Surface area and volume

Transcription

1 2 Surfe re nd volume This hpter dels with lulting the surfe res nd volumes of right prisms nd ylinders. fter ompleting this hpter you should e le to: solve prolems involving the surfe res nd volumes of right retngulr nd tringulr prisms lulte the surfe res nd volumes of ylinders solve prolems involving the surfe res nd volumes of omposite solids. NSW Syllus referenes: 5.2 &G re nd surfe re, 5.2 &G Volume Outomes: 5.2-1W, 5.2-2W, G, G SURNT & GOTRY CG242 02_LY_I_10SB_51_52_22693_PPS.indd 15 15/10/2014 8:36 m

2 Dignosti test 6 The surfe re of this tringulr prism is: Questions 1 to 3 refer to the prism elow. B 26 D 18.9 F H.1 m C m 12. G 1 The fe tht orresponds to the fe BF is: DCHG C HD B BFGC D DCGH 2 The fe tht orresponds to the fe CBFG is: C The surfe re of ue of side length 8.7 m is: C B HD D DGCH 3 The fe tht orresponds to the fe BCD is: re of the wrpping to the nerest m2 is: m 4 The surfe re of the net shown is: 8 hoolte r is shown elow. The surfe B HFG D DCGH B m2 D FHG C FGH B m2 D BF C DH 8.3 m 37.1 m m 80 m2 60 m2 3 m 100 m2 150 m2 C 151 m2 B m2 D 152 m2 9 This solid is mde C 232 m2 B 322 D The surfe re of the net shown is: 16. SURNT & GOTRY m from 1 m3 ues. The volume of the solid is: 18 m3 B 29 m3 C 16 m3 D 30 m3 10 The ross-setion of this solid is: n ovl B ylinder C n ellipse D irle 11 solid tht hs irulr ross-setion is C B D lled : pyrmid C retngulr prism B ylinder D ox Insight themtis 10 stges 5.1/5.2 ustrlin Curriulum 02_LY_I_10SB_51_52_22693_PPS.indd 16 15/10/2014 8:36 m

3 14 The volume of this omposite solid is: 12 The volume of the solid shown is: 61.4 m3 B m3 C m3 D m3 re 56.9 m m 13 The volume of this solid to the nerest m3 is: 709 m3 B 708 m3 C 705 m3 D 710 m m3 C 1200 m m B 1350 m3 D 960 m3 15 The volume of ue of side length 8.7 m is: C m B m3 D m The dignosti test questions refer to outomes CG210 nd CG218. X 1 re review 130 pply ngle of setor = 130 Find the re of setor of irle y ompring its ngle with the ngle of full irle, 360. setor ngle re of setor = 360 re of irle setor ngle (of setor) = re of irle 360 SURNT & GOTRY θ πr2 = = π m2 11. Think Solve Find the re of this setor to 1 deiml ple. xerise 2 1 Clulte the re of eh setor orret to 1 deiml ple. 5.2 m m 11. Chpter 2 Surfe re nd volume 02_LY_I_10SB_51_52_22693_PPS.indd /10/2014 8:36 m

4 d e f m m 17.5 m SURNT & GOTRY 2 Find the res of the following shpes. 4 m d e f X 2 Drw net for this retngulr prism, showing the lengths of its edges. Clulte the surfe re of the prism. Solve/Think pply Drw the net, identify the fes, nd trnsfer the edge lengths from Bk the solid to the net. Clulte the re of eh fe Left Bottom Right side side nd sum these res m 12 m 1 Front Top 9 m 10 km 3 m 6 m = (ottom + top) + (front + k) + (left side + right side) = (9 4) 2 + (9 6) 2 + (6 4) 2 = mm 5 m 5 mm 10 km 8 mm 16 mm 9 m 18 Insight themtis 10 stges 5.1/5.2 ustrlin Curriulum

5 3 For eh of the following retngulr prisms: i Drw net of eh prism nd mrk its edge lengths. ii Clulte the surfe re. d e f X 3 Drw net of this tringulr prism, mrking its edge lengths. Clulte the surfe re of the prism. Solve/Think pply Drw the net, identify the fes nd trnsfer the edge lengths from the solid to the net. Clulte the re of eh fe nd sum these res. 7 m = re of 2 tringles + re of 3 retngles 1_ = ( ) = m 4 Clulte the surfe re of eh of the following tringulr prisms. 5 m 3 m 10 m 3.3 m 10 m 3.6 m 7 mm 3.3 m 2.4 m 50 mm 12 m 3 m 2. 9 m 1.8 m 10 mm SURNT & GOTRY Chpter 2 Surfe re nd volume 19

6 1 d 9 m X 4 SURNT & GOTRY Clulte the length of the unknown edge of this tringulr prism. Drw net of the prism. Clulte its surfe re. Solve/Think By Pythgors theorem: x 2 = = 52 x = m (1 deiml ple) pply Clulte the unknown edge using Pythgors theorem. Drw the net nd lulte the surfe re s efore. 1_ = ( 2 6 4) = m 2 5 For eh tringulr prism: i Find the length of the unknown edge. ii Clulte the surfe re. 3 m 7.2 m x 11 m 7 m 7.2 m x 11 m 5 m 12 m x 8 m 15 mm 10 mm x 3 mm 20 Insight themtis 10 stges 5.1/5.2 ustrlin Curriulum

7 B Surfe res of right ylinders The formul for the surfe re of ylinder n e developed y utting the ylinder nd lying it out flt. The net then gives formul for the surfe re. Rememer: re of irle is π r2. The urved prt forms retngle of length 2πr nd redth h. = 2(re of irle) + re of retngle = 2 π r2 + 2π r h 2π r = 2π r2 + 2π rh The surfe re of losed ylinder is: = 2π r2 + 2π rh r h h The surfe re of ylinder open t oth ends is: = 2π rh 2π r r Cut X 1 SURNT & GOTRY Find the surfe re of this losed ylinder. 1 Solve Surfe re = 2π r2 + 2π rh = 2π π m2 (1 deiml ple) Think Rdius = Height = 1 pply For ylinder losed t oth ends: = 2π r 2 + 2π r h Chpter 2 Surfe re nd volume 02_LY_I_10SB_51_52_22693_PPS.indd /10/2014 8:36 m

8 xerise 2B 1 Complete to find the surfe re of this losed ylinder. Surfe re = 2π r2 + 2π = 2π 2 + 2π 8.7 m2 (1 deiml ple) 3.2 m 8.7 m 2 Find the surfe res of these losed ylinders to the nerest whole numer. 1.4 m 12 m 112 mm 4 mm e f 10 m 1.3 m X 2 8m m d Find the surfe re of this open ylinder. Solve/Think Surfe re = 2 π 8 23 = m2 3 Find the surfe res of the following open ylinders. 13 m SURNT & GOTRY Think Rdius = Height = 23 m 23 m pply For n open ylinder: = 2π rh 20 m 2 3 m 20 m 4 Determine how muh pint is required to over the outside of ylindril ontiner 12 m long with dimeter 10 m if eh litre of pint overs 15 m2. Whih hs the greter surfe re: ylinder of length 1 nd rdius, or ylinder of length nd rdius 10 m? 22 Insight themtis 10 stges 5.1/5.2 ustrlin Curriulum 02_LY_I_10SB_51_52_22693_PPS.indd 22 15/10/2014 8:36 m

9 5 Find the surfe re, orret to 1 deiml ple where neessry, of: n open n with rdius of nd height of 1 n open-ended pipe of 10 m rdius nd 5 m long the losed solid shown elow. 16 m 14 m 6 Determine the ost of pinting the exterior wlls nd top of ylindril whet silo tht is 40 m high nd 20 m in dimeter, given tht eh litre of pint osts $7.25 nd overs 8 m2. 7 Find the ost of mking 125 ylindril tennis ll ontiners X 3 tht hve dimeter 7 m nd height 21 m, given tht the metl osts $4.50 per squre metre (metl se ut open t the top). n open ylinder of rdius of hs urved surfe re of 1000 m2. Find its height. Solve pply To solve 1000 = 16π h, divide oth sides y 16π. Sustitute the given informtion into = 2π rh nd solve the resulting eqution = 2 π 8 h = 16π h 1000 h = 16π = 19.9 m (1 deiml ple) Think 8 Find the height of n open ylinder of rdius 10 m nd urved surfe re of 2000 m2. 9 Find the rdius of n open ylinder of height nd urved surfe re of 1500 m2. SURNT & GOTRY C Volumes with uniform ross-setions h h h The volume of right prism (or ylinder) is given y: V= h where is the re of the se (or ross-setionl re) nd h is the perpendiulr height. Chpter 2 Surfe re nd volume 02_LY_I_10SB_51_52_22693_PPS.indd /10/2014 8:36 m

10 X 1 Find the volumes of these solids. 17 m m Solve Think pply V = = 253 re of se = 17 m2 For prisms nd ylinders use V = h. V = (7.5 6) 4 or (7.5 4) 6 or (6 4) 7.5 = 180 m3 re of se = Choose ny retngle s the se. V = (π 52) 8 = m3 (1 deiml ple) re of se = πr2 = π 52 m2 The se is irle. xerise 2C 1 Complete to find the volume of this prism. V= h where is the re of nd h is the height. V = 25 = m3 2 Clulte the volumes of these solids d m2 SURNT & GOTRY 10 m 12 m 3 Clulte the re of the se nd hene find the volume of eh solid. 2m 8m 7m 12 m 7 m 24 Insight themtis 10 stges 5.1/5.2 ustrlin Curriulum 02_LY_I_10SB_51_52_22693_PPS.indd 24 15/10/2014 8:36 m

11 d 12 m 7 m e 12 m 20 m f 1 m X 2 1 Clulte the volume of eh solid orret to 1 deiml ple. 18 m 22 m Solve V= h 1 = _ 2 πr2 h 1 = _ 2 π Think pply Dimeter = 18 m so rdius = 9 m The se is semiirle so the re of the irle must e hlved. Clulte the re of the se first. ultiply y the height, whih must e perpendiulr to the se. The solid does not hve to stnd on the se. = m3 V= h 80 = πr2 h = π = m3 The se is setor. θ πr2 where θ = 80. = 360 The height is 15 m. SURNT & GOTRY 4 Complete to find the volume of this solid orret to 1 deiml ple. V= h = r2 h 360 = π = m Chpter 2 Surfe re nd volume 02_LY_I_10SB_51_52_22693_PPS.indd /10/2014 8:36 m

12 5 Clulte the volume of eh solid. 3.8 m 50 m 80 d 0.8 m 12 m 6. e f 7 m 3 m 32 m g h 10 m i m 7 m m 9 m D Volumes of omposite solids X 1 Clulte the volume of this omposite solid. Solve Cylinder: V = πr h = π 52 8 = m3 Cue: V = h = = 1000 m3 Totl volume = = 1623 to nerest m3 SURNT & GOTRY m Think The solid is mde up of ylinder nd ue. Cylinder: Rdius = 10 2 = Height = Cue: l = = h = 10 m pply Brek the omposite solid into simpler solids nd find the volume of eh one seprtely. Comine the volumes to give the nswer. Insight themtis 10 stges 5.1/5.2 ustrlin Curriulum 02_LY_I_10SB_51_52_22693_PPS.indd 26 15/10/2014 8:36 m

13 xerise 2D 1 Complete to find the volume of this omposite solid. 6m 10 m 4m 8m 12 m The solid is nd retngulr prism. Cylinder: Dimeter = 6 m Rdius = V = πr2 = π 2 = m3 Retngulr prism: V = h = (12 ) 4 = m3 Totl volume = + = m3 to the nerest whole numer 8m 2 Clulte the volume of eh omposite solid. 12 m d e 0.2 m 20 m 8m 15 m f 2 m 2m 0.4 m 0.2 m 0.5 m g 0.3 m i 4m 10 m 6m 60 2m 5m 5m 10 m h 2m 4m 7 m 7m Chpter 2 Surfe re nd volume 02_LY_I_10SB_51_52_22693_PPS.indd 27 SURNT & GOTRY 12 m 27 15/10/2014 8:36 m

14 Surfe res of omposite solids X 1 Clulte the surfe re of the solid shown. 5.7 m 7 m 10 m 1 re of front fe = _ = 70. Totl surfe re = (10 6) + 2 (10 7) = pply Totl surfe re = front + k + 4 sides + ottom Find the totl surfe re y summing the res of ll the fes of the solid. 2 Think Solve xerise 2 1 Complete to find the surfe re of this solid. Totl surfe re = front + k + side + 4 retngles re of front fe = = m Totl surfe re = = m2 2 Clulte the surfe res of the following solids. 10 m 1 10 m 9 m 9 m SURNT & GOTRY 7 m 9 m 10 m d e 3 m f 2 m 28 7 m m 10 m Insight themtis 10 stges 5.1/5.2 ustrlin Curriulum 02_LY_I_10SB_51_52_22693_PPS.indd 28 15/10/2014 8:36 m

15 g h i 2m 2m m 5m F Prolems with surfe re nd volume xerise 2F 1 sheet of rdord 1200 mm y 1000 mm hs squres of 300 mm side-length 300 mm ut from eh orner. The sides re folded up to form n open retngulr ox. Clulte its internl surfe re. Wht is the volume of the ox? 1000 mm 1200 mm 2 rport nd workshop re overed y flt retngulr roof 3.6 m y 11.2 m. ll the rin tht flls on the roof is olleted in wter tnk. If 3 mm of rin flls on the roof, how muh wter will e olleted in the tnk? (1 m3 = 1000 L) 3 The ross-setion of this losed rinwter tnk is shown eside it. 0.8 m 2.8 m 2m Clulte the re of this ross-setion. Hene lulte the volume of the tnk. Wht is the pity of the tnk if 1 m3 holds 1000 L? The tnk is ompletely mde from sheet steel tht osts $40/m2. Wht is the ost of the steel to mke this tnk? 4 The digrm shows the design for onrete drivewy. Clulte its re. onrete ontrtor hrges $70/m2 to supply nd ly onrete. How muh will he hrge for this jo? Give the nswer to the nerest dollr. If the onrete needs to e 100 mm deep, lulte the volume of onrete needed, in ui metres. SURNT & GOTRY d 1m 12 m 1m Chpter 2 Surfe re nd volume 02_LY_I_10SB_51_52_22693_PPS.indd /10/2014 8:36 m

16 5 The ylindril roller for riket pith is 1.5 m wide nd hs rdius of 0.3 m. Clulte the urved surfe re of the roller. Wht is the minimum numer of revolutions the roller would hve to mke to roll the riket pith one if the pith is 20 m long nd 3 m wide? (Ignore ny revolutions needed to turn the roller round.) 6 kyrd swimming pool hs dimensions s shown. Clulte the volume of the pool. How long will it tke to fill the pool with wter from 12 m 1m 4m 1.8 m grden hose tht n supply wter t the rte of 7.5 L/min? (Use 1 m3 holds 1000 L of wter.) Wht is the ost of filling the pool if wter osts $2.75/kL? 7 hollow iron pipe is 2 m long. Its externl dimeter is 10 m nd it is 1 m thik. Clulte the weight of the pipe if iron weighs 8.2 g/m3. 8 fish tnk hs retngulr se 40 m y 20 m. Wter is poured in to height of 2. Wht is the volume of wter in the tnk? If further 2 litres of wter is poured into the tnk, y how muh will the wter level rise? 9 pontoon with se 3 m y 3 m is floting on lke. When mn swims out nd lims onto it the pontoon sinks 1 m. If 1 L of wter weighs 1 kg, wht is the weight of the mn? (Hint: rhimedes priniple tells us tht the weight of the mn is equl to the weight of wter displed.) m3 lok of metl is mde into wire of dimeter 1 mm. How long will the wire e? SURNT & GOTRY 11 greenhouse with the dimensions 30 shown is to e overed on the top nd sides only (not the front nd k) with shde loth. The shde loth omes in 20 m rolls nd is 1.8 m wide. 2.4 m 3.6 m 8m Clulte the numer of liner metres of shde loth needed. How mny rolls will e needed? Insight themtis 10 stges 5.1/5.2 ustrlin Curriulum 02_LY_I_10SB_51_52_22693_PPS.indd 30 15/10/2014 8:36 m

17 Lnguge in mthemtis Johnn Kepler ( ) Johnn Kepler ws orn in the Germn town of Wurttemerg. s hild he ws smll nd suffered from ill helth, ut he ws reognised s eing intelligent. He ws given sholrship to ttend the University of Tuingen, where he studied first for the Luthern ministry nd then siene. He studied under mster in stronomy who elieved in, nd tught, the Copernin theory tht rth rotted round its own xis nd round the Sun. Kepler tught mthemtis in Grz from In 1600 he went to Prgue nd eme ssistnt to Tyho Brhe, n importnt stronomer. fter Brhe s deth, Kepler sueeded him s stronomer nd mthemtiin to the emperor. Kepler hd ess to Brhe s extensive reords of oservtions nd lultions. Kepler elieved in the Copernin theory, nd eme one of the founders of modern stronomy. He developed three fundmentl lws of plnetry motion, now known s Kepler s Lws, in These proposed, mong other things, tht the Sun ws t the entre of our plnetry system, nd tht the orits of the plnets were elliptil rther thn irulr. Sixty yers lter these lws helped Newton to develop his Universl Lw of Grvittion. Kepler lso suggested tht tides re used y the oon s grvittionl pull on the ses. He produed tles giving the positions of the Sun, oon nd plnets, whih were used for out 100 yers. In 1611 he proposed n improved refrting telesope, nd lter he suggested refleting telesope tht ws developed y Newton. 1 d e f How old ws Kepler when he died? When nd where did Kepler teh mthemtis? Desrie the development of Kepler s ides onerning plnetry motion. Reserh Kepler s three lws. For how long were Kepler s tles of positions of the Sun, oon nd plnets used? How re tides formed? SURNT & GOTRY 2 Rerrnge these words to form sentene. irle semiirle hlf is of. is of qurter qudrnt irle. my wy thn Composite more in res one e found. 3 Use every third letter to find the sentene. WD O HN T S F R B K B F Q L HTG UZ XF TWX D V H H S B Y F T U Y H U O Q I J I H O R J S D N R O HG BW Y G I B B Q P H V K F J F C R K O G H G I K N D H O H F J I O S W Z T I I O P D NU D S NW R P D I D F L C F H L S L H S D T J G T Y OL P FHD CGO S Chpter 2 Surfe re nd volume 02_LY_I_10SB_51_52_22693_PPS.indd /10/2014 8:36 m

18 Terms re qudrnt irle qudrilterl omposite rdius dimeter right formul setor prism semiirle Chek your skills 1 The re of this setor is losest to: 3.87 m2 B C 15. D m m 2 The re of this shpe is losest to: 5.85 m2 B 9.02 m2 C m2 D 14.8 m The surfe re of this prism is: 102 B 182 C 192 m2 D m 1.8 m 4 The surfe re of this losed ylinder is: B m2 C m2 D m2 5 The volume of this solid is: 573 B 143 C 113 D m SURNT & GOTRY 10 m Use this digrm for questions 6 nd 7. 6 The volume of this omposite solid is: m3 B C m D m3 7 The surfe re of the solid is: m2 B C m D m 8m 10 m 6m Insight themtis 10 stges 5.1/5.2 ustrlin Curriulum 02_LY_I_10SB_51_52_22693_PPS.indd 32 15/10/2014 8:36 m

19 8 lidded wooden ox, 1 8., is to e lquered inside nd out with two ots of lquer. Ignoring the thikness of the wood, the totl re to e lquered is: 537 m2 B 2142 C 1072 D 2682 If you hve ny diffiulty with these questions, refer to the exmples nd questions in the setions listed in the tle. Question Setion B C D F 2 Review set 1 Clulte the shded res orret to 1 deiml ple m 10 m 4.8 m 2 Clulte the surfe re of eh prism. 7 m m 3 Clulte the surfe re nd volume of this losed ylinder. 4 Clulte the volumes of the following solids. 1 3 m 3 m 5 Clulte the surfe re nd volume of this solid m SURNT & GOTRY 7 m 2m 2m 5m Chpter 2 Surfe re nd volume 02_LY_I_10SB_51_52_22693_PPS.indd /10/2014 8:36 m

20 2B Review set 1 Deorh s fmily room is shown opposite. Clulte the ost of rpet-tiling the room if the rpet tiles osts $ per squre metre. 1.7 m 1.6 m 4.4 m 2.3 m SURNT & GOTRY 2 door wedge shped s shown is to e pinted. Wht is the totl re to e pinted? 3 Clulte the surfe re nd volume of losed ylinder with dimeter 2.4 m nd height 1.8 m. 4 Clulte the surfe re of this solid. 5 Clulte the volume of this solid. 6 The ross-setion of this rinwter tnk is shown eside it. 2.5 m 1 Clulte the re of this ross-setion. Hene lulte the volume of the tnk. Wht is the pity of the tnk if 1 m 3 holds 1000 L? d The tnk ws mde from sheet steel tht osts $45/m 2. Wht ws the ost, to the nerest dollr, of the steel used to mke this tnk? 9 m 5 m 3 m 9 m 3 m 4 m 2 m 1.2 m 2.5 m 34 Insight themtis 10 stges 5.1/5.2 ustrlin Curriulum

21 2C Review set 1 Clulte the re of eh shpe orret to 1 deiml ple. 4.5 m m 6m 8.2 m d 4.5 m 7 mm 15 mm 8.3 m 2.1 m 2.1 m 2 T he rmy shed shown is to e pinted in mouflge 5m 5m olours. Wht re is to e mouflged? Clulte the volume of the shed. 4.2 m 15 m 8m 3 Clulte the surfe re of this losed ylinder. 5m 4 Clulte the surfe re nd volume of eh solid m 1 4m 9 m 9m 5m 5 hollow steel pipe is 5 m long. Its externl dimeter is 20 m nd it is 1. thik. Clulte the weight of SURNT & GOTRY the pipe to the nerest grm given tht steel weighs 8.2 g/m3. Chpter 2 Surfe re nd volume 02_LY_I_10SB_51_52_22693_PPS.indd /10/2014 8:36 m

22 Review set 2D 1 river delt is shped roughly like qudrnt, s shown. Clulte the popultion of the delt if 225 people per squre kilometre live there. 2 Clulte the surfe re of eh prism. 7.2 km 10 m 12 m 3 m 3 Clulte the surfe re nd volume of this open ylinder. 20 m 0.8 m 4 Clulte the surfe re nd volume of eh losed solid m 9 m 5 Clulte the surfe re nd volume of this solid m 6 greenhouse with the dimensions shown is to e SURNT & GOTRY overed on the top nd sides only (not the front nd k) with shde loth. The shde loth omes in 15 m rolls nd is 1.8 m wide. Clulte the numer of liner metres of shde loth needed. How mny rolls will e needed? m 4.8 m 10 m Insight themtis 10 stges 5.1/5.2 ustrlin Curriulum 02_LY_I_10SB_51_52_22693_PPS.indd 36 15/10/2014 8:36 m