formul yourself if neessry. WORKED EXAMPLE 12 Clulte the surfe re of n open yliner with rius of 6.5 m n height of 10.8 m. Give your nswer orret to 1 eiml ple. THINK WRITE 1 Write the formul. SA = π r 2 + 2π π rh 2 Sustitute the vlues of r n h. = π (6.5) 2 + 2 π 6.5 10.8 3 Clulte the surfe re. = 573.8 m 2 A sphere is roun 3-imensionl shpe, n the only mesurement given is the rius (r). The surfe re of sphere n e lulte using the formul: SA = 4π π r 2 r WORKED EXAMPLE 13 Clulte the surfe re of the sphere rwn on the right. Give the nswer orret to 1 eiml ple. 2.7 m Chpter 2 Further pplitions of re n volume 59
THINK WRITE 1 Write the formul. SA = 4π π r 2 2 Sustitute the vlue of r. = 4 π (2.7) 2 3 Clulte the surfe re. = 91.6 m 2 REMEMBER 1. The surfe re of lose yliner is foun using the formul SA = 2πr π r 2 + 2πrh π. 2. If the yliner is n open yliner, the surfe re formul eomes SA = πrπ r 2 + 2πrh π. 3. The surfe re of sphere is foun using the formul SA = 4πr π r 2. The Atomium, Brussels EXERCISE 2D Digitl o SkillSHEET 2.5 o-1310 Cirumferene of irle Surfe re of yliners n spheres 1 WE 11 Clulte the surfe re of lose yliner with rius of 5 m n height of 11 m. Give your nswer orret to 1 eiml ple. 2 Clulte the surfe re of eh of the lose yliners rwn elow. Give eh nswer orret to 1 eiml ple. 3 m 12 m 1.1 m 1.6 m 20 m 5 m e f r 20 m r = 5 m 5.9 m 5.9 m 2.3 m 1.5 m 3 Clulte the surfe re of lose yliner with imeter of 3.4 m n height of 1.8 m. Give your nswer orret to 1 eiml ple. 4 WE 12 Clulte the surfe re of n open yliner with rius of 4 m n height of 16 m. Give your nswer orret to the nerest whole numer. 60 Mths Quest Generl Mths HSC Course
5 Clulte the surfe re of eh of the following open yliners. Give eh nswer orret to 1 eiml ple. 13.3 m 22 m 30 m r r = 4.1 m 9.6 m 20 m e f 23.2 m 50 m 3.2 m 2.4 m 4 m 4 m 6 Fin the outsie surfe re of yliner open t oth ens with rius of 5 m n height of 10 m. 7 A n of fruit is me of stinless steel. The n hs rius of 3.5 m n height of 7 m. A lel is to e wrppe roun the n. Clulte the mount of steel neee to mke the n (orret to the nerest whole numer). Clulte the re of the lel (orret to the nerest whole numer). 8 WE 13 Clulte the surfe re of sphere with rius of 3 m. Give your nswer orret to the nerest whole numer. 9 Clulte the surfe re of eh of the spheres rwn elow. Give eh nswer orret to 1 eiml ple. 8 m 2.1 m 14 m e f 1 m 3.4 m 1.8 m 10 Clulte the surfe re of sphere with imeter of 42 m. Give your nswer orret to the nerest whole numer. 11 MC An open yliner hs imeter of 12 m n height of 15 m. Whih of the following lultions gives the orret surfe re of the yliner? A π 6 2 + 2 π 6 15 B 2 π 6 2 + 2 π 6 15 C π 12 2 + 2 π 12 15 D 2 π 12 2 + 2 π 12 15 Chpter 2 Further pplitions of re n volume 61
12 MC Whih of the following figures hs the gretest surfe re? A A lose yliner with rius of 5 m n height of 10 m B An open yliner with rius of 6 m n height of 10 m C A yliner open t oth ens with rius of 7 m n height of 10 m D A sphere with rius of 6 m 13 An open yliner hs imeter n height of 12 m. Clulte the surfe re of the yliner (orret to the nerest whole numer). A sphere sits extly insie this yliner. Clulte the surfe re of this sphere (orret to the nerest whole numer). Further evelopment 14 A ylinril n is to ontin three tennis lls eh hving imeter of 6 m. Clulte the surfe re of eh ll. The three lls fit extly insie the n. Stte the rius n height of the n. The n is open n me of stinless steel, exept the top whih will e plsti. Clulte the re of the plsti li (orret to the nerest whole numer). Clulte the mount of stinless steel in the n (orret to the nerest whole numer). e Clulte the re of pper lel tht is to e wrppe roun the n (orret to the nerest whole numer). 15 Clulte the surfe re of the hemisphere rwn elow given tht it is open t the se. 22 m Give your nswer orret to the nerest hunre m 2. 16 Clulte the outsie surfe re of the hemisphere in question 15 if it is lose t the se. 17 Fin the surfe re of the lrgest sphere tht n e ple insie the yliner right. 8 m 4 m 18 A tennis ll hs imeter of 7 m. Clulte the surfe re of tennis ll nister tht is to hol four tennis lls. 19 Soer lls hve imeter of 30 m. The soer ll is to e ple in the smllest possile ui ox. Clulte the surfe re of this ox. Clulte the perentge mteril sve y pling the ll in spheril ox of imeter 30 m. Give your nswer to the nerest whole numer. 62 Mths Quest Generl Mths HSC Course
Digitl o EXCEL Spresheet o-1311 Volume Computer Applition 1: Minimising surfe re Aess the spresheet Volume from the Mths Quest Generl Mthemtis HSC Course ebookplus. A ylinril rink ontiner is to hve pity of 1 litre (volume = 1000 m 3 ). We re going to lulte the most ost effiient imensions to mke the ontiner. To o this, we wnt to mke the ontiner with s little mteril s possile, in other wors we wnt to minimise the surfe re of the yliner. The spresheet shoul look s shown elow. 1. In ell B3 enter the volume of the yliner, 1000. 2. In ell A6 enter rius of 1. In ell A7 enter rius of 2 n so on up to rius of 20. 3. The formul tht hs een entere in ell B6 will give the height of the yliner orres poning to the rius for the given volume. 4. The surfe re of eh possile yliner is in olumn D. Use the hrting funtion on the spresheet to grph the surfe re ginst the rius. 5. Wht re the most ost-effiient imensions of the rink ontiner? Digitl os Investigtion o-2747 Pkging Chllenge exerise Use one of the other worksheets to fin the most effiient imensions to mke retngulr prism of volume 1000 m 3 n one of volume 200 m 3. Chpter 2 Further pplitions of re n volume 63
2E Volume of omposite solis Mny soli shpes re ompose of two or more regulr solis. To lulte the volume of suh figure, we nee to etermine the est metho for eh prtiulr prt. Mny irregulr shpes my still e prisms. A prism is shpe in whih every ross-setion tken prllel to the se shpe is equl to tht se shpe. The formul for the volume of prism is: V = Ah where A is the re of the se shpe n h is the height. Rememer tht the se of the prism is not neessrily the ottom. The se is the shpe tht is onstnt throughout the prism n will usully e rwn s the front of the prism. This mens tht the height will e rwn perpeniulr to the se. To lulte the volume of ny prism, we first lulte the re of the se n then multiply y the height. WORKED EXAMPLE 14 Fin the volume of the figure rwn on the right. 4 m THINK 1 Divie the front fe into two retngles. WRITE 4 m 12 m 12 m 6 m 10 m 3 m A 1 A2 10 m 2 Clulte the re of eh. Α 1 = 4 12 A 2 = 6 6 = 48 m 2 = 36 m 2 3 A the res together to fin the vlue of Α. A = 48 + 36 = 84 m 2 4 Write the formul. V = A h 6 m 5 Sustitute A = 84 n h = 3. = 84 3 6 Clulte. = 252 m 3 If the shpe is not prism, you my nee to ivie it into two or more regulr 3- imensionl shpes. You oul then lulte the volume y fining the volume of eh shpe seprtely. You will nee to use importnt volume formuls tht pper on the formul sheet: Cone: V = 1 πrπ r 2 h 3 Cyliner: V = πrπ r 2 h Pyrmi: V = 1 Ah Sphere: V = 4 πr 3 3 π 3 64 Mths Quest Generl Mths HSC Course
WORKED EXAMPLE 15 Clulte the volume of the figure rwn on the right, orret to 2 eiml ples. 2.4 m 1.2 m THINK WRITE 1 The shpe is yliner with hemisphere on top. 2 Write own the formul for the volume of yliner. V = πrπ r 2 h 3 Sustitute r = 1.2 n h = 2.4. = π (1.2) 2 2.4 4 Clulte the volume of the yliner. = 10.857 m 3 5 Write own the formul for the volume of hemisphere. (This is the formul for the volume of sphere ivie y 2.) V = 4 πrπ 3 2 3 6 Sustitute r = 1.2. = 4 3 π (1.2)3 2 7 Clulte the volume of the hemisphere. = 3.619 m 3 8 A the two volumes together. Volume = 10.857 + 3.619 = 14.48 m 3 In mny ses volume question my e presente in the form of prtil prolem. WORKED EXAMPLE 16 A wter storge tnk is in the shpe of ue of sie length 1.8 m, surmounte y yliner of imeter 1 m with height of 0.5 m. Clulte the pity of the tnk, orret to the nerest 100 litres. THINK WRITE 1 Drw igrm of the wter tnk. 1 m 0.5 m 1.8 m 2 Clulte the volume of the ue using the formul V = s 3. 3 Clulte the volume of the yliner using the formul V = πrπ r 2 h. V = s 3 = 1.8 3 = 5.832 m 3 V = πrπ r 2 h = π 0.5 2 0.5 = 0.393 m 3 Chpter 2 Further pplitions of re n volume 65
4 A the volumes together. Volume = 5.832 + 0.393 = 6.225 m 3 5 Clulte the pity of the tnk using 1 m 3 = 1000 L. Cpity = 6.225 1000 = 6225 L 6 Give n nswer in wors. The pity of the tnk is pproximtely 6200 litres. REMEMBER 1. To fin the volume of ny prism, use the formul V = A h, where A is the re of the se n h is the height. 2. Importnt volume formuls: Cone: V = 1 πrπ r 2 h 3 Pyrmi: V = 1 3 Ah Cyliner: V = πrπ r 2 h Sphere: V = 4 πrπ 3 3 where r = rius, h = perpeniulr height, A = re of se. 3. For other shpes, lulte the volume of eh prt of the shpe seprtely, then together eh prt t the en. 4. Rememer to egin wore or prolem question with igrm n finish with wor nswer. EXERCISE 2E Volume of omposite solis Digitl o SkillSHEET 2.6 o-1312 Volume of ues n retngulr prisms 1 WE 14 Look t the figure rwn on the right. Fin the re of the front fe. Use the formul V = A h to lulte the volume of the prism. 18 m 6 m 20 m 5 m 4 m 2 Fin the volume of the following prisms (to 2 eiml ple). 18 mm 15 m [Bse re: 35 mm 2 ] 14 mm 6 mm 26.5 mm [Bse re: 28 m 2 ] 15 mm 8 mm 6 mm 66 Mths Quest Generl Mths HSC Course
3 Clulte the volume of eh of the figures rwn elow. 5 m 12 m 15 m 10 m 20 m 4 m 5 m 12 m 20 m 0.7 m 12 m 25 m 2.3 m 40 m 3 m 2.1 m 0.4 m e 0.5 m f 4 m 0.6 m 5 m 1 m 2 m 1.5 m Digitl o SkillSHEET 2.7 o-1313 Volume of tringulr prisms 4 WE 15 Consier the figure on the right. The shpe onsists of ue with squre pyrmi on top. Wht is the volume of the ue? Wht is the volume of the squre pyrmi? Wht is the totl volume of this figure? 2 m 1.5 m 5 The figure on the right is yliner with one mounte on top. Clulte the volume of the yliner, orret to the nerest m 3. Clulte the volume of the one, orret to the nerest m 3. Wht is the totl volume of the figure? 40 m 50 m 12 m 6 Clulte the volume of eh of the figures rwn elow, orret to 1 eiml ple. Digitl o SkillSHEET 2.8 o-1314 Volume of yliners r 34 m r =12 m 3 m 5 m 50 m Chpter 2 Further pplitions of re n volume 67
7 MC Whih of the figures rwn elow is not prism? A B Digitl o SkillSHEET 2.9 o-1315 Volume of sphere C D 8 MC The volume of the figure rwn on the right is losest to: A 718 m 3 B 1437 m 3 C 2155 m 3 D 2873 m 3 7 m 14 m 9 A fish tnk is in the shpe of retngulr prism. The se mesures 45 m y 25 m. The tnk is fille to epth of 15 m. Clulte the volume of wter in the tnk in m 3. Given tht 1 m 3 = 1 ml lulte, in litres, the mount of wter in the tnk. 10 WE 16 A hemispheril wine glss of rius 2.5 m is joine to yliner of rius 1 m n height 5 m. The glss then rests on soli se. Drw igrm of the wine glss. Clulte the pity of the glss, to the nerest 10 ml. How mny glsses of wine n e poure from 1 litre ottle? 11 The figure on the right is the ross-setion of onrete pipe use s sewge outlet. Clulte the re of ross-setion of the pipe, orret to 2 eiml ples. Clulte the mount of onrete neee to mke 10 m length of this pipe. 3 m 2.5 m 12 A ommemortive riket ll hs imeter of 7 m. It is to e preserve in ui se tht will llow 5 mm on eh sie of the ll. Wht will the sie length of the ui se e? Clulte the mount of empty spe insie the se, to the nerest whole numer. Clulte the perentge of spe insie the se oupie y the ll, to the nerest whole numer. 68 Mths Quest Generl Mths HSC Course
Digitl o SkillSHEET 2.10 o-1316 Volume of pyrmi Further evelopment 13 A imon is ut into the shpe of two squre-se pyrmis s shown on the right. Eh mm 3 of the imon hs mss of 0.04 g. Clulte the mss of the imon. 6 mm 6 mm 14 Fin the volume of these ojets (to the nerest whole unit). r = 9 m 8 m 5 m 12 m 5 m 6 m 2 m 5 m 2.5 m 3 m 3 m e f 2 m 20 m 4.2 m 12 m 35 m 5 m 12 m g h 19 m 21 m 21 m 60 m 3 m 1.6 m 15 The meiine up on the right hs the shpe of one with imeter of 4 m n height of 5 m (not inluing the up s se). Fin the volume of the one to the nerest millilitre, where 1 m 3 = 1 ml. 14 m 15 m Chpter 2 Further pplitions of re n volume 69
16 Tennis lls hve imeter of 6.5 m n re pkge in yliner tht n hol four tennis lls. Assuming the lls just fit insie yliner, fin: the height of the ylinril n the volume of the n (to 1 eiml ple) the volume of the four tennis lls (to 1 eiml ple) the volume of the n oupie y ir e the frtion of the n s volume oupie y the lls. 17 MC The rtio of the volume of sphere to tht of yliner of similr imensions, s shown in the igrm, is est expresse s: r A C 4 3 3 4 B 2 3 D 3 2 r Digitl os Investigtion o-2748 Mximising volume 2F 18 A moel eroplne is ontrolle y tethere string of 10 metres length. The opertor stns in the mile of n ovl. (Give ll nswers to the nerest whole unit.) Wht is the mximum re of the ovl oupie y the plne in flight? If the plne n e mnoeuvre in hemispheril zone, fin: i the surfe re of the irspe tht the plne n oupy ii the volume of irspe tht is neee y the opertor for ontrolling the plne. Repet prt with new ontrol string length of 15 metres. Error in mesurement As we sw in the preliminry ourse, ll mesurements re pproximtions. The egree of ury in ny mesurement is restrite y the ury of the mesuring evie n the egree of prtility. We hve previously seen tht the mximum error in ny mesurement is hlf of the smllest unit of mesurement. This error is ompoune when further lultions suh s surfe re or volume re me. WORKED EXAMPLE 17 In the retngulr prism on the right, the length, reth n height hve een mesure, orret to the nerest entimetre. Clulte the volume of the retngulr prism. Clulte the gretest possile error in the volume. 8 m 20 m 15 m THINK WRITE Clulte the volume of the retngulr prism. 1 Write the smllest possile imensions of the prism. V = l w h = 20 15 8 = 2400 m 3 Smllest possile imensions: l = 19.5, w = 14.5, h = 7.5 2 Clulte the smllest possile volume. V = l w h = 19.5 14.5 7.5 = 2120.625 m 3 70 Mths Quest Generl Mths HSC Course
3 Write the lrgest possile imensions of the prism. Lrgest possile imensions: l = 20.5, w = 15.5, h = 8.5 4 Clulte the lrgest possile volume. V = l w h = 20.5 15.5 8.5 = 2700.875 m 3 5 Clulte the mximum error. Mximum error = 2700.875 2400 = 300.875 m 3 As n e seen in the ove exmple, possile error of 0.5 m in the liner mesurement ompouns to n error of 300.875 m 3 in the volume mesurement. Errors in mesurement will ompoun errors in ll further lultions. WORKED EXAMPLE 18 A swimming pool is uilt in the shpe of retngulr prism with length of 10.2 m, with of 7.5 m n epth of 1.5 m. The floor n the sies of the pool nee to e emente. Clulte the re tht is to e emente. The onreter inorretly mesure the length of the pool s 9.4 m. Clulte the error in the re lultion. Clulte the perentge error (orret to 1 eiml ple) in the re lultion. THINK WRITE 1 Clulte the re of the pool floor. Are of floor = 10.2 7.5 = 76.5 m 2 2 Clulte the re of the ens. Are of ens = 7.5 1.5 = 11.25 m 2 3 Clulte the re of the sies. Are of sies = 10.2 1.5 = 15.3 m 2 4 Clulte the totl re to e emente. Totl re = 76.5 + 2 11.25 + 2 15.3 = 129.6 m 2 1 Use the inorret mesurement to repet ll the ove lultions. Tutoril int-2414 Worke exmple 18 Are of floor = 9.4 7.5 = 70.5 m 2 Are of ens = 7.5 1.5 = 11.25 m 2 Are of sies = 9.4 1.5 = 14.1 m 2 Totl re = 70.5 + 2 11.25 + 2 14.1 = 121.2 m 2 Chpter 2 Further pplitions of re n volume 71
2 Fin the ifferene etween the two nswers. Error = 129.6 121.2 = 8.4 m 2 Write the error s perentge of the orret nswer. Perentge error = 84. 100% 129. 6 = 6.5% REMEMBER 1. 2. 3. All mesurements re pproximtions. The ury of ny mesurement is limite y the instrument use n the most prtil egree of ury. The mximum error in ny liner mesurement is hlf the smllest unit use. Any error me in liner mesurement will ompoun when use in further lultions suh s those for surfe re or volume. EXERCISE 2F Digitl o SkillSHEET 2.11 o-1317 Error in liner mesurement Error in mesurement 1 WE 17 In the figure on the right eh mesurement hs een tken to the nerest entimetre. Clulte the volume of the figure. Clulte the mximum error in the volume lultion. 16 m 2 The rius of irle is mesure s 7.6 m, orret to 1 eiml ple. Wht is the mximum possile error in the mesurement of the rius? Clulte the re of the irle. Give your nswer orret to 1 eiml ple. Clulte the mximum possile error in the re of the irle. Clulte the mximum possile error in the re of the irle s perentge of the re. 3 A ue hs sie length of 16 mm, orret to the nerest millimetre. Clulte the volume of the ue. Clulte the smllest possile volume of the ue. Clulte the lrgest possile volume of the ue. Clulte the mximum possile perentge error in the volume of the ue. e Clulte the surfe re of the ue. f Clulte the smllest possile surfe re of the ue. g Clulte the lrgest possile surfe re of the ue. h Clulte the mximum possile perentge error in the surfe re of the ue. 4 A yliner hs rius of 4 m n height of 6 m with eh mesurement eing tken orret to the nerest entimetre. Clulte the volume of the yliner (orret to the nerest whole numer). Clulte the smllest possile volume of the yliner (orret to the nerest whole numer). Clulte the lrgest possile volume of the yliner (orret to the nerest whole numer). Clulte the gretest possile perentge error in the volume of the yliner. 5 For the yliner in question 4, lulte the gretest possile perentge error in the surfe re of the yliner. 12 m 6 m 72 Mths Quest Generl Mths HSC Course
Digitl o WorkSHEET 2.2 o-1318 6 The rius of sphere is 1.4 m with the mesurement tken orret to 1 eiml ple. Clulte the volume of the sphere, orret to 1 eiml ple. Clulte the mximum possile error in the volume of the sphere. Clulte the mximum perentge error in the volume. Clulte the surfe re of the sphere, orret to 1 eiml ple. e Clulte the mximum possile error in the surfe re of the sphere. f Clulte the mximum perentge error in the surfe re. 7 WE 18 An open ylinril wter tnk hs rius of 45 m n height of 60 m. Clulte the pity of the tnk, in litres (orret to the nerest whole numer). If the tnk s rius is given s 50 m, orret to the nerest 10 m, lulte the error in the pity of the tnk. Clulte the perentge error in the pity of the tnk. 8 A retngulr prism hs imensions 56 m 41 m 17 m. Clulte the volume of the prism. Clulte the surfe re of the prism. If the imensions re given to the nerest 10 m, wht will the imensions of the prism e given s? Clulte the perentge error in the volume when the imensions re given to the nerest 10 m. e Clulte the perentge error in the surfe re when the imensions re given to the nerest 10 m. 9 The four wlls of room re to e pinte. The length of the room is 4.1 m n the with is 3.6 m. Eh wll is 1.8 m high. Clulte the re to e pinte. One litre of pint will pint n re of 2 m 2. Eh wll will nee two ots of pint. Clulte the numer of litres of pint require to omplete this jo. Krl inorretly mesures the length of the room to e 3.9 m. If Krl oes ll her lultions using this inorret mesurement, how mny litres will she e short of pint t the en of the jo? 10 The imensions of retngulr house re 16.6 m y 9.8 m. Simon tkes the imensions of the house to the nerest metre for ll his lu ltions. Wht imensions oes Simon use? Simon plns to floor the house in slte tiles. Wht is the re tht nees to e tile? The tiles ost $27.50/m 2 n Simon uys n extr 10% to llow for utting n rekge. Clulte the ost of the tiles. How muh extr hs Simon spent thn woul hve een neessry h he use the originl mesurements of the house? Chpter 2 Further pplitions of re n volume 73
Further evelopment 11 The imensions of retngulr ourtyr re 20 metres y 12 metres, orret to the nerest metre. The re is to e pve with pvers tht re squres of sie length 50 m. Clulte the numer of pvers tht will e neee to ensure tht the entire ourtyr is pve, llowing for possile mesurement error in the ourtyr mesurements. If this numer of pvers re orere wht woul e the mximum numer of pvers tht oul e left over t the en of the jo? 12 The re of squre is mesure s eing 4900 m 2, orret to the nerest 100 m 2. Fin: the sie length of the squre the mximum possile sie length (orret to 1 eiml ple) the minimum possile sie length (orret to 1 eiml ple) the mximum perentge error in the sie length (orret to 2 eiml ples). 13 The volume of sphere is foun to e 23 000 m 3, orret to the nerest 1000 m 3. Fin the mximum perentge error in: the volume of the sphere the rius of the sphere. Give your nswers orret to 2 eiml ples. 14 A yliner hs given rius of 10 m n height of 30 m, orret to the nerest entimetre. Fin the volume of the yliner, orret to the nerest m 3. A liqui is to e poure into the yliner. The liqui n expn y s muh s 10% in hot wether. Allowing for possile error in mesurement, wht is the mximum mount of the liqui tht n e poure into the yliner suh tht none will spill in the event of expnsion? Give your nswer orret to the nerest 100 ml. 15 Fin the re of setor of irle of rius 15 m n sutening 74 ο ngle t the entre. Fin the length of the r forme. Fin the mximum perentge error in i the re of the setor ii the r length given tht the rius ws mesure to the nerest entimetre n the ngle ws mesure to the nerest egree. 16 The figure elow is of fiel. The re is to e pproximte using Simpson s rule. 5 m 30 m 67 m 30 m 18 m Estimte the re. Given tht eh mesurement is tken orret to the nerest metre, fin the smllest possile re of the fiel. Fin the lrgest possile re of the fiel. Fin the mximum perentge error. 74 Mths Quest Generl Mths HSC Course
SUMMARY Are of prts of the irle The re of irle n e lulte using the formul A = πrπ r 2. The re of setor is foun y multiplying the re of the full irle y the frtion of the irle oupie y the setor. This is lulte y looking t the ngle tht the setor mkes with the entre. An nnulus is the re etween two irles. The re is lulte y sutrting the re of the smller irle from the re of the lrger irle or y using the formul A = π( π R 2 r 2 ), where R is the rius of the lrge irle n r is the rius of the smll irle. The re of n ellipse is lulte using the formul A = ππ, where is the length of the semi-mjor xis n is the length of the semi-minor xis. Are of omposite shpes The re of omposite figure is lulte y iviing the figure into two or more regulr figures. When lulting the re of omposite figure, some sie lengths will nee to e lulte using Pythgors theorem. Simpson s rule Simpson s rule is use to fin n pproximtion for n irregulr re. The formul for Simpson s rule is A h 3 ( f + 4 m + l ). To otin etter pproximtion for n re, Simpson s rule n e pplie twie. This is one y iviing the re in hlf n pplying Simpson s rule seprtely to eh hlf. Surfe re of yliners n spheres The surfe re of lose yliner is foun y using the formul SA = 2πr π r 2 + 2πrh π. If the yliner is n open yliner, the surfe re is foun using SA = πrπ r 2 + 2πrh π. The surfe re of sphere is lulte using the formul SA = 4πr π r 2. Volume of omposite solis The volume of soli prisms is lulte using the formul V = A h. The volume of one is foun using the formul V = 1 πrπ r 2 h. 3 The volume of yliner is foun using the formul V = πrπ r 2 h. The volume of sphere is foun using the formul V = 4 πrπ 3. 3 The volume of pyrmi is foun using the formul V = 1 Ah. 3 Other solis hve their volume lulte y iviing the soli into regulr soli shpes. Error in mesurement All mesurements re pproximtions. The mximum error in ny mesurement is hlf the smllest unit use. Any error in mesurement will ompoun when further lultions using the mesurement nee to e me. Chpter 2 Further pplitions of re n volume 75
CHAPTER REVIEW MULTIPLE CHOICE 1 Whih of the following lultions will orretly give the re of the ellipse rwn elow? A π 6.2 2 B π 8.5 2 6.2 mm C π 10.8 2 D π 10.8 6.2 10.8 mm 2 The fiel rwn elow is to hve its re pproximte y two pplitions of Simpson s rule. The vlue of h is: A 16 B 20 C 40 D 80 80 m 3 The figure rwn elow is n open yliner. Whih of the lultions elow will orretly give the surfe re of the yliner? A π 5 2 + 2 π 5 20 B 2 π 5 2 + 2 π 5 20 C π 10 2 + 2 π 10 20 20 m D 2 π 10 2 + 2 π 10 20 10 m 4 A lose yliner is mesure s hving rius of 1.2 m n height of 1.4 m. The mximum error in the lultion of the surfe re is: A 1.2 m 2 B 1.5 m 2 C 1.6 m 2 D 19.6 m 2 SHORT ANSWER 10 m 30 m 15 m 25 m 20 m 1 Clulte the re of eh of the irles elow. Give eh nswer orret to 1 eiml ple. 2 Clulte the re of eh of the figures elow. Give eh nswer orret to 1 eiml ple. 30 92 mm 237 12.5 m 4.8 m 3 Clulte the re of eh of the nnuluses elow. Give eh nswer orret to 1 eiml ple. 81 mm 34 m 94 mm 17 m 3.7 m 1.3 m 4 Clulte the re of eh of the ellipses elow, orret to 1 eiml ple. 30 mm 3.7 m 52 mm 45 mm 9.2 m 1.7 m 11.4 m 76 Mths Quest Generl Mths HSC Course
7 m 3.6 m 5 Clulte the re of the figure elow. 35 m 10 m 10 m 12 m 15 m 6 Clulte the re of eh of the figures elow. Where pproprite, give your nswer orret to 2 eiml ples. 4.1 m 0.7 m 3.9 m 0.9 m 1.5 m 3 m 1.5 m 6 m 14 m 62 m 42 m 96 m 2 m 31 m 57 m 57 m 29 m 9 By iviing the re shown elow into two setions, use Simpson s rule to fin n pproximtion for the re. 27 m 62 m 50 m 30 m 30 m 25 m 11 m 19 m 10 Use Simpson s rule twie to fin n pproximtion for the re elow. 36 m 7 Use Simpson s rule to pproximte the re elow. 50 m 13 m 42 m 42 m 21 m 8 Use Simpson s rule to fin n pproximtion for eh of the res elow. 33 m 15 m 44 m 38 m 9 m 15 m 15 m 15 m 23 m 11 Clulte the surfe re of eh of the lose yliners rwn elow, orret to 1 eiml ple. 7 m 10 m 4 m 25 m 30 m 36 m 24 m 1.1 m 60 m Chpter 2 Further pplitions of re n volume 77
12 Clulte the surfe re of n open yliner with imeter of 9 m n height of 15 m. Give your nswer orret to the nerest whole numer. 13 Clulte the surfe re of sphere with: rius of 5 m rius of 2.4 m imeter of 156 mm. Give eh nswer orret to the nerest whole numer. 14 Clulte the volume of the soli rwn elow. 0.5 m 15 m 12 m 9 m 3 m 3 m 3 m 10 m 16 Clulte the volume of the figure rwn on the right, orret to 2 eiml ples. 9 m 15 m 3.1 m 2.7 m 1.9 m 0.6 m 15 Clulte the volume of eh of the solis rwn elow. Where neessry, give your nswer orret to the nerest whole numer. 19 m 12 m 17 A sphere hs imeter of 16 m when mesure to the nerest entimetre. Stte the mximum error me in the mesurement of the rius. Clulte the volume of the sphere. Answer orret to the nerest whole numer. Clulte the mximum perentge error in the volume of the sphere. 18 An luminium soft rink n hs imeter of 8 m n height of 10 m. Clulte the pity of the n, in millilitres, orret to the nerest 10 millilitres. The mhine tht uts the luminium for the n is mistkenly set to 12 m. Clulte the perentge error in the pity of the n (orret to the nerest whole numer). 17 m 22 m 20 m 40 m 10 m EXTENDED RESPONSE 1 The figure on the right shows setion of onrete ringe pipe. Clulte the re of the nnulus, orret to 1 eiml ple. Clulte the volume of onrete neee to mke 5 m length of this pipe (orret to 1 eiml ple). Clulte the volume of wter tht will flow through the 5 m length of the pipe (in litres, to the nerest 100 L). 2.5 m 1.5 m 78 Mths Quest Generl Mths HSC Course
Clulte the surfe re of 5 m setion of pipe (orret to the nerest m 2 ). (Hint: Inlue the re of the insie of the pipe.) 2 The igrm on the right shows the ross-setion of river. Use two pplitions of Simpson s rule to fin the pproximte re of the river s ross-setion. If the river flows with this ross-setion for pproximtely 800 m, lulte the volume of the river. The length of the river hs een pproximte to the nerest 100 m. Clulte the mximum perentge error in lulting this volume. 5.1 m 60 m 9.2 m 4.9 m Digitl o Test Yourself o-1319 Chpter 2 Chpter 2 Further pplitions of re n volume 79
ACTIVITIES Are you rey? Digitl os (pge 40) SkillSHEET 2.1 (o-1303): Are of irle. SkillSHEET 2.2 (o-1304): Ares of squres, retngles n tringles. SkillSHEET 2.6 (o-1312): Volume of ues n retngulr prisms. SkillSHEET 2.7 (o-1313): Volume of tringulr prisms. SkillSHEET 2.8 (o-1314): Volume of yliners. SkillSHEET 2.9 (o-1315): Volume of sphere. SkillSHEET 2.10 (o-1316): Volume of pyrmi. SkillSHEET 2.11 (o-1317): Error in liner mesurement. 2A Are of prts of the irle Tutoril WE 3 int-2411: Lern how to lulte the re of n nnulus. (pge 42) Digitl os SkillSHEET 2.1 (o-1303): Are of irle. (pge 43) 2B Are of omposite shpes Tutoril WE 6 int-2412: Lern how to lulte the re of omposite shpe. (pge 47) Digitl os SkillSHEET 2.2 (o-1304): Ares of squres, retngles n tringles. (pge 48) SkillSHEET 2.3 (o-1305): Using Pythgors theorem. (pge 49) Spresheet: (o-1306): Pythgors. (pge 49) GC progrm Csio (o-1307): Mensurtion. (pge 49) 2C Simpson s rule Tutoril WE 10 int-2413: Lern how to pply Simpson s rule. (pge 54) Digitl os SkillSHEET 2.4 (o-1308): Sustitution into formuls. (pge 55) WorkSHEET 2.1 (o-1309): Perform lultions involving re. (pge 57) 2D Surfe re of yliners n spheres Digitl os SkillSHEET 2.5 (o-1310): Cirumferene of irle. (pge 60) Spresheet (o-1311): Volume. (pge 63) Investigtion (o-2747): Pkging. (pge 63) 2E Volume of omposite solis Digitl os SkillSHEET 2.6 (o-1312): Volume of ues n retngulr prisms. (pge 66) SkillSHEET 2.7 (o-1313): Volume of tringulr prisms. (pge 67) SkillSHEET 2.8 (o-1314): Volume of yliners. (pge 67) SkillSHEET 2.9 (o-1315): Volume of sphere. (pge 68) SkillSHEET 2.10 (o-1316): Volume of pyrmi. (pge 69) Investigtion: (o-2748): Mximising volume. (pge 70) 2F Error in mesurement Tutoril WE 18 int-2414: Lern how to lulte perentge error. (pge 71) Digitl os SkillSHEET 2.11 (o-1317): Error in liner mesurement. (pge 72) WorkSHEET 2.2 (o-1318): Perform lultions involving volume. (pge 73) Chpter review Digitl o Test Yourself (o-1319): Tke the en-of-hpter test to test your progress. (pge 79) To ess ebookplus tivities, log on to www.jplus.om.u 80 Mths Quest Generl Mths HSC Course