Surface area and volume

Transcription

1 Topic 6 Surfce re nd volume 6.1 Overview Why lern this? Humns must mesure! How much pint or crpet will you need to redecorte your edroom? How mny litres of wter will it tke to fill the new pool? How fr is it to the end of the universe? These re just few exmples of where mesurements skills re needed. Mesuring tools hve dvnced significntly in their cpcity to mesure extremely smll nd extremely lrge mounts, leding to mny rekthroughs in medicine, engineering, science, rchitecture nd stronomy. Wht do you know? 1 Think List wht you know out mesurement. Use thinking tool such s concept mp to show your list. pir Shre wht you know with prtner nd then with smll group. 3 shre As clss, crete thinking tool such s lrge concept mp wheel tht shows your clss s knowledge of mesurement. Lerning sequence 6.1 Overview 6. Are 6.3 Totl surfce re 6.4 Volume 6.5 Review ONLINE ONLY

2 Wtch this video The story of mthemtics: Austrlin megfun SEARCHLIGHT ID: eles-1845

3 6. Are The re of figure is the mount of surfce covered y the figure. The units used for re re mm, cm, m, km or h (hectres), depending upon the size of the figure. l h = (or 10 4 ) m There re mny rel-life situtions tht require n understnding of the re concept. Some re, the re to e pinted, the floor re of room or house, how much lnd one hs nd how mny tiles re needed for wll. It is importnt tht you re fmilir with converting units of re. Using re formuls The re of mny plne figures cn e found y using formul. The tle elow shows the formul for the re of some common shpes. Shpe 1. Squre Formul A = l l. Rectngle l A = lw w 3. Tringle A = 1 h h 4. Trpezium A = 1 ( + )h h 5. Circle r A = πr 6. Prllelogrm h A = h 30 Mths Quest A

4 yx mesurement AND geometry Shpe Formul 7. Sector A = θ 360 πr θ r 8. Kite (including rhomus) A = 1 xy, where x nd y re digonls. 9. Ellipse A = π, where nd re the lengths of the semi-mjor nd semi-minor xes respectively. Note: A clcultor uses stored vlue for π of pproximtely Before clcultors were in common usge, π ws often tken to e pproximtely or You 7 re dvised to use the π utton on your clcultor rther thn or Heron s formul If the lengths of ll three sides of tringle re known, its re, A, cn e found y using Heron s formul: A =!s(s ) (s ) (s c) where, nd c re the lengths of the three sides nd s is the semi-perimeter where s = + + c c. eles-0177 WORKED EXAMPLE 1 TI csio Find the res of the following plne figures, correct to deciml plces. c 3 cm 5 cm 6 cm cm 5 cm cm THINK 1 Three side lengths re known, ut not the height. In this cse pply Heron s formul. WRITE A =!s(s )(s )(s c) Identify the vlues of, nd c. = 3, = 5, c = 6 3 Clculte the vlue of s, the semi-perimeter of the tringle. s = + + c = = 14 = 7 Topic 6 Surfce re nd volume 31

5 4 Sustitute the vlues of,, c nd s into Heron s formul nd evlute, correct to deciml plces. A =!7(7 3)(7 5)(7 6) =!7 4 1 =!56 = 7.48 cm 1 The shpe shown is n ellipse. Write the pproprite re formul. Identify the vlues of nd (the semi-mjor nd semi-minor xes). 3 Sustitute the vlues of nd into the formul nd evlute, correct to deciml plces. c 1 The shpe shown is sector. Write the formul for finding the re of sector. A = π = 5, = A = π 5 = 31.4 cm c A = θ 360 πr Write the vlue of θ nd r. θ = 40, r = 15 3 Sustitute nd evlute the expression, correct to deciml plces. A = π 15 = cm Ares of composite figures A composite figure is figure mde up of comintion of simple figures. The re of composite figure cn e clculted y: clculting the sum of the res of the simple figures tht mke up the composite figure clculting the re of lrger shpe nd then sutrcting the extr re involved. WORKED EXAMPLE Find the re of ech of the following composite shpes. C AB = 8 cm EC = 6 cm FD = cm A B 9 cm A THINK E F D B 1 ACBD is qudrilterl tht cn e split into two tringles: ΔABC nd ΔABD. D C cm E 5 cm H 10 cm G WRITE Are ACBD = Are ΔABC + Are ΔABD 3 Mths Quest A

6 Write the formul for the re of tringle contining se nd height. A tringle = 1 h 3 Identify the vlues of nd h for ΔABC. ΔABC: = AB = 8, h = EC = 6 4 Sustitute the vlues of the pronumerls into the formul nd, hence, clculte the re of ΔABC. 5 Identify the vlues of nd h for ΔABD. Are of ΔABC = 1 AB EC = = 4 cm ΔABD: = AB = 8, h = FD = 6 Clculte the re of ΔABD. Are of ΔABD = 1 AB FD = 1 8 = 8 cm 7 Add the res of the two tringles together to find the re of the qudrilterl ACBD. 1 One wy to find the re of the shpe shown is to find the totl re of the rectngle ABGH nd then sutrct the re of the smller rectngle DEFC. Write the formul for the re of rectngle. 3 Identify the vlues of the pronumerls for the rectngle ABGH. 4 Sustitute the vlues of the pronumerls into the formul to find the re of the rectngle ABGH. 5 Identify the vlues of the pronumerls for the rectngle DEFC. 6 Sustitute the vlues of the pronumerls into the formul to find the re of the rectngle DEFC. 7 Sutrct the re of the rectngle DEFC from the re of the rectngle ABGH to find the re of the given shpe. Are of ACBD = 4 cm + 8 cm = 3 cm Are = Are ABGH Are DEFC A rectngle = l w Rectngle ABGH: l = = 0 w = 10 Are of ABGH = 0 10 = 00 cm Rectngle DEFC: l = 5, w = Are of DEFC = 5 = 10 cm Are = = 190 cm Topic 6 Surfce re nd volume 33

7 Exercise 6. Are INDIVIDUAL PATHWAYS REFLECTION How re perimeter nd re different ut fundmentlly relted? Prctise Questions: 1, 3 5, 8, 9, 11, 1, 14 consolidte Questions: 1 6, 8 10, 1, 14, 16, 18 Individul pthwy interctivity int-4593 mster Questions: 1 9, 1 19 Where pproprite, give nswers correct to deciml plces. FLUENCY 1 Find the res of the following shpes. c doc cm 1 cm 4 cm 15 cm doc cm d 1 cm e f 8 cm 15 cm 8 mm 13 mm 18 cm 7 mm g 18 cm h 6 m 7 m i 15 cm 10 cm Express the re in questions 1e nd 1g in terms of π. 3 WE1 Use Heron s formul to find the re of the following tringles correct to deciml plces. 5 cm 16 cm 8 cm 3 cm 6 cm 1 cm 34 Mths Quest A

8 4 WE1 Find the re of the following ellipses. Answer correct to 1 deciml plce. 9 mm 1 mm 4 mm 5 mm 5 WE1c Find the re of the following shpes, i stting the nswer exctly; tht is, in terms of π nd ii correct to deciml plces mm c cm 1 cm MC A figure hs n re of out 64 cm. Which of the following cnnot possily represent the figure? A A tringle with se length 16 cm nd height 8 cm B A circle with rdius 4.51 cm C A rectngle with dimensions 16 cm nd 4 cm D A squre with side length 8 cm E A rhomus with digonls 16 cm nd 4 cm 7 MC The re of the qudrilterl shown elow right is to e clculted. Which of the following lists ll the lengths required to clculte the re? A AB, BC, CD nd AD B AB, BE, AC nd CD C BC, BE, AD nd CD D AC, BE nd FD E AC, CD nd AB B E F C A D 8 WE Find the re of the following composite shpes. 0 cm 40 m 15 cm 8 m Topic 6 Surfce re nd volume 35

9 c 8 cm 3 cm cm d 4 cm 3.8 m.1 m e f 18 cm 8 cm 5 cm 1 cm 9 Find the shded re in ech of the following. 16 m 8 m m m 3 cm 7 cm c d 8 m 3 m 40 5 m e 8 m f 15 m m 5 m 7.5 m 13 m 7 m 5 m 36 Mths Quest A

10 UNDERSTANDING 10 A sheet of crdord is 1.6 m y 0.8 m. The following shpes re cut from the crdord: circulr piece with rdius 1 cm rectngulr piece 0 cm y 15 cm tringulr pieces with se 30 cm nd height 10 cm tringulr piece with side length 1 cm, 10 cm nd 8 cm. Wht is the re of the remining piece of crdord? 11 A rectngulr lock of lnd, 1 m y 8 m, is surrounded y concrete pth 0.5 m wide. Find the re of the pth. 1 Concrete sls 1 m y 0.5 m re used to cover footpth 0 m y 1.5 m. How mny sls re needed? 13 A city council uilds 0.5 m wide concrete pth round the grden s shown elow. 5 m 1 m 8 m 3 m Find the cost of the jo if the workmn chrges $40.00 per m. 14 A tennis court used for doules is m wide, ut singles court is only 8.3 m wide, s shown in the digrm. 8.3 m m 6.40 m m Wht is the re of the doules tennis court? Wht is the re of the singles court? c Wht percentge of the doules court is used for singles? Give your nswer to the nerest whole numer. 15 Ron the excvtor opertor hs 100 metres of rricde mesh nd needs to enclose n re to work in sfely. He chooses to mke rectngulr region with dimensions x nd y. Write n eqution tht connects x, y nd the perimeter. Write y in terms of x. c Write n eqution for the re of the region in terms of x. d Fill in the tle for different vlues of x. x Are (m ) e Cn x hve vlue more thn 50? Why? f Sketch grph of re ginst x. g Determine the vlue of x tht mkes the re mximum. Topic 6 Surfce re nd volume 37

11 h Wht is the vlue of y for mximum re? i Wht shpe encloses the mximum re? j Clculte the mximum re. Ron decides to choose to mke circulr re with the rricde mesh. k Wht is the rdius of this circulr region? l Wht re is enclosed in this circulr region? m How much extr re does Ron now hve compred to his rectngulr region? REASONING 16 Dn hs purchsed country property with lyout nd dimensions s shown in the digrm. Show tht the property hs totl re of h. N Dn wnts to split the property in hlf (in terms of re) y uilding stright-lined fence running either north south or est west through the property. Assuming the cost of the 5000 m fencing is fixed mount per liner metre, justify where the 000 m fence should e uilt (tht is, how mny metres from the top left-hnd corner nd in which direction), to minimise 1000 m the cost. 17 In question 15, Ron the excvtor opertor could choose to enclose rectngulr or circulr re with 100 m of rricde mesh. In this cse, the circulr region resulted in lrger sfe work re. Show tht for 150 m of rricde mesh, circulr region gin results in lrger sfe work re s opposed to rectngulr region. Show tht for n metres of rricde mesh, circulr region will result in lrger sfe work re s opposed to rectngulr region. PROBLEM SOLVING 18 ABC is sclene tringle with se length of 80 cm nd perpendiculr height of 40 cm. A right-ngled tringle, AED, is nestled within ABC such tht DE is 10 cm to the left of the perpendiculr height, s shown. Find the lengths of the sides lelled x nd y if the shorter side of the two is 0 cm less thn the longer side nd the res of the two shded regions re the sme. D C 1500 m 40 cm y A x 80 cm E 10 cm B 38 Mths Quest A

12 19 Proving the segment formul Prove the formul for the re of segment using the fct tht re of the segment = re of sector ABC re of tringle ACD. Using trigonometry, show tht AD = sin θ r. Show tht CD = cos θ r. r sin θ cos θ c Show tht the re of tringle ACD is. Note tht this formul is the sme if θ is in degrees or rdins. d Finlly, show tht the re of the segment (in purple) is r π θ 360 sin θ cos θ if θ is in degrees. A r θ C θ D r B CHALLENGE 6.1 doc Totl surfce re The totl surfce re (TSA) of solid is the sum of the res of ll the fces of tht solid. TSA of rectngulr prisms nd cues The formul for finding the TSA of rectngulr prism (cuoid) is: TSA = (lh + lw + wh) A specil cse of the rectngulr prism is the cue, where ll sides re equl (l = w = h). TSA = 6l l w h TSA of spheres nd cylinders Sphere: TSA = 4πr r l Note: The mthemtics required to prove the formul for the totl surfce re of sphere is eyond the scope of Yer 10. int-78 Topic 6 Surfce re nd volume 39

13 Cylinder: TSA = πr(r + h) or πr + πrh r h The formul for the TSA of cylinder is found from the re of the net s shown. TSA = πr + πr + πrh = πr + πrh = πr(r + h) r A = πr πr A = πrh h r A = πr WORKED EXAMPLE 3 Find the totl surfce re of the solids, correct to the nerest cm. r = 7 cm 50 cm r 1.5 m THINK WRITE 1 Write the formul for the TSA of sphere. TSA = 4πr Identify the vlue for r. r = 7 3 Sustitute nd evlute. TSA = 4 π cm 616 cm 1 Write the formul for the TSA of cylinder. Identify the vlues for r nd h. Note tht the units will need to e the sme. TSA = πr(r + h) r = 50 cm, h = 1.5 m = 150 cm 3 Sustitute nd evlute. TSA = π 50 ( ) cm 6 83 cm TSA of cones The totl surfce re of cone cn e found y considering its net, which is comprised of smll circle nd sector of lrger circle. l r = rdius of the cone l = slnt height of the cone l r r 40 Mths Quest A

14 The sector is frction of the full circle of rdius l with circumference πl. The sector hs n rc length equivlent to the circumference of the se of the cone, πr. The frction of the full circle represented y the sector cn e found y writing the rc length s frction of the circumference of the full circle, πr πl = r l. Are of sector = frction of the circle πl = r l πl = πrl SA = A circulr se + A curved surfce Therefore, = πr + πrl = πr(r + l) Cone: TSA = πr(r + l) or πl + πrl WORKED EXAMPLE 4 csio Find the totl surfce re of the cone shown. TI THINK WRITE 1 Write the formul for the TSA TSA = πr(r + l) of cone. Stte the vlues of r nd l. r = 1, l = 15 3 Sustitute nd evlute. TSA = π 1 (1 + 15) = cm 1 cm 15 cm TSA of other solids TSA cn e found y summing the res of ech fce. The res of ech fce my need to e clculted seprtely. Check the totl numer of fces to ensure tht none re left out. WORKED EXAMPLE 5 Find the totl surfce re of the squre-sed pyrmid shown. 5 cm THINK 1 There re five fces: The squre se nd four identicl tringles. Find the re of the squre se. 3 Drw nd lel one tringulr fce nd write the formul for finding its re. WRITE/DRAW TSA = Are of squre se + re of four tringulr fces Are of se = l, where l = 6 Are of se = 6 = 36 cm h 5 cm 6 cm 3 cm Are of tringulr fce = 1 h; = 6 Topic 6 Surfce re nd volume 41

15 4 Find the height of the tringle, h, using Pythgors theorem. 5 Clculte the re of the tringulr fce y sustituting = 6 nd h = 4. 6 Clculte the TSA y dding the re of the squre se nd the re of four identicl tringulr fces together. = c, where = h, = 3, c = 5 h = 5 3 h = 5 9 h = 16 h = 4 cm Are of tringulr fce = = 1 cm TSA = = = 84 cm TSA of composite solids Composite solids re formed when two or more simple solids re joined together. The TSA of composite solid is clculted y summing the res of the solid s externl fces. WORKED EXAMPLE 6 Find the totl surfce re of the solid shown correct to 1 deciml plce. 6 cm THINK 1 The solid shown hs 9 fces five identicl squres nd four identicl tringles. Find the re of one squre fce with the side length 10 cm. 3 Drw tringulr fce nd work out its height using Pythgors theorem. WRITE/DRAW TSA = 5 re of squre + 4 re of tringle A squre = l, where l = 10 A = 10 A = 100 cm 6 cm h 10 cm 5 cm = c, where = h, = 5, c = 6 h = 6 5 h = 36 5 h = 11 h = cm (or with rounding, h = 3.3) 4 Mths Quest A

16 4 Find the re of one tringulr fce. A tringle = 1 h, where = 10, h = = = cm (or, with rounding, A tringle = 16.6 cm ) 5 Find the TSA of the solid y dding the re of 5 squres nd 4 tringles together. TSA = = cm (or = 566 using the previously rounded vlue) Note: Rounding is not done until the finl step. If h hd een rounded to 3.3 in step 3 nd this vlue used in steps 4 nd 5, the deciml plce vlue of the TSA would hve een lost. It is importnt to relise tht rounding too erly cn ffect the ccurcy of results. WORKED EXAMPLE 7 The silo shown t right is to e uilt from metl. The top portion of the silo is cylinder of dimeter 4 m nd height 8 m. The ottom prt of the silo is cone of slnt height 3 m. The silo hs circulr opening of rdius 30 cm on the top. Wht re of metl (to the nerest m ) is required to uild the silo? If it costs $1.50 per m to cover the surfce with n nti-rust mteril, 4 m 8 m 3 m how much will it cost to cover the silo completely? THINK 1 The surfce re of the silo consists of n nnulus, the curved prt of the cylinder nd the curved section of the cone. To find the re of the nnulus, sutrct the re of the smll circle from the re of the lrger circle. Let R = rdius of smll circle. 3 The middle prt of the silo is the curved prt of cylinder. Find its re. (Note tht in the formul TSA cylinder = πr + πrh, the curved prt is represented y πrh.) WRITE TSA = re of nnulus + re of curved section of cylinder + re of curved section of cone Are of nnulus = A lrge circle A smll circle = πr πr where r = 4 = m nd R = 30 cm = 0.3 m. Are of nnulus = π π 0.3 = 1.8 m Are of curved section of cylinder = πrh where r =, h = 8. Are of curved section of cylinder = π 8 = m Topic 6 Surfce re nd volume 43

17 4 The ottom prt of the silo is the curved section of cone. Find its re. (Note tht in the formul TSA cone = πr + πrl, the curved prt is given y πrl.) 5 Find the totl surfce re of the silo y finding the sum of the surfce res clculted ove. Are of curved section of cone = πrl where r =, l = 3. Are of curved section of cone = π 3 = m TSA = = m 6 Write the nswer in words. The re of metl required is 13 m, correct to the nerest squre metre. To find the totl cost, multiply the totl surfce re of the silo y the cost of the nti-rust mteril per m ($1.50). Cost = 13 $1.50 = $ Exercise 6.3 Totl surfce re INDIVIDUAL PATHWAYS REFLECTION Why is clculting the totl surfce re of composite solid more difficult thn for simple solid such s rectngulr prism or cylinder? Prctise Questions: 1 4, 6 e, 7, 10, 1 consolidte Questions: 1 4, 6, 7, 9 1, 15, 18 Individul pthwy interctivity int-4594 mster Questions: 1 8, FLUENCY Note: Where pproprite, give the nswers correct to 1 deciml plce. 1 Find the totl surfce res of the solids shown. c 1 cm d m 15 cm 1.5 m doc cm 8 cm 0 cm 3 m WE3 Find the totl surfce re of the solids shown elow. r = 3 m 1 cm c 0.5 m d 1 cm r 30 cm.1 m 44 Mths Quest A

18 3 WE4 Find the totl surfce re of the cones elow. 8 cm 14 cm 0 cm 1 cm 4 WE5 Find the totl surfce re of the solids elow. 1 cm.5 m 15 cm 1.5 m c 9.1 cm d 5.1 cm 8 cm 14 cm 7. cm 6 cm 7 cm 10 cm 5 Find the surfce res of the following. A cue of side length 1.5 m A rectngulr prism 6 m 4 m.1 m c A cylinder of rdius 30 cm nd height 45 cm, open t one end d A sphere of rdius 8 mm e An open cone of rdius 4 cm nd slnt height 10 cm f A squre pyrmid of se length 0 cm nd slnt edge 30 cm 6 WE6 Find the totl surfce re of the ojects shown. 10 cm 8 cm 5 cm 1 cm 5 cm 0 cm 0 cm 1 cm 35 cm c d 5 cm cm 3 cm.5 cm 3 cm Topic 6 Surfce re nd volume 45

19 e f 5 cm 10 cm 3.5 cm 0 cm 1 cm 15 cm 7 MC A cue hs totl surfce re of 384 cm. The length of the edge of the cue is: A 9 cm B 8 cm C 7 cm D 6 cm E 5 cm UNDERSTANDING 8 Open cones re mde from nets cut from lrge sheet of pper 1. m 1.0 m. If cone hs rdius of 6 cm nd slnt height of 10 cm, how mny cones cn e mde from the sheet? (Assume there is 5% wstge of pper.) 9 A steel girder is to e pinted. Clculte the re of the surfce to e pinted. cm 5 cm 0 cm 10 cm cm cm 1 cm 10 WE7 The greenhouse shown elow is to e uilt using shde cloth. It hs wooden door of dimensions 1. m 0.5 m. Find the totl re of shde cloth needed to complete the greenhouse. Find the cost of the shde cloth t $6.50 per m..5 m 5 m 3 m 11 A cylinder is joined to hemisphere to mke cke holder, s shown elow. The surfce of the cke holder is to e chromed t 5.5 cents per cm. Find the totl surfce re to e chromed. Find the cost of chroming the cke holder. 15 cm 10 cm 46 Mths Quest A

20 1 A soccer ll is mde up of numer of hexgons sewn together cm on its surfce. Ech hexgon cn e considered to hve dimensions s shown in the digrm. y x Clculte θ. Clculte the vlues of x nd y exctly. c Clculte the re of the trpezium in the digrm. θ d Hence, determine the re of the hexgon. e If the totl surfce re of the soccer ll is 19!3 cm, how mny hexgons re on the surfce of the soccer ll? 13 Determine the exct totl surfce re of sphere with rdius! metres. An inverted cone with side length 4 metres is plced on top of the sphere so tht the centre of its se is 0.5 metres ove the centre of the sphere. Find the rdius of the cone exctly. c Find the re of the curved surfce of the cone exctly. d Wht re the exct dimensions of ox tht could precisely fit the cone connected to the sphere? REASONING Complete the following question without the id of clcultor. 14 The tle shown elow is to e vrnished (including the se of ech leg). The tle top hs thickness of 180 mm nd the cross-sectionl dimension of the legs is 50 mm y 50 mm. 60 cm 80 cm 70 cm A friend completes the clcultion s shown. Assume there re no simple clculting errors. Anlyse the working presented nd justify if the TSA clculted is correct. Tle top (inc. leg ses) 0.96 ( ) Legs ( ) Tle top edging (( )) TSA 1.88 m 15 A shower recess with dimensions 1500 mm (ck wll) y 900 mm (side wll) needs to hve the ck nd two side wlls tiled to height of m. Clculte the re to e tiled in m. Justify tht 180 tiles (including those tht need to e cut) of dimension 0 cm y 0 cm will e required. Disregrd the grout nd ssume tht once tile is cut, only one piece of the tile cn e used. c Evlute the chepest option of tiling; $1.50/tile or $39.50/ox, where ox covers 1 m, or tiles of dimension 30 cm y 30 cm costing $3.50/tile. Topic 6 Surfce re nd volume 47

21 16 If the surfce re of sphere to tht of cylinder is in the rtio 4 : 3 nd the sphere hs rdius of 3, show tht if the rdius of the cylinder is equl to its height, then the rdius of the cylinder is 3!3. PROBLEM SOLVING Frustum of cone 17 A frustum of cone is cone with the top V sliced off (see the drwing on the right). x θ x A B When the curved side is opened up, it t s cretes shpe, ABYX, s shown in the s s πt s digrm. X r Write n expression for the rc length πr XY in terms of the ngle θ. Write nother expression for the rc length AB in terms of the sme ngle θ. Show tht, in π(r t) rdins, θ =. s st i Using the ove formul for θ, show tht x = (r t). ii Use similr tringles to confirm this formul. c Determine the re of sectors AVB nd XVY nd hence determine the re of ABYX. Add the res of the circles to the re of ABYX to determine the TSA of frustum. 18 Tin is re-covering footstool in the shpe of cylinder with dimeter 50 cm nd height 30 cm. She lso intends to cover the se of the cushion. Y doc-54 int-1150 She hs 1 m of fric to mke this footstool. When clculting the re of fric required, llow n extr 0% of the totl surfce re to cter for sems nd pttern plcings. Explin whether Tin hs enough mteril to cover the footstool. 6.4 Volume The volume of 3-dimensionl oject is the mount of spce it tkes up. The volume is mesured in units of mm 3, cm 3 nd m 3. Volume of prism The volume of ny solid with uniform cross-sectionl re is given y the formul: V = AH, where A is the cross-sectionl (or se) re nd H is the height of the solid. Cue Volume = AH = re of squre height l = l l = l 3 48 Mths Quest A

22 Rectngulr prism h Volume = AH = re of rectngle height = lwh l w Cylinder r h Volume = AH = re of circle height = πr h Tringulr prism Volume = AH = re of tringle height = 1 h H h H WORKED EXAMPLE 8 Find the volumes of the following shpes. 14 cm TI csio 5 cm 0 cm 4 cm 10 cm THINK 1 Write the formul for the volume of the cylinder (prism). WRITE V = AH = πr h Identify the vlue of the pronumerls. r = 14, h = 0 3 Sustitute nd evlute. V = π cm 3 1 Write the formul for the volume of tringulr prism. Identify the vlue of the pronumerls. (Note h is the height of the tringle nd H is the depth of the prism.) V = 1 h H = 4, h = 5, H = 10 3 Sustitute nd evlute. V = = 100 cm 3 Topic 6 Surfce re nd volume 49

23 WORKED EXAMPLE 9 Wht effect will douling ech of the side lengths of cue hve on its volume? Wht effect will hlving the rdius nd douling the height of cylinder hve on its volume? THINK 1 Write the formul for the volume of the cue. Identify the vlue of the pronumerl. Note: Douling is the sme s multiplying y. WRITE V = l 3 l new = l 3 Sustitute nd evlute. V new = (l) 3 4 Compre the nswer otined in step 3 with the volume of the originl shpe. = 8l 3 5 Write your nswer. Douling ech side length of cue increses the volume y fctor of 8; tht is, the new volume will e 8 times s lrge s the originl volume. 1 Write the formul for the volume of the cylinder. Identify the vlue of the pronumerls. Note: Hlving is the sme s dividing y. 3 Sustitute nd evlute. 4 Compre the nswer otined in step 3 with the volume of the originl shpe. V = πr h r new = r, h new = h V new = π r h = π r 4 h = πr h = 1 πr h 5 Write your nswer. Hlving the rdius nd douling the height of cylinder decreses the volume y fctor of ; tht is, the new volume will e hlf the originl volume. 50 Mths Quest A

24 Volume of spheres The volume of sphere of rdius, r, cn e clculted using the formul V = 4 3 πr3. r WORKED EXAMPLE 10 Find the volume of sphere of rdius 9 cm. Answer correct to 1 deciml plce. THINK WRITE 1 Write the formul for the volume of sphere. V = 4 3 πr3 Identify the vlue of r. r = 9 3 Sustitute nd evlute. V = 4 π 3 93 = cm 3 Volume of pyrmids Pyrmids re not prisms s the cross-section chnges from the se upwrds. The volume of pyrmid is one-third the volume of n equivlent prism with the sme se re nd height. Volume of pyrmid = 1 3 AH H Bse Are of se = A Since cone is pyrmid with circulr cross- section, the volume of cone is one-third the volume of cylinder with the sme se re nd height. Volume of cone = 1 3 AH = 1 3 πr h h r WORKED EXAMPLE 11 TI csio Find the volume of ech of the following solids. 10 cm 1 cm 8 cm 8 cm Topic 6 Surfce re nd volume 51

25 THINK WRITE 1 Write the formul for the volume of cone. V = 1 3 πr h Identify the vlues of r nd h. r = 8, h = 10 3 Sustitute nd evlute. V = 1 π = cm 3 1 Write the formul for the volume of pyrmid. V = 1 3 AH Find the re of the squre se. A = l where l = 8 A = 8 = 64 cm 3 Identify the vlue of H. H = 1 4 Sustitute nd evlute. V = = 56 cm 3 Volume of composite solids A composite solid is comintion of numer of solids. The volume of ech smller solid component cn e clculted seprtely. The volume of composite solid is clculted y summing the volumes of ech of the smller solid components. WORKED EXAMPLE 1 Clculte the volume of the composite solid shown. 3 m 1.5 m THINK 1 The given solid is composite figure, mde up of cue nd squre-sed pyrmid. WRITE V = Volume of cue + Volume of pyrmid Find the volume of the cue. V cue = l 3 where l = 3 V cue = 3 3 = 7 m 3 3 Write the formul for finding the volume of squre-sed pyrmid. V squre-sed pyrmid = 1 3 AH 5 Mths Quest A

26 4 Find the re of the squre se. A = l = 3 = 9 m 5 Identify the vlue of H. H = Sustitute nd evlute the volume of the pyrmid. 7 Find the totl volume y dding the volume of the cue nd pyrmid. V squre-sed pyrmid = = 4.5 m 3 V = = 31.5 m 3 Cpcity Some 3-dimensionl ojects re hollow nd cn e filled with liquid or some other sustnce. The mount of sustnce which continer cn hold is clled its cpcity. Cpcity is essentilly the sme s volume ut is usully mesured in ml, L nd kl where 1 ml = 1 cm 3 1L = 1000 cm 3 1kL = 1m 3. WORKED EXAMPLE 13 Find the cpcity (in litres) of cuoidl qurium, which is 50 cm long, 30 cm wide nd 40 cm high. THINK 1 Write the formul for the volume of rectngulr prism. Identify the vlues of the pronumerls. WRITE V = lwh l = 50, w = 30, h = 40 3 Sustitute nd evlute. V = = cm 3 4 Stte the cpcity of the continer in millilitres, using 1 cm 3 = 1 ml. 5 Since 1 L = 1000 ml, to convert millilitres to litres divide y = ml = 60 L 6 Give worded nswer. The cpcity of the fish tnk is 60 L. Topic 6 Surfce re nd volume 53

27 Exercise 6.4 Volume INDIVIDUAL PATHWAYS REFLECTION Volume is mesured in cuic units. How is this reflected in the volume formul? Prctise Questions: 1 4, 6 8, 9, 10, 13, 14, 0 consolidte Questions: 1 8, 10 1, 14, 16, 19, 0,, 5 mster Questions: 1 18, 0 6 Individul pthwy interctivity int-4595 FLUENCY 1 Find the volumes of the following prisms. doc cm 4. m doc-540 c 1 cm d 15 cm 0 cm 4. cm 3 cm 7.5 cm Clculte the volume of ech of these solids. 18 mm 15 cm [Bse re: 5 mm ] [Bse re: 4 cm ] 3 WE8 Find the volume of ech of the following. Give ech nswer correct to 1 deciml plce where pproprite. 14 cm.7 m 1 cm 1.5 m c 10 cm d 1 mm 7 cm 8 mm 8 cm 6 mm 54 Mths Quest A

28 e f 6.5 m 45 cm cm 7.1 m 4 WE10 Find the volume of sphere (correct to 1 deciml plce) with rdius of: 1. m 15 cm c 7 mm d 50 cm. 5 Find the volume of ech of these figures, correct to deciml plces. 30 cm 1.4 m c d 4.6 m 18 mm 6 WE11 Find the volume of ech of the following cones, correct to 1 deciml plce. 10 cm 0 mm mm 6 cm 7 WE11 Find the volume of ech of the following pyrmids. 1 cm 4 cm 4 cm 10 cm 30 cm Topic 6 Surfce re nd volume 55

29 8 WE1 Clculte the volume of ech of the following composite solids correct to deciml plces where pproprite. 10 cm 8 cm 5 cm 1 cm 5 cm 0 cm 0 cm 1 cm 35 cm c d 5 cm cm 3 cm.5 cm 3 cm e f 5 cm 3.5 cm 0 cm 10 cm 1 cm 15 cm UNDERSTANDING 9 WE9 Wht effect will tripling ech of the side lengths of cue hve on its volume? Wht effect will hlving ech of the side lengths of cue hve on its volume? c Wht effect will douling the rdius nd hlving the height of cylinder hve on its volume? d Wht effect will douling the rdius nd dividing the height of cylinder y 4 hve on its volume? e Wht effect will douling the length, hlving the width nd tripling the height of rectngulr prism hve on its volume? 10 MC A hemisphericl owl hs thickness of cm nd n outer dimeter of 5 cm. If the owl is filled with wter the cpcity of the wter will e closest to: A 1.56 L B L C L D L E.445 L 5 cm cm 56 Mths Quest A

30 11 Tennis lls of dimeter 8 cm re pcked in ox 40 cm 3 cm 10 cm, s shown. How much spce is left unfilled? 1 WE13 A cylindricl wter tnk hs dimeter of 1.5 m nd height of.5 m. Wht is the cpcity (in litres) of the tnk? 13 A monument in the shpe of rectngulr pyrmid (se length of 10 cm, se width of 6 cm, height of 8 cm), sphericl glss ll (dimeter of 17 cm) nd conicl glsswre (rdius of 14 cm, height of 10 cm) re pcked in rectngulr prism of dimensions 30 cm y 5 cm y 0 cm. The extr spce in the ox is filled up y pcking mteril. Wht volume of pcking mteril is required? 14 A swimming pool is eing constructed so tht it is the upper 8 m prt of n inverted squre-sed pyrmid. Clculte H. Clculte the volume of the pool. c How mny 6 m 3 4 m ins will e required to tke the dirt wy? d How mny litres of wter re required to fill this pool? e How deep is the pool when it is hlf-filled? 15 A soft drink mnufcturer is looking to repckge cns of soft drink to minimise the cost of pckging while keeping the volume constnt. Consider cn of soft drink with cpcity of 400 ml. If the soft drink ws pckged in sphericl cn: i find the rdius of the sphere ii find the totl surfce re of this cn. If the soft drink ws pckged in cylindricl cn with rdius of 3 cm: i find the height of the cylinder ii find the totl surfce re of this cn. c If the soft drink ws pckged in squre-sed pyrmid with se side length of 6 cm: i find the height of the pyrmid ii find the totl surfce re of this cn. d Which cn would you recommend the soft drink mnufcturer use for its repckging? Why? 3 m H Topic 6 Surfce re nd volume 57

31 16 The volume of cylinder is given y the formul V = πr h. Trnspose the formul to mke h the suject. A given cylinder hs volume of 1600 cm 3. Find its height if it hs rdius of: i 4 cm ii 8 cm. c Trnspose the formul to mke r the suject. d Wht restrictions must e plced on r? Why? e A given cylinder hs volume of 1800 cm 3. Find its rdius if it hs height of: i 10 cm ii 15 cm. 17 A toy mker hs enough ruer to mke one super-ll of rdius 30 cm. How mny lls of rdius 3 cm cn he mke from this ruer? 18 A mnufcturer plns to mke cylindricl wter tnk to hold 000 L of wter. Wht must the height e if he uses rdius of 500 cm? Wht must the rdius e if he uses height of 500 cm? c Wht will e the surfce re of ech of the two tnks? Assume the tnk is closed cylinder nd give your nswer in squre metres. 19 The ncient Egyptins knew tht the volume of the frustum of squre-sed pyrmid ws given y the formul V = 1 3 h(x + xy + y ), lthough how they discovered this is uncler. (A frustum is the prt of cone or pyrmid tht is left when the top is cut off.) Find the volume of the frustum shown elow. y h x Wht would e the volume of the missing portion of the squre-sed pyrmid shown? 5 m 4 m 6 m 58 Mths Quest A

32 REASONING 0 Archimedes is considered to e one of the three gretest mthemticins of ll time (long with Newton nd Guss). He discovered severl of the formuls used in this chpter. Inscried on his tomstone ws digrm of his proudest discovery. It shows sphere inscried (fitting exctly) into cylinder. Show tht volume of the cylinder volume of the sphere = surfce re of the cylinder surfce re of the sphere. 1 Mrion hs mixed together ingredients for cke. The recipe requires king tin tht is cylindricl in shpe with dimeter of 0 cm nd height of 5 cm. Mrion only hs tin s shown elow left nd muffin try consisting of 4 muffin cups. Ech of the muffin cups in the try is portion of cone s shown in the digrm elow. Should Mrion use the tin or the muffin try? Explin. 1 cm 8 cm 4 cm 4 cm 10 cm 15 cm 8 cm Nthniel nd Andrew re going to the snow for survivl cmp. They pln to construct n igloo, consisting of n entrnce nd hemisphericl structure, s shown. Nthniel nd Andrew re sked to redrw their plns nd increse the size of the livele region (hemisphericl structure) so tht the totl volume (including the entrnce) is douled. How cn this e 1.5 m 1 m chieved? 3 Sm is hving his 16th irthdy prty nd wnts to mke n ice trough to keep drinks cold. He hs found squre piece of sheet metl with side length of metres. He cuts squres of side length x metres from ech corner, then ends the sides of the remining sheet. When four squres of the pproprite side length re cut from the corners the cpcity of the trough cn e mximised t 588 litres. Explin how Sm should proceed to mximise the cpcity of the trough. 4 The Hstings fmily house hs rectngulr roof with dimensions 17 m 10 m providing wter to three cylindricl wter tnks, ech with rdius of 1.5 m nd height of.1 m. Show tht pproximtely 18 millimetres of rin must fll on the roof to fill the tnks. 1.5 m Topic 6 Surfce re nd volume 59

33 PROBLEM SOLVING 5 Six tennis lls re just contined in cylinder s the lls touch the sides nd the end sections of the cylinder. Ech tennis ll hs rdius of R cm. Express the height of the cylinder in terms of R. Find the totl volume of the tennis lls. c Find the volume of the cylinder in terms of R. d Show tht the rtio of the volume of the tennis lls to the volume of the cylinder is : 3. 6 A frustum of squre-sed pyrmid is squre pyrmid with the top sliced off. H is the height of the full pyrmid nd h is the height of the frustum. H x x h X X doc-6733 Find the volume of the lrge pyrmid which hs squre se side of X cm. Find the volume of the smll pyrmid which hs squre se side of x cm. c Show tht the reltionship etween H nd h is given y H = Xh X x. d Show tht the volume of the frustum is given y 1 3 h(x + x + Xx). CHALLENGE Mths Quest A

34 ONLINE ONLY 6.5 Review The Mths Quest Review is ville in customisle formt for students to demonstrte their knowledge of this topic. The Review contins: Fluency questions llowing students to demonstrte the skills they hve developed to efficiently nswer questions using the most pproprite methods Prolem Solving questions llowing students to demonstrte their ility to mke smrt choices, to model nd investigte prolems, nd to communicte solutions effectively. A summry of the key points covered nd concept mp summry of this topic re ville s digitl documents. Review questions Downlod the Review questions document from the links found in your ebookplus. Lnguge int-841 int-84 int-3593 re cpcity circle composite figure cone cross-section cue cylinder ellipse fce hemisphere prllelogrm prism pyrmid rectngle rhomus sector semi-perimeter sphere squre surfce trpezium tringle volume Link to ssesson for questions to test your rediness FOR lerning, your progress AS you lern nd your levels OF chievement. ssesson provides sets of questions for every topic in your course, s well s giving instnt feedck nd worked solutions to help improve your mthemticl skills. The story of mthemtics is n exclusive Jcrnd video series tht explores the history of mthemtics nd how it helped shpe the world we live in tody. Austrlin megfun (eles-1845) tells the story of some of the lrgest nd most impressive nimls to ever wlk the Erth. Severl of these nimls re introduced s we question wht led to the extinction of Austrlin megfun. Topic 6 Surfce re nd volume 61

35 <INVEStigtion> for rich tsk or <mesurement AND geometry> for puzzle rich tsk So close! Mesurement errors When we mesure quntity y using scle, the ccurcy of our mesurement depends on the mrkings on the scle. For exmple, the ruler shown cn mesure oth in centimetres nd millimetres. Mesurements mde with this ruler would hve ± 0.5 mm dded to the mesurement. The quntity ± 0.5 is clled the tolernce of mesurement or mesurement error. Tolernce of mesurement = 1 size of smllest mrked unit For mesurement of 5.6 ± 0.5 mm, the lrgest possile vlue is 5.6 cm mm = 5.65 cm, nd the smllest vlue is 5.6 cm 0.5 mm = 5.55 cm. 6 Mths Quest 810 for + Victori 10A Austrlin Curriculum edition

36 1 For the thermometer scle t right: determine the temperture stte the mesurement with its tolernce c determine the lrgest nd smllest possile vlues. Clculte the lrgest nd smllest vlues for: 156. ± ± ± ± 0.1. Significnt figures in mesurement 30 A significnt figure is ny non zero-digit, ny zero ppering etween two non-zero digits, ny triling zeros in numer contining deciml point, nd ny digits in the deciml plces. For 5 exmple, the numer hs 7 significnt figures, wheres 300 hs 1 significnt figure. The numer of significnt figures is n expression of the ccurcy of mesurement. The greter 0 the numer of significnt figures, the more ccurte the mesurement. For exmple, fst food 15 chin clims it hs sold hmurgers, not The first mesurement hs only 1 significnt figure nd is very rough pproximtion of the ctul numer sold, which hs 10 significnt figures. Reducing the numer of significnt figures is process tht is similr to rounding. Rounding nd mesurement error in clcultions When you perform clcultions, it is importnt to keep s mny significnt digits s prcticl, nd to perform ny rounding s the finl step. For exmple, clculting y rounding to significnt figures efore multiplying gives = 180, compred with 180 if the rounding is crried out fter the multipliction. Clcultions tht involve numers from mesurements contining errors cn result in nswers with even lrger errors. The smller the tolernces, the more ccurte the nswers will e. 3 Clculte y: i first rounding ech numer to significnt figures ii rounding only the nswer to significnt figures. Compre the two results. Error in re nd volume resulting from n error in length mesurement The side length of cue is mesured nd incorrectly recorded s 5 cm. The ctul size is 6 cm. The effect of the length mesurement error used on clcultions of the surfce re is shown elow. Complete the clcultions for volume. Error used in length mesurement = 1 cm Surfce re clculted with incorrectly recorded vlue = 5 6 = 150 cm Surfce re clculted with ctul vlue = 6 6 = 16 cm Percentge error = 100% 30.5% 6 4 Complete similr clcultion for the volume of the cue using the incorrectly recorded length. Wht conclusion cn you mke regrding errors when the numer of dimensions increse? Give three exmples of prcticl sitution where n error in mesuring or recording would hve potentilly disstrous impct. C Topic 6 Surfce re nd volume 63

37 <INVEStigtion> mesurement AND for geometry rich tsk or <mesurement AND geometry> for puzzle CODE PUZZLE Austrlin inventions! The nswers to the mesurement prolems give the puzzle s code cm 8 cm 10 cm Are = A cm 3 m Volume = G m 3 ( d.p.) 700 cm Volume = O m 3 m 400 cm mm 1 m 11 m 8 m 8 cm 5 mm Volume = B mm 3 mm Are = H m re = 3 cm Are = P cm ( d.p.) 3 m Are = C m ( d.p.) Volume = I cm 3 3 cm 4 cm 15 mm 6 m 5 m Volume = R m 3 8 m 3 m 1 m 1 m Volume = D m 3 Are = E cm 9 cm 8 cm 5 cm 15 cm Volume = F cm 3 ( d.p.) Are = K mm Are = L cm Volume = M m 3 Are = N m 3 m 1 m 5 m 13 m 0 mm Are = T mm Volume = X cm 3 ( d.p.) Are = Y m 16 mm cm 5 cm 6 cm 10 m 6 m 8 m Mths Quest 10 8 for + Victori 10A Austrlin Curriculum edition

38 Activities 6.1 Overview Video The story of mthemtics (eles-1845) 6. Are elesson Heron s formul (eles-0177) Interctivity IP interctivity 6. (int-4593): Are Digitl docs SkillSHEET (doc-536): Conversion of re units SkillSHEET (doc-537): Using formul to find the re of common shpe WorkSHEET 6.1 (doc-541): Are 6.3 Totl surfce re Interctivities TSA sphere (int-78) IP interctivity 6.3 (int-4594): Totl surfce re Digitl docs SkillSHEET (doc-538): Totl surfce re of cues nd rectngulr prisms WorkSHEET 6. (doc-54): Surfce re To ccess ebookplus ctivities, log on to 6.4 Volume Interctivities Mximising the volume of cuoid (int-1150) IP interctivity 6.4 (int-4595): Volume Digitl docs SkillSHEET (doc-539): Conversion of volume units SkillSHEET (doc-540): Volume of cues nd rectngulr prisms WorkSHEET 6.3 (doc-6733): Volume 6.5 Review Interctivities Word serch (int-841) Crossword (int-84) Sudoku (int-3593) Digitl docs Topic summry (doc-1371) Concept mp (doc-137) Topic 6 Surfce re nd volume 65

39 Answers topic 6 Surfce re nd volume Exercise 6. Are 1 16 cm 48 cm c 75 cm d 10 cm e cm f 73.5 mm g cm h 1 m i 75 cm Prt e = 5π cm ; prt g = 81π cm cm 7.64 cm mm mm 5 i 1π cm ii cm i 69π mm ii mm c i 61π cm ii cm 6 E 7 D cm m c 5 cm d 30.4 m e 78 cm f cm cm m c m d m e m f 37.5 m cm 11 1 m $ m m c 75% = x + y y = 50 x c Are = 50x x d x Are (m ) e No, impossile to mke rectngle. f Are x g x = 5 h y = 5 i Squre j 65 m k r = 15.9 m l m m m 16 Students work m; horizontl 17 Circulr re, m ; rectngulr re, m Circulr re, 1 n m ; rectngulr (squre) re, 4π 1 16 n m. Circulr re is lwys 4 or 1.7 times lrger. π 18 x = 50 cm, y = 30 cm 19 Techer to check Chllenge Exerctise 6.3 Totl surfce re cm 384 cm c 1440 cm d 7 m m cm c 8. m d 45.4 cm cm 50.7 cm cm 9.4 m c cm d 4.1 cm m 90 m c cm d mm e 15.7 cm f cm cm cm c 75 cm d 70.4 cm e cm f cm 7 B cm m $ cm $ θ = 10 x = 1; y =!3 c 3!3 cm d 6!3 cm e π m!7 m c!7π m d!7! The clcultion is correct m Bck wll = 80 tiles Side wll = 50 tiles = 180 tiles c Chepest: 30 cm y 30 cm, $69.50; 0 cm y 0 cm (individully) $70; 0 cm y 0 cm (oxed) $ r = 3!3 17 Arc length XY = (x + s)θ Arc length AB = xθ i x = πt θ = st r t ii x x + s = t r c Are of sector AVB = x θ Are of sector XVY = (s + x) θ sθ(s + x) Are of ABYX = TSA of frustum = π(t + r sθ(s + x) ) + 18 The re of mteril required is 1.04 m. If Tin is creful in plcing the pttern pieces, she my e le to cover the footstool. Exercise 6.4 Volume 1 7 cm m 3 c 3600 cm 3 d 94.5 cm mm cm cm m 3 c 80 cm 3 d 88 mm 3 e 91.6 m 3 f cm m cm 3 c mm 3 d cm cm m 3 c mm 3 d cm mm cm cm cm cm 3 c cm 3 d cm 3 e cm 3 f cm 3 9 V new = 7l 3, the volume will e 7 times s lrge s the originl volume. V new = 1 8 l, the volume will e 1 of the originl volume. 8 c V new = πr h, the volume will e twice s lrge s the originl volume. 66 Mths Quest A

40 d V new = π r h, the volume will remin the sme. e V new = 3lwh, the volume will e 3 times s lrge s the originl vlue. 10 E cm L cm 3 14 H = 6 m 11 m 3 c 19 ins d L e 1.95 m from floor 15 i 4.57 cm ii 6.5 cm i cm ii 33.7 cm c i cm ii cm d Sphere. Costs less for smller surfce re. 16 h = V πr i 31.8 cm ii 8.0 cm V c Å πh d r 0, since r is length e i 7.6 cm ii 6. cm cm cm c A = m, A = 1.01 m m m 3 0 Answers will vry. 1 Required volume = cm 3 ; tin volume = 1500 cm 3 ; muffin try volume = cm 3. Mrion could fill the tin nd hve smll mount of mixture left over, or she could lmost fill 14 of the muffin cups nd leve the remining cups empty. Increse rdius of hemisphericl section to 1.9 m. 3 Cut squres of side length, s = 0.3 m or m from the corners. 4 Volume of wter needed; 30.9 m 3. 5 H = 1R 8πR 3 c 1πR 3 d 8 : 1 = : X H 1 3 x (H h) c, d Check with your techer. Chllenge scoops Investigtion Rich tsk 1 The temperture reding is 6.5 C. The smllest unit mrk is 1 C, so the tolernce is 0.5. c Lrgest possile vlue = 7 C, smllest possile vlue = 6 C Lrgest vlue = 37.8, smllest vlue = Lrgest vlue = , smllest vlue = i ii The result for i hs 4 significnt figures, wheres ii hs only 1 significnt figure fter rounding. However, ii is closer to the ctul vlue ( ). 4 Volume using the incorrectly recorded vlue = 15 cm 3 Volume using the ctul vlue = 16 cm 3 The percentge error is 4.1%, which shows tht the error compounds s the numer of dimensions increses. Check with your techer. Code puzzle Bionic er implnt Blck ox flight memory recorder Topic 6 Surfce re nd volume 67