10 AREA AND VOLUME 1. Before you start. Objectives

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1 10 AREA AND VOLUME 1 The Tower of Pis is circulr bell tower. Construction begn in the 1170s, nd the tower strted lening lmost immeditely becuse of poor foundtion nd loose soil. It is 56.7 metres tll, with dimeter t the bse of 15.5 metres, nd there re 97 steps to the top. The tower continues to sink bout 1 mm ech yer. Objectives In this chpter you will: solve problems involving perimeters nd res know nd use the formule for the circumference nd re of circle drw the nets, elevtions nd plns for vriety of 3D shpes work out the volume of cubiods, prisms nd cylinders. Before you strt You need to know: how to mesure or clculte the perimeters of rectngles nd tringles how to use the formul for the re of rectngle wht circle, semicircle nd qurter circle re, nd be ble to nme the prts of circle nd relted terms how to drw circles nd rcs to given rdius. 145

2 Chpter 10 Are nd volume Are of tringles, prllelogrms nd trpeziums Objectives You know nd cn use the formul for the re of tringle. You know nd cn use the formul for the re of prllelogrm. You know nd cn use the formul for the re of trpezium. Why do this? Zoologists t gme reserves need to know the res of different sections of their reserve, so tht they know how mny nimls it cn ccommodte. Get Redy 1. The digrm shows rectngle. The length of the rectngle is 9 cm. The perimeter of the rectngle is 8 cm. Work out the width nd the re of the rectngle. 9 cm 9 cm Key Point The re of D shpe is mesure of the mount of spce inside the shpe. Are of tringle Key Points The digrm below shows tringle ABC. A rectngle hs been drwn round the tringle. The inside of the rectngle hs been split into four tringles. 1 C 4 Tringles 1 nd re congruent so re tringle 1 re tringle. 3 Also re tringle 3 re tringle 4. A D The length of the rectngle is the bse of the tringle nd the width of the rectngle is the perpendiculr height of the tringle. B height h This mens tht the re of tringle ABC is hlf the re of the rectngle. A bse b Are of the rectngle bse height So to find the re of tringle, work out hlf of its bse its height. B 146 re bse perpendiculr height

3 Are of tringle 1_ bse height height (h) A 1_ bh 10.1 Are of tringles, prllelogrms nd trpeziums bse (b) Exmple 1 Work out the re of the tringle. 4 cm Are of tringle 1 bse height cm Are cm² 14 cm² Do not forget to put the units of the nswer. Exminer s Tip The height of tringle is its verticl or perpendiculr height. Are of prllelogrm Key Points Here re two congruent tringles. The tringles cn be put together to form prllelogrm. The two tringles hve equl res so the re of the prllelogrm is twice the re of one of the tringles. Are of one tringle 1_ bse height Are of prllelogrm 1_ bse height bse height Are of prllelogrm bse height A bh height (h) bse (b) Exmple Work out the re of the prllelogrm. Are 8 9 mm² 7 mm² 9 mm Are of prllelogrm bse height. As the lengths re in millimetres, the units of the re re mm². 8 mm 147

4 Chpter 10 Are nd volume 1 Exercise 10A Questions in this chpter re trgeted t the grdes indicted. D 1 Work out the res of these tringles nd prllelogrms. b c 8 cm 9 m d 10 cm 4 m e f 7cm 6 mm 1 cm 1 cm 9 mm 9 cm Copy nd complete this tble. Shpe Bse Height Are Tringle Tringle 10 cm Tringle 8 cm 4 cm Prllelogrm 8 cm 4 cm Prllelogrm 7 cm 5 3 A rectngle hs length of 7 cm nd n re of 3². Work out the width of the rectngle. b A squre hs n re of 144 cm². Work out the length of side of the squre. Are of trpezium Key Points Here is trpezium. The trpezium is split into two tringles by digonl. Are of trpezium re of yellow tringle re of pink tringle. 1 Are bh b h h b h Are h 1 Are of trpezium 1_ 1_ 1_ h bh ( b)h 148

5 Are of trpezium 1_ sum of prllel sides distnce between them. A 1_ ( b)h 10.1 Are of tringles, prllelogrms nd trpeziums h b Exmple 3 Work out the re of the trpezium. 7 cm 11 cm Exminer s Tip 13 cm Remember tht unless the question tells you to tke mesurements from digrm you should not do so s digrms re not ccurtely drwn. Are 1 (7 13) cm Are of trpezium 1 sum of prllel sides distnce between them. Work out the brckets first. Exercise 10B 1 Work out the re of ech of these trpeziums. b 3 cm 18 m C 8 cm 8 m 11 cm 6 m c 9 cm d 10 cm 13 cm 1 7 cm 149

6 Chpter 10 Are nd volume Problems involving perimeter nd re Objectives You cn find the re nd perimeter of more complicted shpe mde from simpler shpes. You cn solve problems involving perimeters nd res. Why do this? A lot of houses seen from the side re pentgon shpe, so pinter would need to work out the re of pentgon to get the right mount of pint. Get Redy 1. Write down the formul for the re of: rectngle b squre c tringle d prllelogrm e trpezium. Key Point The perimeter or re of compound shpe cn be found by splitting the shpe into its simpler prts. A0 A03 Exmple 4 Work out the re of this pentgon cm B 8 cm Split the pentgon into rectngle A nd tringle B. 10 cm A 1 The height of the tringle is The bse of the tringle is 8 cm. 8 cm The rectngle hs length 8 cm nd width 10 cm. Are of rectngle A cm Are of tringle B cm Are of pentgon cm Are of rectngle length width Are of tringle 1 bse height Are of pentgon re of A re of B 150

7 10. Problems involving perimeter nd re Exmple 5 A rectngulr wll is 450 cm long nd 300 cm high. The wll is to be tiled. The tiles re squres of side 50 cm. How mny tiles re needed? A0 A03 wll 300 cm No digrm is given with this question so it is good ide to drw one. 450 cm 50 cm tile 50 cm Method 1 Number of tiles needed for the length Number of tiles needed for the height Number of tiles needed One wy to nswer questions like this is to work out how mny tiles re needed for the length nd how mny re needed for the height. So there re 6 rows of tiles, ech with 9 tiles. Number of tiles number of tiles in ech row number of rows. Method Are of wll cm² cm² Are of tile cm² Number of tiles The other wy to nswer this question is to divide the re of the wll by the re of tile. But remember tht you should not use clcultor nd the rithmetic is esier in the first method. Exercise 10C 1 The digrm shows the floor pln of room. Work out the perimeter of the floor. Give the units of your nswer. b Work out the re of the floor. Give the units of your nswer. 5 m 5 m 3 m D 9 m Krl wnts to mke rectngulr lwn in his grden. He wnts the lwn to be 30 m by 10 m. Krl buys rectngulr strips of turf 5 m long nd 1 m wide. Work out how mny strips of turf Krl needs to buy. 3 A wll is 300 cm by 50 cm rectngle. The wll is to be tiled. The tiles re squres of side 50 cm. Work out how mny tiles re needed. 4 A rectngle is 9 cm by 4 cm. A squre hs the sme re s the rectngle. Work out the length of side of the squre. 151

8 Chpter 10 Are nd volume 1 D 5 Keith is going to wllpper his living room nd his bedroom. Here re the floor plns of these rooms. 5 m living room 4 m 4 m bedroom 5 m 8 m m Work out the re of the floor in: i Keith s living room ii Keith s bedroom. b Work out the perimeter of the floor in Keith s living room. To work out the number of rolls of wllpper he needs, Keith uses this chrt. Keith is going to use stndrd rolls of wllpper. Stndrd rolls of wllpper re pprox 10 m long How mny rolls for the wlls Distnce round the room including doors & windows Wll height 10 m 33 ft 1 m 39 ft 14 m 46 ft 16 m 5 ft 18 m 59 ft 0 m 66 ft m 7 ft 4 m 79 ft.3 m m m m m The height of the wlls in Keith s living room is.5 m. c Find how mny rolls of wllpper Keith needs for his living room. The height of the wlls in Keith s bedroom is.6 m. d Find the number of rolls of wllpper Keith needs for his bedroom. C 6 Here is qudrilterl. 7 cm 1 4 cm 0 cm Work out the perimeter of the qudrilterl. b Work out the re of the qudrilterl. 15

9 10.3 Circumference nd re of circle 7 Work out the re of the yellow shded region in this digrm. 8 cm 9 cm C 1 cm 8 A kite hs digonls of length 10 cm nd 0 cm. Work out the re of the kite Circumference nd re of circle Objectives You cn work out the circumference of circle. You cn work out the re of circle. You cn solve problems involving circles, including semicircles nd qurter circles. Why do this? To fit new tyre on the wheel of your bike, you my need to know the circumference of the wheel to find the correct size. Get Redy 1. Drw circle of rdius. For this circle, drw nd lbel clerly: rdius b dimeter c chord d sector e n rc f segment g tngent. Key Points For ll circles circumference of circle (pi). dimeter of circle This vlue cnnot be found exctly. To 3 deciml plces, circumference of circle dimeter of circle C r C d C d d C Exminer s Tip Clcultor exm ppers hve the following instruction bout, If your clcultor does not hve button, tke the vlue of to be 3.14 unless the question instructs otherwise. Wtch Out! It is importnt not to confuse the dimeter with the rdius. 153

10 Chpter 10 Are nd volume 1 Exmple 6 Work out the circumference of circle with: dimeter 8.7 cm b rdius 3.1 m. Give your nswers correct to 3 significnt figures. Exminer s Tip Remember tht the circumference is pproximtely 3 times the dimeter or 6 times the rdius. C Circumference 7.3 cm Use C d with d 8.7 cm. Use the button or Write down t lest 4 figures of the clcultor disply. Give the nswer correct to 3 significnt figures. The units re the sme s the dimeter (cm). b C Circumference 19.5 m The dimeter cn be worked out from d r so d nd then use C d. Or use C r with r 3.1 m. The units re the sme s the rdius (m). Exmple 7 The circumference of circle is 84.3 cm. Work out the rdius of the circle. Give your nswer correct to 3 significnt figures r r r 84.3 ( ) Use C r with C 84.3 cm s the rdius is given in the question. Divide both sides by nd write down t lest 4 figures of the clcultor disply. Rdius 13.4 cm Give the nswer correct to 3 significnt figures. The units re the sme s the circumference (cm). Wtch Out! Be creful when dividing by on clcultor. It is best to use brckets. Exercise 10D In this exercise, if your clcultor does not hve button, tke the vlue of to be Give nswers correct to 3 significnt figures unless question sys differently. D 1 Work out the circumference of circle with dimeter: 7 cm b 1.9 mm c 5. d 40 cm e 1.9 m The rdius of bsketbll net hoop is 3 cm. Work out the circumference of bsketbll net hoop. A netbll hoop hs rdius of 19 cm. b Work out how much longer is the circumference of bsketbll net hoop thn the circumference of netbll hoop. 154

11 10.3 Circumference nd re of circle 3 The circumference of CD is 37.7 cm. Work out the rdius of the CD. C 4 The dimeter of the front wheel of Michel s bicycle is 668 mm. Work out the circumference of the wheel. Give your nswer in cm correct to the nerest cm. Michel rides his bicycle. b Work out the distnce cycled when the wheel mkes 1000 complete turns. Give your nswer in km correct to significnt figures. The distnce Michel rides his bicycle is 6 km. c Work out the number of complete turns mde by this wheel. 5 The length of the minute hnd of wtch is 1. cm. Work out the distnce moved by the point end of the hnd in 1 hour. b Work out the distnce moved by the point end of the hnd in: i 6 hours ii 0 minutes. 6 A circulr tble hs rdius of 6. Work out the circumference of the tble. The circumference of circulr tblecloth is 5 m. The tblecloth is put symmetriclly on the tble so tht the distnce from the tble to the edge of the tblecloth is the sme ll round the tble. b Work out the distnce from the tble to the edge of the tblecloth. 7 The digrm shows shpe mde from semicircle, rectngle nd n equilterl tringle. The rectngle hs length 18 cm nd width 10 cm. Work out the perimeter of the shpe. 18 cm 10 cm B Are of circle Key Points To find the re of circle mens to find the re enclosed by the circle. Here is circle tht hs been divided into four equl wedges or sectors. The sectors re then rrnged s r shown to form prllelogrm-like shpe. r r The length shown s r is hlf the circumference, r, of the circle. The re of the circle is the sme s the re of the shpe. Here is wht hppens when the circle is divided into more sectors. r r 155

12 Chpter 10 Are nd volume 1 The shpe looks more like prllelogrm nd s the number of sectors increses the prllelogrm becomes more like rectngle. r r The width of this rectngle is equl to hlf of the circumference of the originl circle nd the height of the rectngle is equl to the rdius of the circle. Are of circle re of rectngle r r r Tking A s the re of circle nd r s the rdius of the circle, A r Tht is Are rdius rdius Exmple 8 Work out the re of circle with: rdius of 9 cm b dimeter of 1.8 m. Give your nswers correct to 3 significnt figures. A Use A r with r 9 cm. Write down t lest 4 figures of the clcultor disply. Are 54 cm Give the nswer correct to 3 significnt figures. As the units of the rdius re cm, the units of the re re cm. b Rdius 1.8 m 6.4 m A Are 19 m Divide the dimeter by to get the rdius. Write down t lest 4 figures of the clcultor disply. Give the nswer correct to 3 significnt figures. As the units of the rdius re m, the units of the re re m. Exminer s Tip When the dimeter of circle is given, to work out the re of the circle first find the rdius by dividing the dimeter by. Exmple 9 Work out the rdius of circle with re r Use A r with A 4. r r Work out the vlue of r by dividing both sides by. Tke the squre root to find the vlue of r. Rdius 3.83 cm 156

13 10.3 Circumference nd re of circle Exercise 10E In this exercise, if your clcultor does not hve button, tke the vlue of to be Give nswers correct to 3 significnt figures unless the question sys differently. 1 Work out the re of circle with rdius: 8 cm b 1.7 cm c 8.5 mm d 9.7 cm e 1.6 m Work out the re of circle with dimeter: 4 cm b 8.3 cm c 0.95 m d 58.4 mm e The digrm shows pond surrounded by pth. Work out the re of the blue region of the pond. b Work out the re of the pth. c The pth is mde of shingle tht costs 1.95 per squre metre of pth. Work out the cost of the shingle to mke the pth. 3.5 m.5 m D C 4 The digrm represents the pln of sports field. The field is rectngle with semicirculr ends. The rectngle hs length 100 m nd width 70 m. The semicircles hve dimeter 70 m. Work out the re of the field. 100 m 70 m The field is to be covered in fertiliser tht costs 3p per squre metre. b Use your nswer to prt to work out the cost of the fertiliser for the field. 5 A circle of dimeter 8 cm is cut from piece of yellow crd. The crd is in the shpe of squre of side 11 cm. The crd shown yellow in the digrm is thrown wy. Work out the re of the crd thrown wy. 11 cm 8 cm 11 cm 6 A, B nd C re three circles. Circle A hs rdius nd circle B hs rdius 1 cm. Circle C is such tht re of circle C re of circle A re of circle B. Work out the rdius of circle C. B 7 The digrm shows str mde by removing four identicl qurter circles from the corners of squre of side 30 cm. Work out the re of the str. 30 cm 30 cm 157

14 Chpter 10 Are nd volume Drwing 3D shpes Objective You cn recognise nd drw the net of 3D shpe. Why do this? A mnufcturer of chocolte boxes would hve to consider the nets of different sizes of boxes in order to see how best to pckge their product. Get Redy 1. Sketch these shpes. tringulr prism c cylinder b squre-bsed pyrmid d tringulr-bsed pyrmid Key Points Isometric pper will help you to mke scle drwings of three-dimensionl objects. Isometric pper must be the right wy up i.e. verticl lines down the pge nd no horizontl lines. A net of 3D shpe is D shpe tht cn be folded to mke the 3D shpe. A 3D shpe cn hve more thn one net. This cube hs sides of length. This cuboid hs height 4, length 3 nd width. This prism hs tringulr fce. Shpes cn be joined together Exmple 10 Drw two different nets for this cuboid. cm 3 cm cm 3 cm cm 3 cm There re six different nets tht will mke this cuboid. Wtch Out! 3 cm cm 3 cm cm A 3D shpe my hve mny different nets. The shpe of the net will depend on where the 3D shpe hs been split prt. 158 three-dimensionl net cuboid

15 10.5 Elevtions nd plns Exercise 10F 1 Use isometric pper to drw cuboid with height cm, width 4 cm nd length 3 cm. Sketch six different nets tht will mke cube. 3 Here re the nets of some 3D shpes. Identify the shpes. b c d 4 Drw n ccurte net for ech of these. b cm 3 cm 3 cm 3 cm 4 cm 10.5 Elevtions nd plns Objective You cn drw elevtions nd plns of 3D shpes. Why do this? Architecturl proposls will usully contin plns nd elevtions of the proposed building, to give people n ide of wht the building will look like from ech side. Get Redy 1. Wht would the shpes in question 4, bove, look like if drwn from bove, the side nd the front. Key Points The front elevtion is the view from the front. The side elevtion is the view from the side. The pln is the view from bove. pln side elevtion front elevtion front elevtion side elevtion pln 159

16 Chpter 10 Are nd volume 1 Exmple 11 Drw the front elevtion, side elevtion nd pln of this 3D shpe. There re six cubes in this shpe but you cn see only five of them. There must be cube under the top one. pln Drw the elevtions nd pln like this: 1. Pln t the top.. Front elevtion under the pln. 3. Side elevtion (view from the right) to the right of the front elevtion. front elevtion Exmple 1 side elevtion Sketch the shpe represented by the front nd side elevtions nd pln. pln front front elevtion side elevtion Exercise 10G D 1 Drw the elevtions nd plns of these shpes. b c m d front e 4 cm 5 m 3 m 3 cm cm 3 cm f g cm 4 cm 160

17 10.6 Volume of cubiod Sketch the shpes represented by these elevtions nd plns. b pln pln c pln D front elevtion side elevtion front elevtion side elevtion front elevtion side elevtion 10.6 Volume of cuboid Objective You cn work out the volume of cuboid nd shpes mde from cuboids. Why do this? If you were filling swimming pool you might first hve to consider its volume in order to work out how much wter you would need. Get Redy 1. Work out the volumes of these cuboids. Give the units with your nswers. b 8 cm 4 m 1 cm 6 m 8 m Exmple 13 This shpe is mde from two cuboids. Work out the totl volume of the shpe. 9 m 4 m 9 m Work out the volume of ech cuboid. Use volume of cuboid l w h. m m 3 m 3 m 3 m 4 m Volume m 3 For the lrger cuboid l 9 m, w 3 m nd h 4 m. m m 3 m Volume 3 1 m 3 For the smller cuboid l m, w 3 m nd h m. Totl volume m 3 To work out the totl volume of the shpe dd the volumes of the cuboids. 161

18 Chpter 10 Are nd volume 1 D Exercise 10H 1 These shpes re mde from cuboids. Work out the volumes of the shpes. cm b 4 cm c 3 cm 7 cm cm 3 cm 8 cm 9 cm 9 cm cm cm 9 cm 110 mm 4 cm Here is net of cuboid. Work out the volume of the cuboid. 10 cm 14 cm 18 cm 10.7 Volume of prism Objective You cn work out the volume of prism. Why do this? Sndwiches re often sold in pcks tht re tringulr prisms, so you cn work out how much sndwich you re getting. Get Redy 1. Work out the volume of these shpes. b c Find the volume of hlf shpe b. Key Point Volume of prism re of cross-section length crosssection length 16 prism

19 10.7 Volume of prism Exmple 14 The re of the cross-section of this prism is. The length of the prism is 10 cm. Work out the volume of the prism. 10 cm Use volume of prism re of cross-section length. Here, the re of cross-section nd the length 10 cm. Volume cm 3 Give the unit with your nswer. The unit of re is cm, the length is in cm so the unit of volume is cm 3. Exmple 15 Work out the volume of this prism. 4 cm The cross-section of the prism is tringle. Remember: re of tringle 1 bse height. Here the bse 3 cm nd height 4 cm. 3 cm 6. Are of cross-section Volume of prism cm 3 Use volume of prism re of cross-section length. Here the re of cross-section nd length 6.. Exercise 10I 1 Work out the volumes of these prisms. b C 1 cm mm c d 30 mm 3 cm 1.75 m 0.95 m 0.6 m 8 cm Work out the volumes of these prisms. b 9 cm 1 1 cm 8 cm 3 163

20 Chpter 10 Are nd volume 1 C c 3.3 cm d cm.7 cm cm 3 The re of the cross-section of prism is 4. The volume of the prism is Work out the length of the prism. B 4 Here is prism. Show tht the volume of the prism is 8x 3 cm 3. x x 3x x 5 The digrm shows tringulr prism. The volume of the prism is 45y 3 cm 3. Find n expression for h in terms of y. 5y 4y h 10.8 Volume of cylinder Objective You cn work out the volume of cylinder. Why do this? You could work out the volume of liquid tht your mug cn hold if you wnted to boil only tht exct mount of wter, to sve energy. Get Redy 1. Find the re of these circles: rdius 3 cm b dimeter c rdius 10 cm. Key Point Volume of cylinder re of cross-section length r h where r is the rdius nd h is the height. r h 164 cylinder

21 10.8 Volume of cylinder Exmple 16 Work out the volume of this cylinder. Give your nswer in terms of nd to 3 significnt figures. The cross-section of the cylinder is circle with rdius. Remember: re of circle rdius. Tke s cm Are of cross-section 6 36 Volume of cylinder cm 3 Use volume of cylinder re of cross-section length. Do not round your nswer t this stge. Write down ll the digits on your clcultor disply cm 3 (3 s.f.) Give your finl nswer correct to 3 significnt figures. Exercise 10J 1 Work out the volumes of these cylinders. Give your nswers correct to 3 significnt figures. 4 cm b c 30 mm d 1 cm C 40 mm 300 mm 79 cm Work out the volumes of these cylinders. Give your nswers in terms of. b c 0 cm 0.45 m m 10 cm 3 An ircrft hngr hs semicirculr cross-section of dimeter 0 m. The length of the hngr is 3 m. Work out the volume of the hngr. Give your nswer in terms of. 3 m 0 m 165

22 Chpter 10 Are nd volume 1 B 4 An nnulus hs n externl dimeter of 7.8 cm, n internl dimeter of 6. cm nd length of 6.. Work out the volume of the nnulus. Give your nswer correct to 1 deciml plce. 6. cm 7.8 cm 6. 5 A gold coin hs height of.5 mm nd volume of 000 mm 3. Work out the dimeter of the gold coin. Give your nswer correct to deciml plces. 6 An oil drum hs rdius of 0.9 m nd height of 1.4 m. The oil drum is completely filled with oil. Work out the volume of the oil in the oil drum. Give your nswer correct to 3 significnt figures. Chpter review Are of tringle 1_ bse height. A 1_ bh Are of prllelogrm bse height. A bh Are of trpezium 1_ sum of prllel sides distnce between them. A 1_ ( b)h The perimeter or re of compound shpe cn be found by splitting the shpe into its simpler prts. For ll circles, circumference of circle (pi). dimeter of circle To 3 deciml plces, Circumference of circle d r where d is the dimeter of the circle, nd r is the rdius of the circle. Are of circle r where r is the rdius of the circle. The net of 3D shpe is D shpe tht cn be folded to mke the 3D shpe. A 3D shpe cn hve more thn one net. The front elevtion is the view from the front. The side elevtion is the view from the side. The pln is the view from bove. Volume of prism re of cross-section length. front elevtion pln side elevtion cross section length 166

23 Chpter review Volume of cylinder re of cross-section length r h where r is the rdius nd h is the height. r h Review exercise 1 The digrm shows some nets nd some solid shpes. An rrow hs been drwn from one net to its solid shpe. Drw n rrow from ech of the other nets to its solid shpe. Nov

24 Chpter 10 Are nd volume 1 Find the volume of this prism. Digrm NOT ccurtely drwn represents 1 cm 3 June 08 D 3 Work out the re of the shpe. 9 cm 7 cm Digrm NOT ccurtely drwn 1 cm Nov The digrm shows solid object mde of 6 identicl cubes. Exm Question Report front On centimetre grid, drw the side elevtion of the solid object from the direction of the rrow. 95% of students nswered this question poorly becuse they did not know wht the different types of plns nd elevtions re. b On centimetre grid, drw the pln of the solid object. June 07 5 The digrm shows cuboid. The cuboid hs: volume of 300 cm 3 length of 10 cm width of. Work out the height of the cuboid. 10 cm height Nov 06 6 Boxes re pcked into crtons. A box mesures 4 cm by by 10 cm. A crton mesures 0 cm by 30 cm by 60 cm. The crton is completely filled with boxes. Work out the number of boxes tht will completely fill one crton. box 4 cm 10 cm crton Digrm NOT ccurtely drwn 60 cm 30 cm 0 cm Nov

25 Chpter review 7 Jne mkes chocoltes. Ech box she puts them in hs: volume 1000 cm 3 length 0 cm width 1000 cm. Work out the height of box. Jne mkes 350 chocoltes. Ech box will hold 18 chocoltes. b Work out: i how mny boxes Jne cn fill completely ii how mny chocoltes will be left over. D 8 Here is net of cuboid. Work out: the surfce re b the volume of the cuboid cm 9 cm 9 The digrm shows tringulr prism. 7. C 4. Drw the elevtions nd pln for the prism. b Work out the surfce re of the prism. Give the units with your nswer. 9 cm * 10 Shelim is replcing the skirting bords nd coving in his living room. 1 m Skirting bord cn be bought in: 4 m lengths t m lengths t m lengths t Coving cn be bought in: 3 m lengths t m lengths t.00. FIREPLACE Coving cn be joined together, but skirting bord must not be pieced together s the joins will be noticeble. Find the cost of his mterils for both jobs, minimising the wste. 169

26 Chpter 10 Are nd volume 1 C * 11 Amy hs sved 600 to spend on crpeting her front room. There re four types she likes: 5 m Nturl Twist t per m Medium Blend t per m Hevy Weve t per m Luxury Pile t 4.90 per m. She lso needs to buy underly, which is vilble in two types: 5 m Cushion t.00 per m Super Cushion t 4.00 per m. Fitting is 50 extr. Wht cn she fford to buy? 1 m m 1 A lndscpe contrctor chrges: 40 per squre metre for levelling the ground nd lying pving stones 15 per squre metre for levelling the ground nd sowing grss seed. Clculte the cost of both pving nd seeding the grden shown on the right. house 10 m pving m 10 m grss 1 m 13 A ring-shped flowerbed is to be creted round circulr lwn of rdius.55 m..55 m Roses costing 4.0 re to be plnted pproximtely every 50 cm round this flowerbed. How much money will be needed for roses? 170

27 Chpter review 14 The digrm shows grden tht includes lwn, vegetble ptch, circulr pond nd flowerbed. 1 m All mesurements re shown in metres. The lwn is going to be relid with turf costing 4.60 per squre metre. 4 m How much will this cost? 4 m VEGETABLE PATCH pond, rdius 1 m C 1 m 1.8 m flower bed 15 You re plnning prty for 30 children. You buy some concentrted ornge sqush nd some plstic cups. 10 cm Ech plstic cup will hve 150 ml of drink in it. (150 ml 150 cm 3 ) Check tht the plstic cup shown cn hold 150 ml of drink. Use the formul: volume 5 h d 4 Ech of the 30 children t the prty will hve mximum of three drinks of ornge sqush. Ech plstic cup is to be filled with 150 ml of drink. The sqush needs to be diluted s shown on the bottle lbel. A bottle of concentrted ornge sqush contins 0.8 litres of sqush nd costs 1.5. b How mny bottles of concentrted ornge sqush do you need for the prty? c How much will they cost in totl? 16 8 cm 0 cm The cross-section of the prism in the digrm is trpezium. The lengths of the prllel sides of the trpezium re 8 cm nd. The distnce between the prllel sides of the trpezium is. The length of the prism is 0 cm. Work out the volume of the prism. The prism is mde out of gold. Gold hs density of 19.3 g/cm 3. b Work out the mss of the prism. Give your nswer in kilogrms. 171

28 Chpter 10 Are nd volume 1 B 17 A swimming pool hs cross-sectionl re in the shpe of trpezium, s shown in the digrm. Wter is pumped in t m 3 per minute. Using the dimensions shown in the digrm, find how long it tkes to fill the pool? 1 m 10 m 5 m 3 m 18 A running trck consists of two 60 m strights nd two semicirculr bends of dimeter 60 m. 60 m 60 m Find the length of one lp of this running trck. b The owners of the trck wish to stge thletics meetings nd need it to be exctly 400 m long. This cn be done by just ltering the strights or just widening the bends. Clculte wht djustments would need to be mde. 19 Discs of dimeter cm re cut from metl strip tht is cm by 100 cm. 100 cm cm Wht is the minimum mount of wste mteril? 0 A cylindricl oil tnk hs rdius 60 cm nd length of 180 cm. It is mde from reinforced steel nd is full of oil. The oil hs density of 4.3 g/cm 3. The reinforced steel hs mss of.8 g/cm. Find the totl mss of the tnk nd the oil in kg. 60 cm 180 cm A 1 The solid shpe, shown in the digrm, is mde by cutting hole ll the wy through wooden cube. The cube hs edges of length 7 cm. The hole hs squre cross-section of side cm. Work out the volume of wood in the solid shpe. cm The mss of the solid shpe is 189 grms. 7 cm cm b Work out the density of the wood. 7 cm 7 cm Mrch 009, dpted 17