NSW Syllabus references: 5.2 M&G Area and surface area, 5.2 M&G Volume Outcomes: MA5.2-1WM, MA5.2-2WM, MA5.2-11MG, MA5.2-12MG

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1 SA M PL E 2 Surfce re nd volume This chpter dels with clculting the surfce res nd volumes of right prisms nd cylinders. After completing this chpter you should e le to: solve prolems involving the surfce res nd volumes of right rectngulr nd tringulr prisms clculte the surfce res nd volumes of cylinders solve prolems involving the surfce res nd volumes of composite solids. NSW Syllus references: 5.2 M&G Are nd surfce re, 5.2 M&G Volume Outcomes: MA5.2-1WM, MA5.2-2WM, MA5.2-11MG, MA5.2-12MG ACMMG242

2 Dignostic test Questions 1 to 3 refer to the prism elow. A 6 The surfce re of this tringulr prism is: B F E H G cm cm 37.1 cm 1 The fce tht corresponds to the fce ABFE is: A DCHG C AEHD.1 cm C D A C B BFGC D DCGH B cm2 D The surfce re of cue of side length 2 The fce tht corresponds to the fce CBFG is: B AEHD D DGCH 3 The fce tht corresponds to the fce ABCD is: re of the wrpping to the nerest cm2 is: cm PL B HEFG D DCGH 8 A chocolte r is shown elow. The surfce 4.3 A EFHG C EFGH B cm2 D E A ABFE C DAEH 8. is: A C The surfce re of the net shown is: 22 3 cm 60 cm 2 M 80 cm cm 2 A 150 cm2 C 151 cm2 B cm2 D 152 cm2 9 This solid is mde 2 SA 2 A 282 C 232 cm2 B 322 D The surfce re of the net shown is: cm from 1 cm3 cues. The volume of the solid is: A 13 B 23 C 13 D 30 cm3 10 The cross-section of this solid is: A n ovl B cylinder C n ellipse D circle 11 A solid tht hs circulr cross-section is 2 A C B D Insight Mthemtics 10 stges 5.1/5.2 Austrlin Curriculum clled : A pyrmid C rectngulr prism B cylinder D ox

3 12 The volume of the solid shown is: A 61.3 B cm3 C D The volume of this composite solid is: Are The volume of this solid to the nerest cm3 is: A 703 B 703 C 703 D 73 A 1680 cm3 C 1200 cm cm B 1350 cm3 D 960 cm3 15 The volume of cue of side length 8. is: A C cm B cm3 D E 7.3 cm PL The dignostic test questions refer to outcomes ACMMG210 nd ACMMG218. A Are review EXAMPLE 1 M Find the re of this sector to 1 deciml plce. Apply Angle of sector = 130 Find the re of sector of circle y compring its ngle with the ngle of full circle, 360. sector ngle re of sector = 360 re of circle sector ngle re of circle A (of sector) = 360 θ A = πr = π Think SA Solve 11. Exercise 2A 1 Clculte the re of ech sector correct to 1 deciml plce. 5.2 cm 60 c cm 11. Chpter 2 Surfce re nd volume 17

4 d e f m m 17.5 m 2 Find the res of the following shpes. 4 m c d e f EXAMPLE 2 Drw net for this rectngulr prism, showing the lengths of its edges. Clculte the surfce re of the prism. Solve/Think Apply Drw the net, identify the fces, nd trnsfer the edge lengths from Bck the solid to the net. Clculte the re of ech fce Left Bottom Right side side nd sum these res cm 1 Front Top 10 km 6 m SA = (ottom + top) + (front + ck) + (left side + right side) = (9 4) 2 + (9 6) 2 + (6 4) 2 = mm 5 m 5 mm 10 km 8 mm 16 mm 18 Insight Mthemtics 10 stges 5.1/5.2 Austrlin Curriculum

5 3 For ech of the following rectngulr prisms: i Drw net of ech prism nd mrk its edge lengths. ii Clculte the surfce re. c d e f EXAMPLE 3 Drw net of this tringulr prism, mrking its edge lengths. Clculte the surfce re of the prism. Solve/Think Apply Drw the net, identify the fces nd trnsfer the edge lengths from the solid to the net. Clculte the re of ech fce nd sum these res SA = re of 2 tringles + re of 3 rectngles 1_ = ( ) = cm 4 Clculte the surfce re of ech of the following tringulr prisms. 5 m 3 cm 3.6 m 7 mm 3.3 cm 2.4 m 50 mm 3.3 cm 3 cm m 10 mm Chpter 2 Surfce re nd volume 19

6 c 1 d EXAMPLE 4 Clculte the length of the unknown edge of this tringulr prism. Drw net of the prism. c Clculte its surfce re. Solve/Think By Pythgors theorem: x 2 = = 52 x = cm (1 deciml plce) Apply Clculte the unknown edge using Pythgors theorem. Drw the net nd clculte the surfce re s efore. c SA = ( 1_ 2 6 4) = cm 2 5 For ech tringulr prism: i Find the length of the unknown edge. ii Clculte the surfce re. c 3 cm 7.2 cm x 11 cm 7.2 cm x 11 cm 5 m 12 m x 8 m 15 mm 10 mm x m 20 Insight Mthemtics 10 stges 5.1/5.2 Austrlin Curriculum

7 B Surfce res of right cylinders The formul for the surfce re of cylinder cn e developed y cutting the cylinder nd lying it out flt. The net then gives formul for the surfce re. The curved prt forms rectngle of length 2πr nd redth h. A = 2(re of circle) + re of rectngle = 2 π r 2 + 2π r h = 2π r 2 2π r + 2π rh The surfce re of closed cylinder is: A = 2π r 2 + 2π rh The surfce re of cylinder open t oth ends is: A = 2π rh EXAMPLE 1 Find the surfce re of this closed cylinder. Cut h Rememer: Are of circle is π r 2. 2π r r r h Solve Think Apply Surfce re = 2π r 2 + 2π rh = 2π π (1 deciml plce) Rdius = Height = 1 1 For cylinder closed t oth ends: SA = 2π r 2 + 2π r h Chpter 2 Surfce re nd volume 21

8 Exercise 2B 1 Complete to find the surfce re of this closed cylinder. Surfce re = 2π r 2 + 2π = 2π 2 + 2π 8.7 m 2 (1 deciml plce) 2 Find the surfce res of these closed cylinders to the nerest whole numer. c 8.7 m 3.2 m d e f 30 cm EXAMPLE 2 Find the surfce re of this open cylinder. Solve/Think Think Apply Surfce re = 2 π 8 23 = cm 2 Rdius = Height = 23 cm 3 Find the surfce res of the following open cylinders. c 20 cm 13 cm 1. For n open cylinder: SA = 2π rh 4 Determine how much pint is required to cover the outside of cylindricl continer 12 m long with dimeter 10 m if ech litre of pint covers 15 m 2. Which hs the greter surfce re: cylinder of length 1 nd rdius, or cylinder of length nd rdius? 8 m 1.4 m 4 mm mm 1 23 cm 20 cm 3 cm 2 22 Insight Mthemtics 10 stges 5.1/5.2 Austrlin Curriculum

9 5 Find the surfce re, correct to 1 deciml plce where necessry, of: n open cn with rdius of nd height of 1 n open-ended pipe of rdius nd 5 m long c the closed solid shown elow. 6 Determine the cost of pinting the exterior wlls nd top of cylindricl whet silo tht is 40 m high nd 20 m in dimeter, given tht ech litre of pint costs $7.25 nd covers 8 m 2. 7 Find the cost of mking 125 cylindricl tennis ll continers tht hve dimeter nd height 21 cm, given tht the metl costs $4.50 per squre metre (metl se ut open t the top). EXAMPLE 3 An open cylinder of rdius of hs curved surfce re of 1000 cm 2. Find its height. Solve Think Apply 1000 = 2 π 8 h = 16π h h = π = 19. (1 deciml plce) To solve 1000 = 16π h, divide oth sides y 16π. 8 Find the height of n open cylinder of rdius nd curved surfce re of 2000 cm 2. 9 Find the rdius of n open cylinder of height nd curved surfce re of 1500 cm 2. C 14 m 16 m Sustitute the given informtion into SA = 2π rh nd solve the resulting eqution. Volumes with uniform cross-sections A h The volume of right prism (or cylinder) is given y: V = A h where A is the re of the se (or cross-sectionl re) nd h is the perpendiculr height. A h A h Chpter 2 Surfce re nd volume 23

10 EXAMPLE 1 Find the volumes of these solids. c Solve Think Apply V = Are of se = 1 2 For prisms nd cylinders use = 25 3 V = Ah. V = (7.5 6) 4 or (7.5 4) 6 or (6 4) 7.5 = 180 m 3 c V = (π 5 2 ) 8 = cm 3 (1 deciml plce) Exercise 2C 1 Complete to find the volume of this prism. V = A h where A is the re of nd h is the height. V = 25 = cm 3 2 Clculte the volumes of these solids. c Are of se = Choose ny rectngle s the se. Are of se = πr 2 The se is circle. = π 5 2 cm 2 3 Clculte the re of the se nd hence find the volume of ech solid. c 2 m 3 2 d m m 8 m m 24 Insight Mthemtics 10 stges 5.1/5.2 Austrlin Curriculum

11 d e f EXAMPLE 2 Clculte the volume of ech solid correct to 1 deciml plce. 1 cm 18 m V = A h = 1_ 2 πr2 h 1 = 1_ 2 π = m 3 20 cm V = A h = πr2 h = π = m 3 22 m 80 1 Solve Think Apply Dimeter = 18 m so rdius = 9 m The se is semicircle so the re of the circle must e hlved. The se is sector. θ A = 360 πr2 where θ = 80. The height is 15 m. Clculte the re of the se first. Multiply y the height, which must e perpendiculr to the se. The solid does not hve to stnd on the se. 4 Complete to find the volume of this solid correct to 1 deciml plce. V = A h = 360 r2 h = 360 π 2 40 = cm 3 2 Chpter 2 Surfce re nd volume 25

12 5 Clculte the volume of ech solid. c m 50 m 0.8 m 3.8 m d e f 3 cm D cm 1 g h i 60 Volumes of composite solids EXAMPLE 1 Clculte the volume of this composite solid. Solve Think Apply Cylinder: V = πr 2 h = π = cm 3 Cue: V = Ah = = 1000 cm 3 Totl volume = = to nerest cm The solid is mde up of cylinder nd cue. Cylinder: Rdius = 10 2 = Height = Cue: l = = h = 20 cm 270 Brek the composite solid into simpler solids nd find the volume of ech one seprtely. Comine the volumes to give the nswer. 26 Insight Mthemtics 10 stges 5.1/5.2 Austrlin Curriculum

13 Exercise 2D 1 Complete to find the volume of this composite solid. 6 m 10 m 4 m 12 m 8 m The solid is nd rectngulr prism. Cylinder: Dimeter = 6 m Rdius = V = πr 2 = π 2 = m 3 Rectngulr prism: V = Ah = (12 ) 4 = m 3 Totl volume = + = m 3 to the nerest whole numer 2 Clculte the volume of ech composite solid. 8 m c 12 m 20 m 15 m d 0.2 m e 2 cm f 0.4 m 0.2 m 8 m 2 m 2 m 0.5 m 0. g h i 2 m 60 5 m 4 m 5 m 10 m 7 m 4 m 6 m Chpter 2 Surfce re nd volume 27

14 E Surfce res of composite solids EXAMPLE 1 Clculte the surfce re of the solid shown. 5. Are of front fce = _ Solve Think Apply = Totl surfce re = (10 6) + 2 (10 7) = Exercise 2E 1 Complete to find the surfce re of this solid. Totl surfce re = front + ck + side + 4 rectngles Are of front fce = = 9 2 Totl surfce re = = cm 2 Totl surfce re = front + ck + 4 sides + ottom 2 Clculte the surfce res of the following solids. c d e 3 cm f Find the totl surfce re y summing the res of ll the fces of the solid. 1 2 cm Insight Mthemtics 10 stges 5.1/5.2 Austrlin Curriculum

15 g h i 2 m 5 m 2 m 3. F Prolems with surfce re nd volume Exercise 2F 1 A sheet of crdord 1200 mm y 1000 mm hs squres of side-length 300 mm cut from ech corner. The sides re folded up to form n open rectngulr ox. Clculte its internl surfce re. Wht is the volume of the ox? 2 A crport nd workshop re covered y flt rectngulr roof 3.6 m y 11.2 m. All the rin tht flls on the roof is collected in wter tnk. If m of rin flls on the roof, how much wter will e collected in the tnk? (1 m 3 = 1000 L) 3 The cross-section of this closed rinwter tnk is shown eside it. 2 m 0.8 m Clculte the re of this cross-section. Hence clculte the volume of the tnk. c Wht is the cpcity of the tnk if 1 m 3 holds 1000 L? d The tnk is completely mde from sheet steel tht costs $40/m 2. Wht is the cost of the steel to mke this tnk? 4 The digrm shows the design for concrete drivewy. Clculte its re. A concrete contrctor chrges $70/m 2 to supply nd ly concrete. How much will he chrge for this jo? Give the nswer to the nerest dollr. c If the concrete needs to e 100 mm deep, clculte the volume of concrete needed, in cuic metres mm 300 mm 12 m 1 m 1 m 2.8 m 1000 mm Chpter 2 Surfce re nd volume 29

16 5 The cylindricl roller for cricket pitch is 1.5 m wide nd hs rdius of 0.. Clculte the curved surfce re of the roller. Wht is the minimum numer of revolutions the roller would hve to mke to roll the cricket pitch once if the pitch is 20 m long nd wide? (Ignore ny revolutions needed to turn the roller round.) 6 A ckyrd swimming pool hs dimensions s shown. Clculte the volume of the pool. How long will it tke to fill the pool with wter from grden hose tht cn supply wter t the rte of 7.5 L/min? (Use 1 m 3 holds 1000 L of wter.) c Wht is the cost of filling the pool if wter costs $2.75/kL? 7 A hollow iron pipe is 2 m long. Its externl dimeter is nd it is 1 cm thick. Clculte the weight of the pipe if iron weighs 8.2 g/cm 3. 8 A fish tnk hs rectngulr se 40 cm y 20 cm. 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 much will the wter level rise? 9 A pontoon with se y is floting on lke. When mn swims out nd clims onto it the pontoon sinks 1 cm. If 1 L of wter weighs 1 kg, wht is the weight of the mn? (Hint: Archimedes principle tells us tht the weight of the mn is equl to the weight of wter displced.) m A 50 cm 3 lock of metl is mde into wire of dimeter 1 mm. How long will the wire e? 12 m 4 m A greenhouse with the dimensions shown is to e covered on the top nd sides only (not the front nd ck) with shde cloth. The shde cloth comes in 20 m rolls nd is 1.8 m wide m 3.6 m 8 m 1 m Clculte the numer of liner metres of shde cloth needed. How mny rolls will e needed? 30 Insight Mthemtics 10 stges 5.1/5.2 Austrlin Curriculum

17 Lnguge in mthemtics Johnn Kepler ( ) Johnn Kepler ws orn in the Germn town of Wurttemerg. As child he ws smll nd suffered from ill helth, ut he ws recognised s eing intelligent. He ws given scholrship to ttend the University of Tuingen, where he studied first for the Luthern ministry nd then science. He studied under mster in stronomy who elieved in, nd tught, the Copernicn theory tht Erth rotted round its own xis nd round the Sun. Kepler tught mthemtics in Grz from In 1600 he went to Prgue nd ecme ssistnt to Tycho Brhe, n importnt stronomer. After Brhe s deth, Kepler succeeded him s stronomer nd mthemticin to the emperor. Kepler hd ccess to Brhe s extensive records of oservtions nd clcultions. Kepler elieved in the Copernicn theory, nd ecme 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 centre of our plnetry system, nd tht the orits of the plnets were ellipticl rther thn circulr. Sixty yers lter these lws helped Newton to develop his Universl Lw of Grvittion. Kepler lso suggested tht tides re cused y the Moon s grvittionl pull on the ses. He produced tles giving the positions of the Sun, Moon nd plnets, which were used for out 100 yers. In 1611 he proposed n improved refrcting telescope, nd lter he suggested reflecting telescope tht ws developed y Newton. 1 How old ws Kepler when he died? When nd where did Kepler tech mthemtics? c Descrie the development of Kepler s ides concerning plnetry motion. d Reserch Kepler s three lws. e For how long were Kepler s tles of positions of the Sun, Moon nd plnets used? f How re tides formed? 2 Rerrnge these words to form sentence. circle semicircle A hlf is of. is of qurter qudrnt A circle. c my wy thn Composite more in res one e found. 3 Use every third letter to find the sentence. W D T R F H T G E H Y A U J R N H E G B A V F O E D F S W A A Z R D F H H J O L P M O E B Q A U Z D S F Y O I J R B W A Q A K C G I H J T I I E O P I L L S G F H D E A S K L A X F V B T H Q H S O E Y A P E F R H K O I P D N M U A E C S D T C G O H N F B E T W X H A U E I O D A G I B H A J K G N H O D S N W E A D F L T Y S Chpter 2 Surfce re nd volume 31

18 Terms re circle composite dimeter formul prism qudrnt qudrilterl rdius right sector semicircle Check your skills 1 The re of this sector is closest to: A B 7. 2 C D cm 2 2 The re of this shpe is closest to: A 5.85 m 2 B 9.02 m 2 C m 2 D 14.8 m 2 3 The surfce re of this prism is: A 10 2 B 18 2 C 192 cm 2 D The surfce re of this closed cylinder is: A B cm 2 C cm 2 D cm 2 5 The volume of this solid is: A 57 3 B 14 3 C 11 3 D 5 3 Use this digrm for questions 6 nd 7. 6 The volume of this composite solid is: A cm 3 B C D cm 3 7 The surfce re of the solid is: A B C D m m 55 4 m 6 m 1.8 m 8 m 32 Insight Mthemtics 10 stges 5.1/5.2 Austrlin Curriculum

19 8 A lidded wooden ox, 1 8., is to e lcquered inside nd out with two cots of lcquer. Ignoring the thickness of the wood, the totl re to e lcquered is: A 53 2 B C D If you hve ny difficulty with these questions, refer to the exmples nd questions in the sections listed in the tle. Question Section A B C D E F 2A Review set 1 Clculte the shded res correct to 1 deciml plce. c 3.2 m 2 Clculte the surfce re of ech prism Clculte the surfce re nd volume of this closed cylinder. 4 Clculte the volumes of the following solids. 3 cm 4.8 m 5 Clculte the surfce re nd volume of this solid cm 70 2 m 5 m 2 m Chpter 2 Surfce re nd volume 33

20 2B Review set 1 Deorh s fmily room is shown opposite. Clculte the cost of crpet-tiling the room if the crpet tiles costs $ per squre metre. 1.7 m 1.6 m 4.4 m 2. 2 A door wedge shped s shown is to e pinted. Wht is the totl re to e pinted? 3 Clculte the surfce re nd volume of closed cylinder with dimeter 2.4 m nd height 1.8 m. 4 Clculte the surfce re of this solid. 5 Clculte the volume of this solid. 6 The cross-section of this rinwter tnk is shown eside it. 2.5 m 1 Clculte the re of this cross-section. Hence clculte the volume of the tnk. c Wht is the cpcity of the tnk if 1 m 3 holds 1000 L? d The tnk ws mde from sheet steel tht costs $45/m 2. Wht ws the cost, to the nerest dollr, of the steel used to mke this tnk? 5 m 4 m 2 m 1.2 m 2.5 m 34 Insight Mthemtics 10 stges 5.1/5.2 Austrlin Curriculum

21 2C Review set 1 Clculte the re of ech shpe correct to 1 deciml plce m 4. c 2.1 m 8.2 m 2 The rmy shed shown is to e pinted in cmouflge colours. Wht re is to e cmouflged? Clculte the volume of the shed. 3 Clculte the surfce re of this closed cylinder. 4 Clculte the surfce re nd volume of ech solid m 4.5 m 5 A hollow steel pipe is 5 m long. Its externl dimeter is 20 cm nd it is 1. thick. Clculte the weight of the pipe to the nerest grm given tht steel weighs 8.2 g/cm 3. d 6 m 15 mm 4.2 m 8 m 7 mm 5 m 5 m 9 m 5 m 5 m 4 m 15 m 1.5 m Chpter 2 Surfce re nd volume 35

22 2D Review set 1 A river delt is shped roughly like qudrnt, s shown. Clculte the popultion of the delt if 225 people per squre kilometre live there. 7.2 km 2 Clculte the surfce re of ech prism. 3 cm 3 Clculte the surfce re nd volume of this open cylinder. 4 Clculte the surfce re nd volume of ech closed solid. 1 5 Clculte the surfce re nd volume of this solid. 6 A greenhouse with the dimensions shown is to e covered on the top nd sides only (not the front nd ck) with shde cloth. The shde cloth comes in 15 m rolls nd is 1.8 m wide. Clculte the numer of liner metres of shde cloth needed. How mny rolls will e needed? 0.8 m 20 m 2.7 m m 10 m 36 Insight Mthemtics 10 stges 5.1/5.2 Austrlin Curriculum