10 AREA AND VOLUME 1. Before you start. Objectives


 Avice Rice
 2 years ago
 Views:
Transcription
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 prllelogrmlike 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 squrebsed pyrmid d tringulrbsed pyrmid Key Points Isometric pper will help you to mke scle drwings of threedimensionl 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 threedimensionl 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 crosssection length crosssection length 16 prism
19 10.7 Volume of prism Exmple 14 The re of the crosssection 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 crosssection length. Here, the re of crosssection 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 crosssection 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 crosssection Volume of prism cm 3 Use volume of prism re of crosssection length. Here the re of crosssection 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 crosssection 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 crosssection 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 crosssection of the cylinder is circle with rdius. Remember: re of circle rdius. Tke s cm Are of crosssection 6 36 Volume of cylinder cm 3 Use volume of cylinder re of crosssection 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 crosssection 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 crosssection length. front elevtion pln side elevtion cross section length 166
23 Chpter review Volume of cylinder re of crosssection 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 ringshped 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 crosssection 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 crosssectionl 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 crosssection 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
4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A
Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter
More informationLet us recall some facts you have learnt in previous grades under the topic Area.
6 Are By studying this lesson you will be ble to find the res of sectors of circles, solve problems relted to the res of compound plne figures contining sectors of circles. Ares of plne figures Let us
More informationSurface Area and Volume
Surfce Are nd Volume Student Book  Series J Mthletics Instnt Workooks Copyright Surfce re nd volume Student Book  Series J Contents Topics Topic  Surfce re of right prism Topic 2  Surfce re of right
More informationPythagoras theorem and trigonometry (2)
HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in rightngled tringles. These
More informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More informationGeometry and Measure. 12am 1am 2am 3am 4am 5am 6am 7am 8am 9am 10am 11am 12pm
Reding Scles There re two things to do when reding scle. 1. Mke sure you know wht ech division on the scle represents. 2. Mke sure you red in the right direction. Mesure Length metres (m), kilometres (km),
More informationLines and angles. Name. Use a ruler and pencil to draw: a 2 parallel lines. c 2 perpendicular lines. b 2 intersecting lines. Complete the following:
Lines nd s 1 Use ruler nd pencil to drw: 2 prllel lines 2 intersecting lines c 2 perpendiculr lines 2 Complete the following: drw in the digonls on this shpe mrk the interior s on this shpe c mrk equl
More informationChapter 2 Decimals. (A reminder) In the whole number chapter, we looked at ones, tens, hundreds, thousands and larger numbers. = 1
Chpter 2 Decimls Wht is Deciml? (A reminder) In the whole numer chpter, we looked t ones, tens, hundreds, thousnds nd lrger numers. When single unit is divided into 10 (or 100) its, we hve deciml frctions
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More information10.6 Applications of Quadratic Equations
10.6 Applictions of Qudrtic Equtions In this section we wnt to look t the pplictions tht qudrtic equtions nd functions hve in the rel world. There re severl stndrd types: problems where the formul is given,
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationLesson 12.1 Trigonometric Ratios
Lesson 12.1 rigonometric Rtios Nme eriod Dte In Eercises 1 6, give ech nswer s frction in terms of p, q, nd r. 1. sin 2. cos 3. tn 4. sin Q 5. cos Q 6. tn Q p In Eercises 7 12, give ech nswer s deciml
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationPROBLEMS 13  APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS  APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationQuadrilaterals Here are some examples using quadrilaterals
Qudrilterls Here re some exmples using qudrilterls Exmple 30: igonls of rhomus rhomus hs sides length nd one digonl length, wht is the length of the other digonl? 4  Exmple 31: igonls of prllelogrm Given
More informationAnswer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This printout should hve 17 questions, check tht it is complete. Multiplechoice questions my continue on the net column or pge: find ll choices before mking your
More informationPlane figure geometry
2 lne figure geometry ontents: E F G H I Turning Mesuring ngles lssifying nd nming ngles omplementry nd supplementry ngles ngles in revolution isecting ngles onstructing 9 ngles to line lne shpes oints
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More informationThe Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,
More informationState the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127
ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More information11.1 Kick off with CAS Clculting res with CAS We cn use CAS to define formuls which llow us to quickly nd efficiently clculte the res of different shp
FS O PA G E PR O 11 O R R EC TE D Geometry: similrity nd mensurtion 11.1 Kick off with CAS U N C 11.2 Properties of ngles, tringles nd polygons 11.3 Are nd perimeter I 11.4 Are nd perimeter II 11.5 Gret
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationLesson 8.1 Areas of Rectangles and Parallelograms
Leon 8.1 Are of Rectngle nd Prllelogrm In Eercie 1 4, find the re of the hded region. 1.. 1 cm 1 cm. 17 cm 4. 9 cm 5 cm 1.5 cm 1 cm cm cm 5. Rectngle ABCD h re 684 m nd width 44 m. Find it length. 6. Drw
More informationTriangles, Altitudes, and Area Instructor: Natalya St. Clair
Tringle, nd ltitudes erkeley Mth ircles 015 Lecture Notes Tringles, ltitudes, nd re Instructor: Ntly St. lir *Note: This M session is inspired from vriety of sources, including wesomemth, reteem Mth Zoom,
More informationVolumes of solids of revolution
Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the xxis. There is strightforwrd technique which enbles this to be done, using
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More informationEnd of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.
End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.
More information9.1 PYTHAGOREAN THEOREM (right triangles)
Simplifying Rdicls: ) 1 b) 60 c) 11 d) 3 e) 7 Solve: ) x 4 9 b) 16 80 c) 9 16 9.1 PYTHAGOREAN THEOREM (right tringles) c If tringle is right tringle then b, b re the legs * c is clled the hypotenuse (side
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationExercises in KS3 Mathematics Levels 78. R Joinson
Exercises in KS Mthemtics Levels 78 R Joinson Sumbooks Northwy Chester CH 8BB Exercises in KS Mthemtics  Levels 7 nd 8 First Published 00 Copyright R Joinson nd Sumbooks This pckge of worksheets is sold
More information11. PYTHAGORAS THEOREM
11. PYTHAGORAS THEOREM 111 Along the Nile 2 112 Proofs of Pythgors theorem 3 113 Finding sides nd ngles 5 114 Semiirles 7 115 Surds 8 116 Chlking hndll ourt 9 117 Pythgors prolems 10 118 Designing
More informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More information1+(dy/dx) 2 dx. We get dy dx = 3x1/2 = 3 x, = 9x. Hence 1 +
Mth.9 Em Solutions NAME: #.) / #.) / #.) /5 #.) / #5.) / #6.) /5 #7.) / Totl: / Instructions: There re 5 pges nd totl of points on the em. You must show ll necessr work to get credit. You m not use our
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationQUANTITATIVE REASONING
Guide For Exminees InterUniversity Psychometric Entrnce Test QUNTITTIVE RESONING The Quntittive Resoning domin tests your bility to use numbers nd mthemticl concepts to solve mthemticl problems, s well
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationCOMPONENTS: COMBINED LOADING
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationRadius of the Earth  Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth  dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More informationIntroduction to Mathematical Reasoning, Saylor 111
Frction versus rtionl number. Wht s the difference? It s not n esy question. In fct, the difference is somewht like the difference between set of words on one hnd nd sentence on the other. A symbol is
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More informationChapter G  Problems
Chpter G  Problems Blinn College  Physics 2426  Terry Honn Problem G.1 A plne flies horizonlly t speed of 280 mês in position where the erth's mgnetic field hs mgnitude 6.0µ105 T nd is directed t n
More informationIntroduction. Teacher s lesson notes The notes and examples are useful for new teachers and can form the basis of lesson plans.
Introduction Introduction The Key Stge 3 Mthemtics series covers the new Ntionl Curriculum for Mthemtics (SCAA: The Ntionl Curriculum Orders, DFE, Jnury 1995, 0 11 270894 3). Detiled curriculum references
More informationMechanics Cycle 1 Chapter 5. Chapter 5
Chpter 5 Contct orces: ree Body Digrms nd Idel Ropes Pushes nd Pulls in 1D, nd Newton s Second Lw Neglecting riction ree Body Digrms Tension Along Idel Ropes (i.e., Mssless Ropes) Newton s Third Lw Bodies
More informationLesson 4.1 Triangle Sum Conjecture
Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b 31 82 p 98 q 28 53 y 17 79 23 50 b 4. r, s, 5., y 6. y t t s r 100 85 100 y 30 4 7 y 31 7. s 8.
More informationPlotting and Graphing
Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured
More informationUsing Definite Integrals
Chpter 6 Using Definite Integrls 6. Using Definite Integrls to Find Are nd Length Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: How
More informationFor a solid S for which the cross sections vary, we can approximate the volume using a Riemann sum. A(x i ) x. i=1.
Volumes by Disks nd Wshers Volume of cylinder A cylinder is solid where ll cross sections re the sme. The volume of cylinder is A h where A is the re of cross section nd h is the height of the cylinder.
More informationWorksheet 24: Optimization
Worksheet 4: Optimiztion Russell Buehler b.r@berkeley.edu 1. Let P 100I I +I+4. For wht vlues of I is P mximum? P 100I I + I + 4 Tking the derivtive, www.xkcd.com P (I + I + 4)(100) 100I(I + 1) (I + I
More informationApplications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
More informationSequences and Series
Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic
More informationPicture Match Words Fusion Density Isotope Neutron Atomic Number Structure Components Function Atomic Mass Orbit
Picture Mtch Words Fusion Density Isotope Neutron Atomic Number Structure Components Function Atomic Mss Orbit Mterils copyrighted by the University of Louisville. Eductors re free to use these mterils
More informationKnow the sum of angles at a point, on a straight line and in a triangle
2.1 ngle sums Know the sum of ngles t point, on stright line n in tringle Key wors ngle egree ngle sum n ngle is mesure of turn. ngles re usully mesure in egrees, or for short. ngles tht meet t point mke
More informationWell say we were dealing with a weak acid K a = 1x10, and had a formal concentration of.1m. What is the % dissociation of the acid?
Chpter 9 Buffers Problems 2, 5, 7, 8, 9, 12, 15, 17,19 A Buffer is solution tht resists chnges in ph when cids or bses re dded or when the solution is diluted. Buffers re importnt in Biochemistry becuse
More informationPROBLEM 4.1 SOLUTION. Knowing that the couple shown acts in a vertical plane, determine the stress at (a) point A, (b) point B.
PROBLEM.1 Knowing tht the couple shown cts in verticl plne, determine the stress t () point A, (b) point B. SOLUTON () (b) For rectngle: For cross sectionl re: 1 = bh 1 1 = 1 + + = ()(1.5) + ()(5.5) +
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationCHAPTER 4: POLYGONS AND SOLIDS. 3 Which of the following are regular polygons? 4 Draw a pentagon with equal sides but with unequal angles.
Mthemtis for Austrli Yer 6  Homework POLYGONS AND SOLIDS (Chpter 4) CHAPTER 4: POLYGONS AND SOLIDS 4A POLYGONS 3 Whih of the following re regulr polygons? A polygon is lose figure whih hs only stright
More informationAnswer, Key Homework 4 David McIntyre Mar 25,
Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his printout should hve 18 questions. Multiplechoice questions my continue on the next column or pe find ll choices before mkin your selection.
More informationr 2 F ds W = r 1 qe ds = q
Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study
More informationSymmetry in crystals National Workshop on Crystal Structure Determination Using Powder XRD
Symmetry in crystls Ntionl Workshop on Crystl Structure Determintion Using Powder XRD Muhmmd Sbieh Anwr School of Science nd Engineering Lhore University of Mngement & Sciences (LUMS) Pkistn. (Dted: August
More informationLesson 10. Parametric Curves
Return to List of Lessons Lesson 10. Prmetric Curves (A) Prmetric Curves If curve fils the Verticl Line Test, it cn t be expressed by function. In this cse you will encounter problem if you try to find
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationSection 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables
The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long
More informationAngles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example
2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel
More informationm, where m = m 1 + m m n.
Lecture 7 : Moments nd Centers of Mss If we hve msses m, m 2,..., m n t points x, x 2,..., x n long the xxis, the moment of the system round the origin is M 0 = m x + m 2 x 2 + + m n x n. The center of
More informationDiffraction and Interference of Light
rev 12/2016 Diffrction nd Interference of Light Equipment Qty Items Prt Number 1 Light Sensor CI6504 1 Rotry Motion Sensor CI6538 1 Single Slit Set OS8523 1 Multiple Slit Set OS8523 1 Liner Trnsltor
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationNQF Level: 2 US No: 7480
NQF Level: 2 US No: 7480 Assessment Guide Primry Agriculture Rtionl nd irrtionl numers nd numer systems Assessor:.......................................... Workplce / Compny:.................................
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationTheory of Forces. Forces and Motion
his eek extbook  Red Chpter 4, 5 Competent roblem Solver  Chpter 4 relb Computer Quiz ht s on the next Quiz? Check out smple quiz on web by hurs. ht you missed on first quiz Kinemtics  Everything
More informationSolving BAMO Problems
Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only
More informationEinstein. Mechanics. In Grade 10 we investigated kinematics, or movement described in terms of velocity, acceleration, displacement, and so on.
Cmbridge University Press 9780521683593  Study nd Mster Physicl Sciences Grde 11 Lerner s Book Krin Kelder More informtion MODULE 1 Einstein Mechnics motion force Glileo Newton decelerte moment of
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationPicture Match Words. Strobe pictures. Stopping distance. Following. Safety
Tuesdy: Picture Mtch + Spelling Pyrmid Homework [the hndout for it is two pges down] Mterils: 1 bord + 1 set of words per 2 students (totl: 12 of ech) Routine: () once the Pictionry is completed; pirs
More informationThree squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.
1 The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor SelfChek Quiz Three squres with
More informationaddition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.
APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The
More informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More informationN Mean SD Mean SD Shelf # Shelf # Shelf #
NOV xercises smple of 0 different types of cerels ws tken from ech of three grocery store shelves (1,, nd, counting from the floor). summry of the sugr content (grms per serving) nd dietry fiber (grms
More informationAngles. Angles. Curriculum Ready.
ngles ngles urriculum Redy www.mthletics.com ngles mesure the mount of turn in degrees etween two lines tht meet t point. Mny gmes re sed on interpreting using ngles such s pool, snooker illirds. lck
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationChapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
More informationChapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
More information2 square units. The trapezoid. 15. There are many possible solutions. Sample answer: 16 cm. 16 cm
LESSON 8.1 EXERCISES CHAPTER 8 1. 8 m. Use te Rectngle Are Conjecture. A b (19)(1) 8 m.. 41.85 cm. A b (9.3)(4.5) 41.85 cm. 3. 8 yd. A b, so 96 yd b 1. 9 6 1 8, so b 8 yd. 4. 1 cm. A b, so 73 13. 73 13
More information