11A 11B 11C 11D 11E 11F 11G 11H 11I 11J Chpter Wht you will lern Mesurement systems (Extening) Using n onverting metri lengths (Consoliting) Perimeter (Consoliting) Ares n retngles Are of tringle Are of prllelogrm Are of omposite shpes (Extening) Volume of retngulr prisms Cpity (Consoliting) Mss n temperture (Consoliting) 11Mesurement 1 2 Austrlin urriulum MEASUREMENT AND GEOMETRY Using units of mesurement Estlish the formuls for res of retngles, tringles n prllelogrms n use these in prolem solving (ACMMG159) Clulte volumes of retngulr prisms (ACMMG160) Shpe Drw ifferent views of prisms n solis forme from omintions of prisms (ACMMG161) Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement everywhere Imgine trying to esrie fts out the worl roun us without using ny form of mesurement. We use units of length to esrie istne n egrees Celsius ( C) to esrie temperture. Other units re use for re, volume, time, pity n mss. Here re some exmples of fts tht use ifferent units of mesurement. The Eiffel Tower in Frne is pinte with 50 tonnes of pint every 7 yers. The Gret Wll of Chin is more thn 6000 km long. The Gret Pyrmi of Giz ws uilt roun 2500 BCE n inlues out 2 300 000 loks of stone, eh weighing out 2500 kg. The worl s smllest ountry is Vtin City in Rome, with n re of 0.44 km 2. The mximum temperture uring the y on Mrs is out 20 C. The istne etween the orits of Mrs n the Erth roun the Sun is out 78 000 000 km. The volume of wter in Syney Hrour is out 500 giglitres or fi ve hunre thousn million litres or 0.5 km 3. Online resoures Chpter pre-test Vieos of ll worke exmples Intertive wigets Intertive wlkthroughs Downlole HOTsheets Aess to HOTmths Austrlin Curriulum ourses 581 Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
582 Chpter 11 Mesurement 11A Mesurement systems EXTENDING From out 3000 BCE unit of mesure lle the uit ws use in nient Egypt to mesure lengths. It is known tht the uit, whih is the length of n rm from the elow to the en of the mile finger, ws use to mesure the epth of the Nile River in floo. Other unit mesures se on the humn oy were lso use. Some inlue the igit (with of finger), plm (with of four fingers) n spn (istne from the tip of the thum to the tip of the little finger in n outstrethe hn). 1/3 1/4 1/2 1/2 1 1 1 Royl Cuits Egyptins use their rms n hns to mesure ifferent units. Beuse eh iniviul s rm length n finger with is ifferent, there ws nee to evelop stnr unit of length. The Egyptins efine stnr royl uit (out 524 mm), n this ws represente s stone ro. From this uit the following ivisions were me: 28 igits in uit, 4 igits in plm n 14 igits in spn. Mny of these units of mesurement were pte n evelope y the Bylonins, Greeks, Romns, English n Frenh over mny enturies. The English imperil system, whih ws pte from the Romn n Greek systems, is ommonly use in the Unite Kingom n the Unite Sttes toy, n ws use in Austrli until the 1970s. Mny people toy still prefer to esrie lengths n other mesures using Imperil units, suh s the inh n mile. The metri system ws evelope in Frne in the 1790s n is the universlly epte system toy. The wor metri omes from the Greek wor metron, mening mesure. It is eiml system where length mesures re se on the unit lle the metre. The efinition of the metre hs hnge over time. Originlly it ws propose to e the length of penulum tht ets t rte of one per seon. It ws lter efine s 1/10 000 000 of the istne from the North Pole to the equtor on line on the Erth s surfe pssing through Pris. In 1960, metre eme 1 650 763.73 wve lengths of the spetrum of the krypton-86 tom in vuum. In 1983, the metre ws efine s the istne tht light trvels in 1/299 792 458 seons insie vuum. Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 583 Let s strt: Egyptin trer Imgine you re in nient Egypt n you re tring goos t mrket. You use the Egyptin units: igit (with of finger), plm (with of four fingers) n spn (istne from the top of the thum to the tip of the little finger in n outstrethe hn). Use ruler to fin the metri equivlent of your igit, plm n spn. You purhse w of ppyrus pper tht is 1 igit thik. Whih stuents in the lss woul get the lest pper if they use their own finger with? You purhse owl of grin 1 spn eep. Whih stuent in the lss gets the most grin? You purhse 5 uits of loth. Whih stuent gets the most loth? Anient mesurement systems tht evelope from out 3000 BCE inlue the Egyptin, Bylonin, Greek n Romn systems. The metri system is the ommonly use system toy. Romn system 1 foot = 12 inhes = 16 igits = 4 plms 1 uit = 6 plms 1 pe (oule step) = 5 feet 1 mile = 1000 pes imperil system 1 foot = 12 inhes (1 inh is out 2.5 m) 1 yr = 3 feet (1 yr is out 91.5 m) 1 ro = 16.5 feet 1 hin = 22 yrs 1 furlong = 40 ros 1 mile = 8 furlongs = 1760 yrs (1 mile is out 1.6 km) metri system 1 entimetre (m) = 10 millimetres (mm) 1 metre (m) = 100 entimetres (m) 1 kilometre (km) = 1000 metres (m) Exmple 1 Using mesurement systems How mny feet re there in 1 mile, using the Romn mesuring system? How mny inhes re there in 3 yrs, using the imperil system? SOLUTION 1 mile = 1000 pes = 5000 feet 3 yrs = 9 feet = 108 inhes EXPLANATION There re 1000 pes in Romn mile n 5 feet in pe. There re 3 feet in n imperil yr n 12 inhes in foot. Key ies Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
584 Chpter 11 Mesurement Exmple 1 Exmple 2 Choosing metri lengths Whih metri unit woul e the most pproprite for mesuring these lengths? with of lrge room thikness of glss in winow SOLUTION EXPLANATION metres (m) Using mm or m woul give very lrge numer, n using km woul give numer tht is very smll. millimetres (mm) The thikness of glss is likely to e roun 5 mm. Exerise 11A 1 Complete these numer sentenes. 1 4 4 Romn system i 1 = 12 inhes = 16 = plms ii 1 = 1000 pes imperil system i 1 foot = 12 ii 3 = 1 yr iii = 1760 yrs metri system i 1 m = m ii 1 m = mm iii km = 1000 m 2 List the units of length (e.g. uit), from smllest to lrgest, ommonly use in the Romn system. 3 List the units of length (e.g. inh), from smllest to lrgest, ommonly use in the imperil system. 4 List the units of length (e.g. entimetre), from smllest to lrgest, ommonly use in the metri system. 5 Use the Romn system to stte how mny: feet re in 1 pe feet re in 1 mile plms re in 1 pe e igits re in 1 foot 5 8 5 9 f plms re in 1 foot igits re in 1 pe 5 9(½) UNDERSTANDING FLUENCY Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 585 Exmple 2 6 Use the imperil system to stte how mny: inhes re in 1 foot feet re in 1 yr yrs re in 1 mile e yrs re in 1 hin 7 Use the metri system to stte how mny: millimetres re in 1 entimetre metres re in 1 kilometre e entimetres re in 1 kilometre 8 f f inhes re in 1 yr ros re in 1 furlong entimetres re in 1 metre millimetres re in 1 metre millimetres re in 1 kilometre Whih metri unit woul e the most pproprite for mesuring the following? the istne etween two towns with of smll rill it height of flg pole length of gren hose e with of smll esk f istne ross ity A rill it 9 Choose whih metri unit woul e the most suitle for mesuring the rel-life length inite in eh of these photos. FLUENCY 11A Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
586 Chpter 11 Mesurement 11A e 10 A Romn offers you either 12 plms or 1 pe of loth. Whih option gives you the most loth? 11 The Romn rmy mrhes 5 Romn miles to the next post. How mny pes is the journey? 12 An English riketer runs 1 hin for eh run me. How mny yrs will he run if he mkes 20 runs? 13 Here is the length of 1 mm n 1 m. Use these igrms s guie to estimte the length of these lines. e 14 Estimte the length of eh line or urve, in entimetres. f 10, 11 11 13 1 mm 1 m 12 14 FLUENCY PROBLEM-SOLVING Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 587 15 Complete these tles. metri mm m m km mm 1 1 10 m 10 1 m 1000 1 km 1 Romn Digit Plm Feet Pe Mile Digit 1 1 16 Plm 4 1 Feet 1 Pe 1 Mile 1 15 15, 16 16, 17 imperil Inh Feet Yr Mile Inh 1 1 36 Feet 12 1 Yr 1 Mile 1760 1 16 Why woul it e more iffiult to inlue the imperil units of hins n ros in the tle in Question 15? 17 Generlly speking, why is the metri system esier to use thn either the imperil or Romn systems? Wlking pes 18 The Romn pe involves 2 steps, n 1000 of these pes mke up Romn mile. These units woul hve een use to estimte istnes for the Romn rmies tht spre throughout muh of the worl uring tht time. Estimte how mny pes (i.e. oule steps) you woul tke in 1 kilometre (1000). Clulte how mny pes you woul tke to over 1 kilometre. If eh pe tkes 1 seon, fin how long it woul tke to wlk from Syney to Melourne (out 900 km) non-stop. Convert your nswer to numer of hours. 18 REASONING ENRICHMENT 11A Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
588 Chpter 11 Mesurement Key ies 11B Using n onverting metri lengths CONSOLIDATING To voi the use of very lrge n very smll numers, n pproprite unit is often hosen to mesure length or istne. It my lso e neessry to onvert units of length. For exmple, 150 piees of timer, eh mesure in entimetres, my nee to e ommunite s totl length using metres. Another exmple might e tht 5 millimetres is to e ut from length of timer 1.4 metres long euse it is too wie to fit oor opening tht is 139.5 entimetres wie. Let s strt: How goo is your estimte? In less thn 3 seons, guess the length of your esk, in entimetres. A rpenter my nee to mesure lengths of woo in metres, entimetres n millimetres. Now use ruler to fin the tul length in entimetres. Convert your nswer to millimetres n metres. If you line up ll the lss esks en to en, how mny esks woul e neee to reh 1 kilometre? Explin how you got your nswer. The metre (m) is the si metri unit of length. 1 km = 1000 m 1 m = 100 m 1 m = 10 mm km 1000 1000 m 100 100 Conversion When onverting to smller unit, multiply y power of 10 (i.e. 10, 100, 1000). A smller unit mens you nee more of them, so it mkes sense tht you multiply. When onverting to lrger unit, ivie y power of 10 (i.e. 10, 100, 1000). A lrger unit mens you nee less of them, so it mkes sense tht you ivie. When reing sles, e sure out wht units re showing on the sle. This sle shows 36 mm. mm 1 2 3 4 m m 10 10 mm Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 589 Exmple 3 Converting metri units of length Convert to the units given in rkets. 3 m (m) SOLUTION 3 m = 3 100 m = 300 m 25 600 m = 25 600 100 000 = 0.256 km Exmple 4 Reing length sles 25 600 m (km) EXPLANATION Re the sles on these rulers to mesure the mrke length. SOLUTION 1 m 2 m 3 m 1 m = 100 m Multiply sine you re onverting to smller unit. There re 100 m in 1 m n 1000 m in 1 km n 100 1000 = 100 000. EXPLANATION 5 m 6 m 7 m 25 mm 2.5 m is lso urte. 70 m Eh ivision is 1 of metre, whih is 10 m. 10 Exerise 11B 1 3 3 1 Write own the missing numer or wor in these sentenes. When onverting from metres to entimetres, you multiply y. When onverting from metres to kilometres, you ivie y. When onverting from entimetres to metres, you y 100. When onverting from kilometres to metres, you y 1000. e When onverting to smller unit, you. f When onverting to lrger unit, you. 2 Clulte eh of the following. 100 10 10 100 100 1000 10 100 1000 3 When multiplying y power of 10, in whih iretion oes the eiml point move left or right? When iviing y power of 10, in whih iretion oes the eiml point move left or right? Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400 UNDERSTANDING
590 Chpter 11 Mesurement 11B Exmple 3 Exmple 3 Exmple 4 4 Convert these mesurements to the units shown in rkets. 5 m (mm) 2 m (m) 3.5 km (m) e 40 mm (m) f 500 m (m) g 4200 m (km) i 6.84 m (m) j 0.02 km (m) k 9261 mm (m) h l 26.1 m (m) 472 mm (m) 4230 m (km) 5 A these lengths together n give the result in the units shown in rkets. 2 m n 5 mm (m) 8 m n 2 mm (mm) 2 m n 50 m (m) 7 m n 30 m (m) e 6 km n 200 m (m) f 25 km n 732 m (km) 6 7 Convert to the units shown in the rkets. 3 m (mm) 6 km (m) 0.04 km (m) e 47 000 m (km) g 216 000 mm (km) h 0.5 mm (m) f 2.4 m (mm) 913 000 mm (m) These rulers show entimetres with millimetre ivisions. Re the sle to mesure the mrke length. e g 0 1 2 0 0 0 1 2 1 2 3 4 1 2 3 f h 0 0 0 0 1 2 3 4 5 1 2 3 8 Re the sle on these tpe mesures. Be reful with the units shown! 0 m 1 2 3 4 7(½), 8 4 7(½), 8, 9 1 2 3 4 5 1 2 3 4 11 12 13 km 4 7(½), 8, 9 9 Use sutrtion to fin the ifferene etween the mesurements, n give your nswer with the units shown in rkets. 9 km, 500 m (km) 3.5 m, 40 m (m) 0.2 m, 10 mm (m) FLUENCY Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 591 10 12 12 14 10 Arrnge these mesurements from smllest to lrgest. 38 m, 540 mm, 0.5 m 0.02 km, 25 m, 160 m, 2100 mm 0.003 km, 20 m, 3.1 m, 142 mm 0.001 km, 0.1 m, 1000 m, 10 mm 11 Joe wiens 1.2 m oorwy y 50 mm. Wht is the new with of the oorwy, in entimetres? 12 Three onstrution engineers iniviully hve plns to uil the worl s next tllest tower. The Titn tower is to e 1.12 km tll, the Gign tower is to e 109 500 m tll n the Bign tower is to e 1210 m tll. Whih tower will e the tllest? 13 Steel hin osts $8.20 per metre. How muh oes it ost to uy hin of the following lengths? 1 km 80 m 50 mm 14 A house is 25 metres from liff ove the se. The liff is eroing t rte of 40 mm per yer. How mny yers will pss efore the house strts to fll into the se? 15 Mount Everest is moving with the Ino-Austrlin plte t rte of out 10 m per yer. How mny yers will it tke to move 5 km? 16 A rem of 500 sheets of pper is 4 m thik. How thik is 1 sheet of pper, in millimetres? 17 A snil slithers 2 mm every 5 seons. How long will it tke to slither 1 m? 18 Copy this hrt n fill in the missing informtion. 18 18 100 19 Mny trespeople mesure n ommunite with millimetres, even for long mesurements like timer ems or pipes. Cn you explin why this might e the se? Very smll n lrge units 20 When 1 metre is ivie into 1 million prts, eh prt is lle mirometre (μm). At the other en of the spetrum, light yer is use to esrie lrge istnes in spe. Stte how mny mirometres there re in: i 1 m ii 1 m iii 1 mm iv 1 km km m m mm 10 15 17 18, 19 A virus is 0.000 312 mm wie. How mny mirometres is this? Reserh the length lle the light yer. Explin wht it is n give exmples of istnes using light yers, suh s to the nerest str other thn the Sun. 20 PROBLEM-SOLVING REASONING ENRICHMENT 11B Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
592 Chpter 11 Mesurement Key ies 11C Perimeter CONSOLIDATING The istne roun the outsie of twoimensionl shpe is lle the perimeter. The wor perimeter omes from the Greek wors peri, mening roun, n metron, mening mesure. We ssoite perimeter with the outsie of ll sorts of regions n ojets, like the length of fening surrouning lok of ln or the length of timer require to frme piture. Let s strt: Is there enough informtion? This igrm, whih is not rwn to sle, inlues only 90 ngles n only one sie length is given. Disuss if there is enough informtion given in the igrm to fin the perimeter of the shpe. Wht itionl informtion, if ny, is require? Perimeter, sometimes enote s P, is the istne roun the outsie of two-imensionl shpe. Sies with the sme mrkings re of equl length. The unknown lengths of some sies n sometimes e etermine y onsiering the given lengths of other sies. Exmple 5 Fining the perimeter Fin the perimeter of eh of these shpes. 3 m 5 m This fene mrks the perimeter (i.e. the istne roun the outsie) of pok. 3 m 6 m 10 m 1.6 m 2.8 m 4.1 m P = 1.6 + 1.6 + 2.8 + 4.1 = 10.1 m 5 m 2 m Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 593 SOLUTION EXPLANATION perimeter = 2 5 + 3 = 13 m perimeter = 2 6 + 2 8 Exerise 11C = 28 m There re two equl lengths of 5 m n one length of 3 m. 6 m 3 m 1 These shpes re rwn on 1 m gris. Give the perimeter of eh. 6 2 = 4 m 5 m 3 + 5 = 8m 2 Use ruler to mesure the lengths of the sies of these shpes, n then fin the perimeter. 1, 2 2 2 m UNDERSTANDING Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
594 Chpter 11 Mesurement 11C Exmple 5 3, 4 3 5 3(½), 4, 5 3 Fin the perimeter of these shpes. (Digrms re not rwn to sle.) 3 m 5 m 8 m 6 m e 5 km 6 m 7 m 10 km 10 m f 5 m 0.2 m Exmple 5 4 Fin the perimeter of these shpes. All orner ngles re 90. 4 m 2 m 10 m 9 km 6 km 7 m 3 km 5 m 5 m 10 m 4 m 1 m 2.5 m 5 A squre hs sie length of 2.1 m. Fin its perimeter. A retngle hs length of 4.8 m n with of 2.2 m. Fin its perimeter. An equilterl tringle hs ll sies the sme length. If eh sie is 15.5 mm, fin its perimeter. 8 m 8 m FLUENCY Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 595 6 A grzing pok is to e fene on ll sies. It is retngulr in shpe, with length of 242 m n with of 186 m. If fening osts $25 per metre, fin the ost of fening require. 7 A grss tennis ourt is line with hlk. All the mesurements re shown in the igrm n given in feet. Fin the totl numer of feet of hlk require to o ll the lines of the given tennis ourt. There re 0.305 metres in 1 foot. Convert your nswer to prt to metres. 36 feet 21 feet 8 Only some sie lengths re shown for these shpes. Fin the perimeter. 20 mm 4 m 15 mm 18 m 9 Fin the perimeter of eh of these shpes. Give your nswers in entimetres. 271 mm 7.1 m 168 mm 0.38 m 39 feet 430 mm 10 m 1.04 m 10 A squre pok hs 100 eqully-spe posts tht re 4 metres prt, inluing one in eh orner. Wht is the perimeter of the pok? 11 The perimeter of eh shpe is given. Fin the missing length of eh. 4 m 2 m? P = 11 m P = 20 m? 6, 7 7 9 12 km P = 38 km 12 A retngle hs perimeter of 16 m. Using only whole numers for the length n with, how mny ifferent retngles n e rwn? Do not ount rottions of the sme retngle. 10 12 27 feet? PROBLEM-SOLVING 11C Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
596 Chpter 11 Mesurement 11C 13 Write n lgeri rule (e.g. P = 2 + ) to esrie the perimeter of eh shpe. 14 Write n lgeri rule for the perimeter of eh given shpe. e 15 A squre hs perimeter P. Write n expression for its sie length. A retngle hs perimeter P n with. Write n expression for its length. Piture frmes 16 The mount of timer use to frme piture epens on the outsie lengths of the overll frme. These re then ut t 45 to mke the frme. A squre pinting of sie length 30 m is to e frme with timer of with 5 m. Fin the totl length of timer require for the jo. A retngulr photo with imensions 50 m y 30 m is frme with timer of with 7 m. Fin the totl length of timer require to omplete the jo. 13 13, 14 f 30 m 30 m 14, 15 5 m Kimerley uses 2 m of timer of with 5 m to omplete squre piture frme. Wht is the sie length of the piture? A squre piee of emroiery hs sie length m n is frme y timer of with 4 m. Write n expression for the totl mount of timer use in m. 16 REASONING ENRICHMENT Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 597 11D Ares n retngles Are is mesure in squre units. It is often referre to s the mount of spe ontine insie flt (i.e. plne) shpe; however, urve three-imensionl (3D) solis lso hve surfe res. The mount of pint neee to pint house n the mount of hemil neee to spry pok re exmples of when re woul e onsiere. Let s strt: The 12 m 2 retngle The greter the re of the pok, the more hemils the plne nees to rry. A retngle hs n re of 12 squre entimetres (12 m 2 ). Drw exmples of retngles tht hve this re, showing the length n with mesurements. How mny ifferent retngles with whole numer imensions re possile? How mny ifferent retngles re possile if there is no restrition on the type of numers llowe to e use for length n with? The metri units of re inlue: 1 squre millimetre (1 mm 2 ) 1 mm 1 mm 1 squre entimetre (1 m 2 ) 1 m 2 = 100 mm 2 1 m 1 m 1 squre metre (1 m 2 ) 1 m 2 = 10 000 m 2 1 m 1 m 1 squre kilometre (1 km 2 ) 1 km 2 = 1 000 000 m 2 1 km (Not rwn to sle.) (Not rwn to sle.) 1 km 1 hetre (1 h) 1 h = 10 000 m 2 100 m (Not rwn to sle.) 100 m Key ies Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
598 Chpter 11 Mesurement Key ies The re of retngle is given y the numer of rows multiplie y the numer of olumns. Written s formul, this looks like: A = l w. This lso works for numers tht re not whole. The re of squre is given y: A = l l = l 2 Exmple 6 Counting res Count the numer of squres to fin the re of the shpe rwn on this entimetre gri. SOLUTION A = l w l A = l 2 EXPLANATION 6 m 2 There re 5 full squres n hlf of 2 squres in the tringle, giving 1 more. Exmple 7 Ares of retngles n squres Fin the re of this retngle n squre. 10 mm 4 mm l 2.5 m w 1 of 2 = 1 2 Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 599 SOLUTION EXPLANATION Are = l w Are = l 2 = 10 4 = 40 mm 2 = 2.5 2 = 6.25 m 2 Exerise 11D 1 For this retngle rwn on 1 m gri, fin eh of the following. the numer of single 1 m squres the length n the with length with 2 For this squre rwn on entimetre gri, fin the following. the numer of single 1 m squres the length n the with length with 3 Count the numer of squres to fin the re of these shpes. The re of retngle is the prout of the length n with. The with is the sme s the length, so A = l l = l 2. (2.5) 2 = 2.5 2.5 1 4 4 4 Whih unit of re (mm 2, m 2, m 2, h or km 2 ) woul you hoose to mesure these res? Note tht 1 km 2 is muh lrger thn 1 h. re of n A4 piee of pper re of wll of house re of smll frm re of lrge esert e re of lrge footll ovl f re of nil he UNDERSTANDING Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
600 Chpter 11 Mesurement 11D Exmple 6 Exmple 7 5 6 Count the numer of squres to fin the re of these shpes on entimetre gris. e Fin the re of these retngles n squres. Digrms re not rwn to sle. 20 m 10 m 5, 6(½), 7 10 5, 6(½), 7 11 f 11 mm 3.5 m 2 mm 5, 6(½), 7 11 2 m FLUENCY Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 601 g 1.7 m 5 m 0.8 m e h 1.2 mm 0.9 m f i 2.5 mm 17.6 km 10.2 km 7 Fin the sie length of squre with eh of these res. Use tril n error if you re unsure. 4 m 2 25 m 2 144 km 2 8 There re 10 000 m 2 in one hetre (h). Convert these mesurements to hetres. 20 000 m 2 100 000 m 2 5000 m 2 9 A retngulr soer fiel is to e li with new grss. The fiel is 100 m long n 50 m wie. Fin the re of grss to e li. 10 Glss is to e ut for squre winow of sie length 50 m. Fin the re of glss require for the winow. 11 Two hunre squre tiles, eh mesuring 10 m y 10 m, re use to tile n open floor re. Fin the re of flooring tht is tile. 12, 13 13, 14 12 A squre hs perimeter of 20 m. Fin its re. A squre hs n re of 9 m 2. Fin its perimeter. A squre s re n perimeter re the sme numer. How mny units is the sie length? 13 The rpet hosen for room osts $70 per squre metre. The room is retngulr n is 6 m long y 5 m wie. Wht is the ost of rpeting the room? 14 Troy wishes to pint gren wll tht is 11 m long n 3 m high. Two ots of pint re neee. The pint suitle to o the jo n e purhse only in whole numers of litres n overs n re of 15 m 2 per litre. How mny litres of pint will Troy nee to purhse? 15 A retngulr re of ln mesures 200 m y 400 m. Fin its re in hetres. 13 15 FLUENCY PROBLEM-SOLVING 11D Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
602 Chpter 11 Mesurement 11D 16 Fin the missing length for eh of these retngles. i ii A = 50 m 2 5 m A = 22.5 mm2 2.5 mm?? Explin the metho tht you use for fining the missing lengths of the retngles ove. 17 Explin why the re she here is extly 2 m 2. 18 A squre hs perimeter P m. 16 16, 17 If P = 44 m, fin the re of the squre. If P is unknown, write n expression for the re of the squre, using P. 4 m 1 m P = 44 m 19 A squre hs ll its sie lengths oule. How oes this hnge the re? Investigte n justify your nswer. Are onversions 20 Use this igrm or similr to help nswer the following. i How mny mm 2 in 1 m 2? ii How mny m 2 in 1 m 2? iii How mny m 2 in 1 km 2? Complete the igrm elow. km 2 m 2 m 2 mm 2 Convert these units to the units shown in rkets. i 2 m 2 (mm 2 ) ii 10 m 2 (m 2 ) iii 3.5 km 2 (m 2 ) iv 300 mm 2 (m 2 ) v 21 600 m 2 (m 2 ) vi 4 200 000 m 2 (km 2 ) vii 0.005 m 2 (mm 2 ) viii 1 km 2 (h) ix 40 000 000 m 2 (h) 10 mm A = 1 m 2 10 mm 17 19 20 Guess the re of the pok in the foregroun in squre millimetres. REASONING ENRICHMENT Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 603 11E Are of tringle Put simply, tringle n e onsiere to e hlf retngle, whih is why the formul for the re of tringle looks very muh like the formul for the re of retngle ut with the e ftor of 1. One of the sies of 2 tringle is lle the se (), n the height (h) is the istne etween the se n the opposite vertex. This is illustrte using line tht is perpeniulr (i.e. t 90 ) to the se. Any shpe with ll stright sies (i.e. polygons) n e ivie up into omintion of retngles (or squres) n tringles. This n help to fin res of suh shpes. Let s strt: Hlf retngle Look t these tringles. For eh one, isuss why the re oul e onsiere s hlf retngle. Give resons for eh se. 1 2 3 height se Any re of floor with stright sies n e fille with squre n tringulr tiles. One sie of tringle is lle the se,. The perpeniulr istne from the se to the opposite vertex is lle the height, h. In the thir illustrtion elow, the se nees to e extene to help show the height. h The re of tringle is given y the formul: A = 1 2 h = 1 se height 2 Note tht 1 2 h h h is equivlent to h 2. Key ies Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
604 Chpter 11 Mesurement Exmple 8 Fining res of tringles Fin the re of eh given tringle. 9 m 10 m 9 m SOLUTION re = 1 2 h = 1 2 10 9 = 45 m 2 re = 1 2 h = 1 2 6 9 = 27 m 2 Exerise 11E EXPLANATION 1 For eh of these tringles, wht length woul e use s the se? 3 m 8 m 20 m 2 m 6 m Use the formul n sustitute the vlues for se length n height. The length mesure of 9 m is mrke t 90 to the sie mrke 6 m. So 6 m is the length of the se n 9 m is the perpeniulr height. 1 3 3 6.3 m 7 m 2.1 m 5 m UNDERSTANDING Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 605 Exmple 8 2 For eh of these tringles, wht length woul e use s the height? 11 m 4.1 m 6 m 3 Fin the vlue of A in A = 1 h if: 2 4 1.9 m 3.2 mm 15 mm 11 mm 4.7 mm = 5 n h = 4 = 7 n h = 16 = 2.5 n h = 10 Fin the re of eh tringle given. g 2 m 2.4 m 4 m 1 m 8 m 1.3 m e h 3 mm 10 m 12 m 5 m 4(½), 5, 6 4(½), 5 7 2 mm f i 4 m 16 m 7 km 1.7 m 4 5(½), 6 7 20 m 5 m 7 km UNDERSTANDING FLUENCY 11E Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
606 Chpter 11 Mesurement 11E 5 Fin the re of these tringles, whih hve een rwn on 1 m gris. Give your nswer in m 2. 6 A retngulr lok of ln mesuring 40 m long y 24 m wie is ut in hlf long igonl. Fin the re of eh tringulr lok of ln. 7 A squre pyrmi hs se length of 120 m n tringulr fe of height 80 m. Fin the re of one tringulr fe of the pyrmi. 80 m 120 m FLUENCY Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 607 8 Eh fe of 4-sie ie is tringulr, with se of 2 m n height of 1.7 m. Fin the totl re of ll 4 fes of the ie. 9 A frmer uses fening to ivie tringulr piee of ln into two smller tringles, s shown. Wht is the ifferene in the two res? 8, 9 9, 10 40 m 18 m 26 m 10 A yht must hve two of its sils reple s they hve een mge y reent storm. One sil hs se length of 2.5 m n height of 8 m n the igger sil hs se length of 4 m n height of 16 m. If the ost of sil mteril is $150 per squre metre, fin the totl ost to reple the yht s mge sils. 11 The re of tringle is 10 m 2 n its se length is 4 m. Fin its height. The re of tringle is 44 mm 2 n its height is 20 mm. Fin its se length. 9 11 PROBLEM-SOLVING 11E Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
608 Chpter 11 Mesurement 11E 12 12, 13 12 The mipoint, M, of the se of tringle joins the opposite vertex. Is the tringle re split in hlf extly? Give resons for your nswer. 13 If the vertex C for this tringle moves prllel to the se AB, will the re of the tringle hnge? Justify your nswer. 14 The re of tringle n e foun using the formul A = 1 h. Write own the formul to fin 2 the se,, if you re given the re, A, n height, h. Estimting res with urves 15 This igrm shows she region tht is 1 2 of 3 m2 = 1.5 m 2. Using tringles like the one shown here, n y ounting whole squres lso, estimte the res of these shpes elow. A h C M 13, 14 15 B REASONING ENRICHMENT Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 609 11F Are of prllelogrm Rell tht prllelogrm is qurilterl with two pirs of prllel sies. Opposite sies re of the sme length n opposite ngles re equl. Like tringle, the re of prllelogrm is foun y using the length of one sie (lle the se) n the height (whih is perpeniulr to the se.) Lets strt: Developing the rule Strt this tivity y rwing lrge prllelogrm on loose piee of pper. Ensure the opposite sies re prllel n then use sissors to ut it out. Lel one sie s the se n lel the height, s shown in the igrm. Cut long the otte line. Now shift the tringle to the other en of the prllelogrm to mke retngle. Now explin how to fin the re of prllelogrm. height The re of prllelogrm is given y: A = h where is the se (one of the sies) n h is the height (perpeniulr to the se). h h 2 m Exmple 9 Fining the re of prllelogrm Fin the re of these prllelogrms. 12 m 5 m 3 m se Key ies Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
610 Chpter 11 Mesurement SOLUTION EXPLANATION A = h = 12 5 = 60 m 2 A = h = 2 3 = 6 m 2 Exerise 11F Choose the given sie s the se (12 m) n note the perpeniulr height is 5 m. Use the given sie s the se (2 m), noting tht the height is 3 m. 1 Copy n omplete the following, using the given vlues of n h. = 5, h = 7 = 20, h = 3 = 8, h = 2.5 A = h = = 35 A = = 20 = A = = 8 2 For eh of these prllelogrms, stte the sie length of the se n the height tht might e use to fin the re. 2 m 4 m 7 m 10 m 6 m 5.8 m 6.1 m e 1.5 m 5 m 1, 2 2 2 m f = 1.8 m 0.9 m 5 m 1.3 m UNDERSTANDING Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 611 Exmple 9 3(½), 4 3(½), 4, 5 3 4(½), 5 3 Fin the re of these prllelogrms. g 4 m 2.1 m 10 m 7 m 1 m 2.5 m 1.8 m e 3 m 7 m 2 m 4 m 2.1 m 7 m 3 m 4 These prllelogrms re on 1 m gris (not to sle). Fin their re. h f i 3 km 16 mm 11 mm 5 The floor of n offie spe is in the shpe of prllelogrm. The longest sies re 9 m n the istne etween them is 6 m. Fin the re of the offie floor. 12 km 15 m 20 mm 2 m FLUENCY 11F Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
612 Chpter 11 Mesurement 11F 6, 7 7, 8 7 9 6 Fin the height of prllelogrm when its: re = 10 m 2 n se = 5 m re = 28 m 2 n se = 4 m re = 2.5 mm 2 n se = 5 mm 7 Fin the se of prllelogrm when its: re = 40 m 2 n height = 4 m re = 150 m 2 n height = 30 m re = 2.4 km 2 n height = 1.2 km 8 A lrge wll in the shpe of prllelogrm is to e pinte with speil re pint, whih osts $20 per litre. Eh litre of pint overs 5 m 2. The wll hs se length of 30 m n height of 10 m. Fin the ost of pinting the wll. 9 A propose retngulr flg for new ountry is yellow with re stripe in the shpe of prllelogrm, s shown. Fin the re of the re stripe. Fin the yellow re. 30 m 70 m 10 Explin why this prllelogrm s re will e less thn the given retngle s re. 10 m 5 m 10 m 11 A prllelogrm inlues green tringulr re, s shown. Wht frtion of the totl re is the green re? Give resons for your nswer. 12 The re of prllelogrm n e thought of s twie the re of tringle. Use this ie to omplete this proof of the rule for the re of prllelogrm. Are = twie tringle re = 2 = 10 10, 11 5 m h 60 m 11, 12 PROBLEM-SOLVING REASONING Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 613 Glss fçe 13 The Puert e Europ (Gte of Europe) towers re twin offie uilings in Mri, Spin. They look like norml retngulr glss-overe skysrpers ut they len towrs eh other t n ngle of 15 to the vertil. Two sies re prllelogrms n two sies re retngles. Eh tower hs vertil height of 120 m, slnt height of 130 m n squre se of sie 50 m. All four sies re overe with glss. If the glss osts $180 per squre metre, fin the ost of overing one of the towers with glss. (Assume the glss overs the entire surfe, ignoring the ems.) 130 m 120 m 50 m 50 m 13 ENRICHMENT 11F Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
614 Chpter 11 Mesurement Key ies 11G Are of omposite shpes EXTENDING The res of more omplite shpes n e foun y iviing them up into more simple shpes, suh s the retngle n tringle. We n see this in n eril view of ny Austrlin ity. Suh view will show tht mny ity streets re not prllel or t right ngles to eh other. As result this uses ity loks to form interesting shpes, mny of whih re omposite shpes me up of retngles n tringles. Let s strt: Diviing ln to fin its re Working out the re of this piee of ln oul e one y iviing it into three retngles, s shown. Cn you work out the re using this metho? Wht is nother wy of iviing the ln to fin its re? Cn you use tringles? Wht is the esiest metho to fin the re? Is there wy tht uses sutrtion inste of ition? Streets, prks n uilings form omplex shpes tht n e me up of tringles n retngles. 30 m 20 m 20 m 10 m Composite shpes re me up of more thn one simple shpe. The re of omposite shpes n e foun y ing or sutrting the res of simple shpes. A squre plus tringle A retngle sutrt tringle Exmple 10 Fining the re of omposite shpes Fin the re of eh of these omposite shpes. 12 m 5 m 9 m 4 m 6 m 30 m Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 615 SOLUTION EXPLANATION A = l w + 1 2 h = 9 5 + 1 2 3 5 = 45 + 7.5 = 52.5 m 2 A = l 2 1 2 h = 6 2 1 2 6 4 = 36 12 = 24 m 2 Exerise 11G Divie the shpe into retngle n tringle n fin the missing lengths. 9 m 3 m 5 m 5 m 9 m Sutrt the tringle ( 1 6 4) t the top of the 2 shpe from the lrger squre (6 6). 1 3 3 6 m 4 m 6 m 6 m 1 Copy these igrms n rw otte line where you might ivie these shpes into two more simple shpes. 2 To fin the re of eh of the following shpes, eie if the esiest metho woul involve the ition of two shpes or the sutrtion of one shpe from nother. UNDERSTANDING Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
616 Chpter 11 Mesurement 11G Exmple 10 Exmple 10 3 Copy n omplete the solutions for the res of these shpes. 1 m 4 5 1 m A = l 2 + = 1 2 + 3 = + 3 m = m 2 2 m A = lw 13 m 5 m 4 m 4 m = 8 1 2 = = m 2 Fin the re of these omposite shpes y ing together the re of simpler shpes. 3 m 10 mm e 4 m 3 m 6 m 21 m 2 m 17 m 8 m 9 m f 3 m 3 km Use sutrtion to fin the re of these omposite shpes. 3 m 6 m 7 m 2 m 4, 5 4, 5 40 mm 5 m 9 m 10 km 10 mm 6 km 10 m 3 m 3 m 4 5(½) UNDERSTANDING FLUENCY Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 617 e 2 m 6 m 14 m 5 m 20 m 6 Fin the res of these omposite shpes. 8 m 4 m 5 m 2 m f 4 m 8 m 9 m 1 m 4 m 9 m 2.5 m 4 m 20 m 7 By fining the missing lengths first, lulte the re of these omposite shpes. 12 m 2 m 5 m 6, 7 7, 8 3.5 m 3.5 m 4 m 2 m 1 m 6 m 2 m 15 m 7 9 FLUENCY PROBLEM-SOLVING 11G Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
618 Chpter 11 Mesurement 11G 8 A wll hs three squre holes ut into it to llow for winows, s shown. Fin the remining re of the wll. 2.4 m 9 A ftory floor, with imensions shown opposite, is to e overe with linoleum. Inluing unerly n instlltion, the linoleum will ost $25 per squre metre. The uget for the jo is $3000. Is there enough money in the 12 m uget to over the ost? 10 10 10 Explin why using sutrtion is sometimes quiker thn using ition to fin the re of omposite shpe. Refer to the igrm s n exmple. 6 m 5 m 1.2 m 11 The 4-sie shpe lle the trpezium hs one pir of 6 m prllel sies. 5 m For the trpezium shown opposite, is it possile to fin the se length of eh tringle on the sies? Justify your 10 m nswer. Cn you ome up with metho for fining the re of this trpezium using the retngle n tringles shown in the igrm? Use igrms to explin your metho. Aing to infinity 12 The squre given opposite, whih hs n re of 1 unit, is ivie to show the res of 1 2, 1 4, 1 8,... Similr to the one shown opposite, rw your own squre, showing s mny frtions s you n. Try to follow the spirl pttern shown. Note: The igger the squre you strt with, the more squres you will e le to show. i Write the next 10 numers in this numer pttern. 1 2, 1 4, 1 8,... ii Will the pttern ever stop? 1 8 1 4 1 32 1 16 1 Wht is the totl re of the strting squre? Wht o your nswers to prts ii n tell you out the nswer to the sum elow? 1 2 + 1 4 + 1 8 + 1 +... (ontinues forever) 8 10 m 10, 11 1 2 12 4 m 1 PROBLEM-SOLVING REASONING ENRICHMENT Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 619 38pt 11A Ext 38pt 11B 38pt 11C 38pt 11D 38pt 11E Progress quiz 1 2 3 4 5 Whih metri unit woul e most pproprite for mesuring the lengths of: the height of the lssroom? the with of your thum the istne from Syney to Cnerr? Convert to the units given in rkets. 4 m (m) 2 m (mm) 3 km (m) e 1.45 km (m) Fin the perimeter of eh of these shpes. 5 m 46 mm 5 m 1.2 m Fin the re of this retngle n squre. 2 m 45 m Fin the re of eh given tringle. 4 m 3 m 4.4 m 5 m 4 m 9 m 5 mm 7 m f 3.5 m (mm) 23 000 m (km) 7 m 8 mm 70 m 32 m 52 m 25 m Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
620 Chpter 11 Mesurement 38pt 11F 38pt 11G Ext 38pt 11C/D 6 7 8 Fin the re of these prllelogrms. 9 m 5 m 8.1 m 11 m Fin the re of these omposite shpes. 5 m 13 m 40 m 15 m 12 m 9 m 41 m The perimeter of squre me of thin wire is 60 m. Fin: the re of this squre 5 m 12 mm 16 mm the re of retngle me with this wire, if its length is twie its with 20 mm 5 m 1 m 1 m the numer of smller squres tht n e me from this wire if eh squre hs n re of 4 squre entimetres Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 621 11H Volume of retngulr prisms The mount of spe insie three-imensionl (3D) ojet is lle volume. Volume is mesure in ui units suh s the ui entimetre, whih is 1 m long, 1 m wie n 1 m high. Just like the topis of length n re, ifferent units n e selete, epening on the size of the volume eing mesure. For exmple, the volume of wter in the se oul e mesure in ui kilometres n the volume of onrete poure from ement mixing truk oul e mesure in ui metres. Let s strt: Volume The Pifi Oen ontins hunres of thousns of ui kilometres of wter. We ll know tht there re 100 m in 1 m, ut o you know how mny ui entimetres re in 1 ui metre? Try to visulise 1 ui metre 1 metre long, 1 metre wie n 1 metre high. Guess how mny ui entimetres woul fit into this spe. Desrie metho for working out the ext nswer. Explin how your metho works. Volume is mesure in ui units. The ommon metri units for volume inlue: ui millimetres (mm 3 ) ui entimetre (m 3 ) 1 mm 1 mm 1 mm 1 m 1 m 1 m (Not rwn to sle.) Key ies Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
622 Chpter 11 Mesurement Key ies ui metre (m 3 ) (Not rwn to sle.) ui kilometre (km 3 ) (Not rwn to sle.) The volume of retngulr prism is given y the formul: V = length with height 1 m 1 km = lwh A retngulr prism is lso lle uoi. l The volume of ue is given y: V = l l l = l 3 l Exmple 11 Fining the volume of uoi Fin the volume of this retngulr prism. SOLUTION V = lwh = 8 3 4 = 96 m 3 EXPLANATION 8 m l l 1 m 1 m V = l 3 3 m 4 m 1 km h w 1 km Use the formul for the volume of retngulr prism, then sustitute the three lengths into the formul. V = lwh Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 623 Exerise 11H 1 For eh of these solis, ount the numer of ui units to fin its volume. e 2 Copy n omplete the working shown for eh of these solis. 4 m V = lwh = 4 = m 3 2 m 3 m 1 m V = lwh f 3 m = 1 = 1, 2 2 6 m 2 km V = lwh = 2 = km 3 2 km 2 km UNDERSTANDING Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
624 Chpter 11 Mesurement 11H Exmple 11 3 Fin the volume of these retngulr prisms. 3 m e g 7 m 7 m 7 m 10 km 10 m 4 m 2 m 1 m f h 3 6 3(½), 4 7 5 m 3 m 1.4 mm 1.5 km 5 km 1 m 2 m 4 A fruit ox is 40 m long, 30 m wie n 20 m high. Fin its volume. 5 A shipping ontiner is 3 m wie, 4 m high n 8 m long. Fin its volume. 6 A short retngulr ruler is 150 mm long, 40 mm wie n 2 mm thik. Fin its volume. 3(½), 4 7 7 There is enough ie on Erth to fill ue of sie length 300 km. Fin the pproximte volume of ie on Erth. FLUENCY Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 625 8 These solis re me up of more thn one retngulr prism. Use ition or sutrtion to fin the volume of the omposite soli. 4 m 2 m 5 m 3 m 8 m 4 m 2 m 2 m 9 m 8, 9 9, 10 10 m 9 A ox mesuring 30 m long, 20 m high n 30 m wie is pke with mthoxes, eh mesuring 5 m long, 2 m high n 3 m wie. How mny mthoxes will fit in the ox? 10 The outsie imensions of lose wooen ox re 20 m y 20 m y 20 m. If the ox is me from woo tht is 2 m thik, fin the volume of ir insie the ox. 11 The re of one fe of ue is 25 m 2. Fin the ue s volume. The perimeter of one fe of ue is 36 m. Fin the ue s volume. 12 12 12 We n fin the re of this she tringle y thinking of it s hlf retngle. Use the sme ie to fin the volume of eh of these solis. 4 m 3 m 9 11 12, 13 PROBLEM-SOLVING REASONING 11H Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
626 Chpter 11 Mesurement 11H 13 Fin the height of these retngulr prisms with the given volumes. Use tril n error if you prefer. i iii 4 m V = 16 m 3 4 m? 2 m? 2 m V = 56 m3 ii iv? 3 m V = 45 m 3 9 m V = 54 m 3 Cn you explin metho tht lwys works for fining the height of retngulr prism? Use V, l n w to write rule for h. Cui onversions 14 The igrm shows 1 m 3 lok tht is ivie into ui millimetres. i How mny mm 3 re there long one ege of the ue? ii How mny mm 3 re there in one lyer of the ue? (Hint: How mny ues sit on the se?) iii How mny lyers of mm 3 re there in the ue? iv Use your nswers from prts i to iii ove to now lulte how mny mm 3 there re in 1 m 3. Use similr metho to lulte the numer of: i m 3 in 1 m 3 ii m 3 in 1 km 3 Complete the igrm shown. km 3 m 3 m 3 mm 3 1 m 2 m 14 1 m 1 m REASONING ENRICHMENT Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 627 11I Cpity CONSOLIDATING Cpity reltes to the volume of flui or gs tht ontiner n hol. For exmple, the pity of wter tnk my e 5000 litres, or frmer s wter llotion might e 300 meglitres (mening 300 million litres). The si unit is the litre, whih ontins 1000 m 3 of spe. Other ommon metri units for pity inlue the millilitre, kilolitre n meglitre. There is ler link etween pity n volume, s they oth relte to the spe oupie y three-imensionl ojet. Let s strt: Wter ontiners Irrigtion hnnels rry meglitres of wter to frms. Wil, Tony n Ethn eh ring ontiner to ollet some wter from fountin. Wil sys his ontiner hols 2 litres. Tony sys his ontiner hols 2000 m 3. Ethn sys his ontiner hols 2000 millilitres. Who n ollet the most wter? Give resons for your nswer. Cpity is the volume of flui or gs tht n ojet n hol. Common metri units inlue: 1 litre (L) = 1000 millilitres (ml) 1 kilolitre (kl) = 1000 litres (L) 1 meglitre (ML) = 1000 kilolitres (kl) Relting volume n pity 1 m 3 = 1 ml 1 m 3 = 1000 L = 1 kl 1000 ML 1000 1000 kl 1000 L 1000 M IL K 1 L (1000 ml) ml 1000 M IL K 1000 m3 Key ies Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
628 Chpter 11 Mesurement Exmple 12 Converting units for pity Convert to the units shown in rkets. 500 ml (L) 3 ML (L) SOLUTION 500 ml = 500 1000 = 0.5 L 3 ML = 3 1000 kl = 3 1000 1000 L = 3 000 000 L Exmple 13 Converting m 3 to litres Fin the pity of this ontiner, in litres. SOLUTION V = 20 10 10 = 2000 m 3 = 2000 ml = 2000 1000 L = 2 L Exerise 11I 1 Complete these sentenes. 1 ml ontins the volume of m 3. 1 L ontins ml. 1 L ontins m 3. 1 kl ontins L. e 1 ML ontins kl. EXPLANATION When onverting to lrger unit, ivie. There re 1000 ml in 1 L. There re 1000 kl in 1 ML n 1000 L in 1 kl. So 1 ML is 1 million litres. 10 m EXPLANATION V = lwh 1 m 3 = 1 ml There re 1000 ml in 1 litre. 20 m 1 3 3 UNDERSTANDING Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 629 Exmple 12 Exmple 12 2 Cirle or write own whih volumes re the sme. 1 L, 10 kl, 1000 ml, 1 m 3, 1000 m 3 1 m 3, 100 L, 1000 L, 1000 ML, 1 kl 3 From options A to F, hoose the pity tht est mthes the given ontiner. tespoon A 18 L up B 250 ml ottle C 10 kl kithen sink D 20 ml e wter tnk E 45 ML f wter in lke F 0.8 L 4 Convert to the units shown in rkets. 2 L (ml) 0.1 L (ml) e 2000 L (kl) f 3500 ml (L) i 0.257 L (ml) j 9320 ml (L) m 0.5 kl (L) n 91 000 kl (ML) g k o 6 ML (kl) 70 000 ml (L) 3.847 ML (kl) 0.42 L (ml) 5 Re these sles to etermine the mount of wter in eh of the ontiners. 25 1 L 4 20 750 3 15 500 2 10 250 1 5 m ml 3 ml h l p 24 kl (L) 2500 kl (ML) 47 000 L (kl) 170 L (kl) 6 A up of 200 ml of wter is e to jug lrey ontining 1 L of wter. Fin the totl volume in: ml L 7 Convert to the units shown in rkets. 6 ML (L) 0.004 kl (ml) 8 A frmer purhses 3.3 ML of wter for her pple orhr. How mny litres is this? 4(½), 5 9 4(½), 5 9 320 000 L (ML) 992 700 ml (kl) 4(½), 5, 6, 7(½), 9(½) UNDERSTANDING FLUENCY 11I Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
630 Chpter 11 Mesurement 11I Exmple 13 9 Fin the pity of eh of these ontiners, in litres. e 10 m 6 m 10 m 10 m 50 m 9 m 15 m 3 m f 8 m 32 m 32 m 5 m 13 m 12 m 21 m 10 A swimming pool in the shpe of retngulr prism hs length 50 m, with 25 m n epth 2 m. Fin the swimming pool s: volume, in m 3 pity, in L 11 A ripping tp leks out 10 ml every minute. 10, 11 11, 12 If there re 50 rips per minute, fin the volume of one rip. Fin the pproximte volume of wter, in litres, tht hs leke from tp fter the following time perios. i 100 minutes ii 1 hour iii 1 y iv 1 yer 12 A ose of 12 ml of meiine is to e tken twie eh y from 0.36 L ottle. How mny ys will it tke to finish the meiine? 12 14 FLUENCY PROBLEM-SOLVING Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
Mesurement n Geometry 631 13 A gs ottle ontins 50 L of liqui gs. If the liqui gs is use t rte of 20 ml per minute, how mny hours will the gs ottle lst? 14 A ity s ms hve 2 million meglitres of wter n the verge ily onsumption of the ity s people is 400 L per y per person. If the ity s popultion is 5 million people, how long will the m supply lst without further wter thment? 15 If x is ny numer, then x litres is the sme s 1000 x = 1000x millilitres euse there re 1000 ml in 1 L. Write expressions for x L in the following units. m 3 m 3 kl ML 16 A retngulr prism hs length l m, with w m n height h m. Write n expression for the pity of the ontiner mesure in: i m 3 ii ml iii L iv kl 15 15 15, 16 A retngulr prism hs length l m, with w m n height h m. Write n expression for the pity of the ontiner mesure in: i m 3 ii L iii kl iv ML Ae epth 17 A ontiner is 10 m long, 5 m wie n 8 m high. Fin the epth of wter when the following mounts of wter re poure in. (Rememer: 1 ml = 1 m 3.) 8 m i 400 ml ii 200 ml iii 160 ml After ing 200 ml, further 30 ml is e. Wht is the inrese in epth? A 1-litre ontiner of milk hs se re of 8 m y 7 m. After 250 ml of milk is poure out, wht is the epth of the milk remining in the ontiner? Give your nswer to the nerest mm. 10 m 17 5 m PROBLEM-SOLVING REASONING ENRICHMENT 11I Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400
632 Chpter 11 Mesurement 11J Mss n temperture CONSOLIDATING The sles for oth mss n temperture re se on the properties of wter. In Frne in 1795, the grm ws efine s eing the weight of 1 m 3 of wter t 0 C. Lter it ws reefine to e the weight t 4 C, s this is onsiere to e the temperture t whih wter is the most ense. So, 1 litre of wter is very lose to 1 kilogrm, whih is the si unit for mss. Other units for mss inlue the tonne, grm n milligrm. A smll r hs mss of out 1 tonne n 20-ent oin hs mss of out 11 grms. Temperture tells us how hot or ol something is. Aners Celsius (1701 1744), Sweish sientist, worke to efine sle for temperture. After his eth, temperture ws offiilly efine y: 0 C (0 egrees Celsius) the freezing point of wter. 100 C (100 egrees Celsius) the oiling point of wter (t one stnr egree of pressure). This is still the ommon unerstning of egrees Celsius. As mentione in Chpter 10, Fhrenheit is nother sle use for temperture. This is investigte further in the Enrihment questions. Let s strt: Choose unit of mss Nme five ojets of whih their mss woul ommonly e mesure in: tonnes kilogrms grms milligrms. Is it possile for the temperture to rop elow 0 C? How is this mesure n n you give exmples of ples or situtions where this might e the se? A smll r hs mss of out 1 tonne. Ie melts t 0 C. Unorrete 3r smple pges Cmrige University Press Greenwoo et l., 2015 978-1-107-56882-2 Ph 03 8671 1400