Examination Revision and Solutions

Transcription

1 Yer 10 Generl Mthemtis 013 Exmintion Revision nd Solutions Pythgors Theorem Trigonometry Mesurement Sttistis Instrutions: We expet tht you omplete this revision efore the finl exmintion nd strongly reommend tht you ommene now. Most of this revision should e ompleted t home. Students re dvised to review previously ompleted SAC's Approprite working must e inluded to gin mximum mrks in the exmintion. Mrks will e deduted in ny question for poor presenttion or the use of inonsistent mthemtil sttements. Refer to the solutions provided if neessry. Sientifi Clultor permitted. Attempt ll questions. Give your nswer to the required ury stted. Mrks will e deduted for missing units

2 PYTHAGORAS' THEOREM Find the length of the hypotenuse, leving your nswer in surd form where pproprite: A x m m x m 5 C ITI 5 m Find the length of the hypotenuse, leving your nswer in surd form where pproprite: 5 m m 8 m x m 0.6 m x m 3 Find the vlue of x in the following tringles, leving your nswer in surd form where pproprite: 6m 8 x m 4 Find the vlue of x in the following tringles, leving your nswer in surd form where pproprite: 6 m x m 4.3 m 7 m 77m x m Find the vlue of x in the following tringles. Where pproprite, leve your nswer in surd form. hk.n.n n 3 m Find the vlue of x in the following tringles. Where pproprite, leve your nswer in surd form. 1 m m 17 m x m 5 m x m 8 Find the vlue of x in the following tringles. Where pproprite, leve your nswer in surd form. 9 Solve for x: 10 m x m 1 m

3 1 Find the vlue of x in the following: 17 Find, orret to deiml ples, the vlue of the unknowns: 19 Find, orret to deiml ples, the vlue of the unknowns: 5 m y r8 m 3 m 0 Find x, orret to deiml ples: 3 Find the length of AB, orret to deiml ples. You my need to drw extr lines on your figure. 9 m rl m 6m A 9 Is {, 3, 5} Pythgoren triple? Show your working. {6, 8, n} is Pythgoren triple. Find n. 30 Is {10, 15, 18} Pythgoren triple? Show your working. {n, 30, 34} is Pythgoren triple. Find n A retngle of length 14 m hs digonl of 19 m. Find the width of the retngle. 38 Domini wlks 15 km due est nd then 10 km due north. Drw digrm to show his ourse. How fr is Domini from his strting point? 19 m 14 m

4 41 An equilterl tringle hs sides of length 18 m. Find: the length of one of its ltitudes the re of the tringle. 1 8 A 4 A siling ship sils 46 km north then 74 km est. Drw fully lelled digrm of the ship's ourse. How fr is the ship from its strting point? 43 A 5 m long ldder lens ginst rik wll. The feet of the ldder re 1.8 m from the se of the wll. How fr up the wll does the ldder reh? 1.8 m 44 A power sttion P, supplies two towns A nd B with power. New underground power lines to eh town re required. The towns re onneted y stright highwy through A nd B nd the power sttion is 4 km from this highwy. Find the length of power line required from P to eh town. Find the totl ost of the new power lines given tht eh kilometre will ost $350. A 13 km 45 kin 47 A le r sends from se level to the top of plteu, s shown in the digrm. How fr does the le r trvel? If it trvels t 3 metres per seond, how long will the journey tke? 800 m Three towns re situted so tht B is 35 km in diretion 5 T from A, nd C is 4 km in diretion 135 T from A. Find the distne from B to C to the nerest 100 metres. A C 50 A plne trvels from A on ering of 66 T for 450 km to B, nd then trvels on ering of 156 T until it rehes C. If A nd C re 600 km prt, wht is the distne from B to C? A 51 A heliopter trvels from se sttion S on true ering of 063 for 143 km for n emergeny resue. It then trvels 157 km on true ering of 153 to n emergeny hospitl H. How fr is the emergeny hospitl from the se sttion S? 53 The slnt height of one is 15 m nd its height is 11 m. Find the dimeter of the se. 11 m 15 m....c1...

5 PYTHAGORAS' THEOREM SOLUTIONS 1 x = + 5 {Pythgors}... x = x = 9 x = {x > -. hypotenuse is 1/ m x = {Pythgors} x =5+5 *. X = 50 x =/ {x > 0}.. hypotenuse is A/7) m x = {Pythgors}... x =5+64 x = 89 x =/ > 0}... hypotenuse is Vg m x = + (0.6) {Pythgors} x = x = 4.36 x =Jfl {x > 0}... hypotenuse is -A L/ m 3 x = 7 +9 {Pythgors}... x = x = 130 x la/j {x > 0} x + 6 = 8 {Pythgors} X + 36 = 64 x = 8 x = N/N {x > 0} 4 x = 6 + (4.3) {Pythgors}... x = x = x =.A/519 {x > 0} X ± 7 = 10 {Pythgors} x + 49 = 100 x = 51 x = {x > 0} 5 x +3 = 4 {Pythgors} x + 9 = x = 7 x = {x > 0} 1 ± = (v75) {Pythgors} X + 8 = x = 7 x = Ar7 Ix > 0} 6 x + 1 = 17 {Pythgors} X = 89 x = 145 x {x > 0} x + = 5 {Pythgors} X + 4 = 5 x = 1 x = AM_ {x > 0} 8 x +10 = 1 {Pythgors} X = 144 x = 44 x = A/474 {x > 9 x + x = 7 {Pythgors} x = 49 x = 1 X = {x > 0} 4 m m x ± = 4 {Pythgors} x + 4 = 16 x = 1.*. x = {x > 0} 17 x = + 1 {Pythgors} x = x = 5 x > 0} x.4 ( d.p.) Also, y = x + 1 {Pythgors} y = y = 6 y = 'Vfi {y > 0} y.45 ( d.p.) 19 x + 5 = 8 {Pythgors}... x + 5 = 64 x = 39 x = A/ {x > 0} x = 6.4 ( d.p.) Also, y = x + 3 {Pythgors} y = y = Vr8 {y > 0} y * ( d.p.) 0 (x) = 4 +4 {Pythgors} 4x = x = 3 x = N/g > 0} x,*.83 ( d.p.)

6 3 A m 9m m 6m AC = 9 m, BC = + 6 = 8 m = {Pythgors} x = X = 145 x = A/1.5 {x > 0} i.e., AB * 1.04 m long = = 13 nd 5 = {, 3, 5} is not Pythgoren triple = n = n n = 100 n = N/11:03 > 0} n = = = 35 nd 18 = {10, 15, 18} is not Pythgoren triple. 37 n + 30 = 34 n = 1156 n = 56 n=/f; {n > 0} n = m x m 14 m x + 14 = 19 {Pythgors}.*. x = 361 x = 165 x = {x > 0} x i.e., the width of the retngle is 1.8 m. 38 AN x km F 101m 15 km x = 10 ± 15 {Pythgors} X = x = 35 x = {x > 0} x i.e., Domini is km from his strting point. 41 h +9 = 18 {Pythgors} h + 81 = m h = 43 h = th > 0} h m i.e., the ltitude is 15.6 m. 4 Are = se x height 1 x 18 x m 74 km 46 km PIFFV-, km strt x = 46 ± 74 {Pythgors} x = x = 759 x = {x > 0} x = the ship is km from strting point. 43 X ± 1.8 = 5 {Pythgors} x = 5 x = 1.76 x = A/1.76 {x > 0} x i.e., ldder rehes 4.66 m up the wll km 45 km A x km 4 km y km n/x m m X = {Pythgors} x = X = 1933 x = {x > 0} x i.e., km of power line is required from P to A. y = 4 ± 45 {Pythgors} y = '. y = 3789 y = > 0} y i.e., km of power line is required from P to B. Totl ost = ( ) x $350 = $4797 i.e., totl ost is pproximtely $48000.

7 47 51 x ITA 650m 800 m x = {Pythgors} x = x = x =./ {x > 0} x i.e., the le r trvels 1031 m, distne time = speed time = 3 * se time * 5 min 44 se km 63 \ x km 153 \ 157 km 7 \ ZSRH = = 90 x = {Pythgors} x = x = x = V {x > 0} x i.e., the emergeny hospitl is 1.4 km from the se sttion. 11 m 15 m km 5 :16) km xkm ZBAC = = 90 x = {Pythgors} x = x = 1801 x = {x > 0} x i.e., the distne from B to C is 4.4 km r CM r + 11 = 15 {Pythgors} r + 11 = 5.. r = 104 r = )/ 1 > 0}.. r r * 0.40 i.e., the dimeter of the se is 0.4 m. ri A 600 km LABC = =90 x = 600 {Pythgors} x = x = x {x > 0} x i.e., the journey from B to C is km.

8 TRIGONOMETRY I find: find: os 0 sino sin90 os 60 C sin30 C sin0 3 find: 4 find: sin60 os 90 os 50 sin50 C tn45 C tn 0 5 Use your lultor to find, orret to 4 deiml ples: sin 43 os 43 tn 43 6 Use your lultor to find, orret to 4 deiml ples: os 18 sin 76 tn 81 7 Use your lultor to find, orret to 4 deiml ples: sin 8 os 16 tn 54 8 Use your lultor to find, orret to 4 deiml ples: tn 64 os 73 sin 3 9 In the digrm longside, nme the: hypotenuse side opposite the ngle mrked side djent to DCYZ. x 10 In the digrm longside, nme the: A hypotenuse side opposite the ngle mrked 0 side djent to BCA. 13 For the right ngled tringle longside, find: sin 0 os tn0 P 14 Find x. Find sin 0, os 0 nd tn O. 15 Find x. Find sin 0, os 0 nd tn O. 16 Find x. Find sin 0, os 0 nd tn O.

9 1 Find, to deiml ples, the unknown length in:,diegi Ao m x m IriFr' Find, to deiml ples, the unknown length in: x m 1 m x m 1.7 m 3 Find, to deiml ples, the unknown length in: m 3.6 mm x m 60' Slm 8 4 Find, to deiml ples, the unknown length in: 1.95 km x km. 111DPIPP- x m 6 11.m 5 Find, to deiml ples, the unknown length in: 9 m x m 35 m 61 x m 6 Find, to deiml ples, the unknown length in: 1 m x m m x m 9 Find, orret to deiml ples, the vlue of 0 if: sin 0 = os 0 = Find, orret to deiml ples, the vlue of 0 if: tn 0 = 5.3 sin 0 = 31 Find, orret to deiml ples, the vlue of 0 if: os 0 = 0 tn 0 = C tn = os iy = 11. sin 0 = Find, to one deiml ple, the mesure of the ngle mrked 0 in: 7 m 6 m r

10 34 Find, to one deiml ple, the mesure of the ngle mrked 0 in: 13 mm 8 mm 7.4m 3. m 35 Find, to one deiml ple, the mesure of the ngle mrked 0 in: 5.7 m 8. m 13.4m 11.5 m 36 Find, to one deiml ple, the mesure of the ngle mrked 0 in: st 8. m 6.1 m m 3 m 37 Find x orret to 3 signifint figures: 11 m x m m 5 m Find x orret to 3 signifint figures: 11.6 m x 0 m m 39 Find x orret to 3 signifint figures: 5.1 m 4.6 m.' irm.v m 40 Find x orret to 3 signifint figures: x mm mm 11.1 m x"

11 45 Find the height of flgpole whih sts shdow of 9.3 in when the Sun mkes n ngle of 63 to the horizontl. 63 A 46 A surveyor stnding t A mesures the ngles sutended y two posts B nd C on the opposite side of nl. The posts re 10 m prt. If the ngle of sight etween the posts is 37, how wide is the nl? m B 10m C nl 47 An eroplne moves from diretly overhed to position where the ngle of elevtion is 10, in 4! 111ft. minutes. Find the speed of the eroplne given m tht it is onstnt nd tht the eroplne is try ' to elling t onstnt height of 4000 m ove the ground ground. A 48 Two people of the sme height stnd t A nd B t the top of two uildings of heights 335 m nd 80 m. The uildings re 185 m prt. Find the ngle of elevtion of B from A.... :: i A m===-lin m 185 m 80m 49 Find the ngle sutended t the entre of irle of rdius 1 m y hord of length 15 m. 5 Find the re of n isoseles tringle whih hs se ngles of 37 nd equl sides of length 15 m. 53 From n oserver 0, the ngles of elevtion to the ottom nd the top of flgpole re 36 nd 38 respetively. Find the height of the flgpole. (Hint: Find AB nd AC.) : A., m A 56 From point A, the ngle of elevtion to the top of tll uilding is 0. On wlking 80 m towrds the uilding the ngle of elevtion is now 3. How tll is the uilding? 57 Find the ering of X from Y if the ering of Y from X is: Find the ering of R from Q if the ering of Q from R is:

12 59 Find the ering of: B from A B from C C from A An orienteer runs in diretion 04 for distne of 7 km nd then in the diretion 13 for 1 km. Find: the distne from the strting point the ering of the finl position from the strting point. 6 Mry is in diretion 337 nd distne 3 km from Sue. Jne is in diretion 067 nd distne 35 km from Sue. Find: the distne of Jne from Mry the ering of Jne from Mry TRIGONOMETRY SOLUTIONS 1 os 0 = 1 sin90 = 1 sin 30 = 0.5 sin 0 = 0 os 60 = 0.5 sin 0 * sin 60 * 0.87 os 50 * 0.64 C tn 45 = 1 4 os 90 = 0 sin 50 * 0.77 tn 0 = 0 5 sin 43 * os 43 4: tn43 * os sin 76 * tn 81 * sin 8 = h os tn tn64 4:.0503 os 73 * 0.94 sin 3 * ZY XY XY 13 sin 0 = r os 0 = tn 0 = p X ± 10 = 13 {Pythgors} X = x = 69 0 X =.10 {X > 0} sin 0 = ' os 0 = 69 ' tn 0 = X = 6 ± 11 {Pythgors} 0 x.. X = x = x =1,/W7 > 0} sin 0 =, os 0 = 4, tn 0 = 1,/7 v X = {Pythgors} X = X = 0 x=/ > sin = os0 =N/P tn0 = 1 1 tn 47 = 10 lorn 10 x tn 47 = x v m 47 x = 10.7 i.e., length 10.7 m os 37 = x x os 37 = x x = i.e., length * m x m 1.7 m 10 BC AC AC

13 sin 71 = - x 1 x m 1 x sin 71 = x x * IPPPIPP- 1 m i.e., length * m os 33 = x os 33 = x.*. x * m i.e., length * m 3 sin 60 = - x x x sin 60 = sin 60 x 4'.31 i.e., length *.31 m 3.6 tn 8 = - x x tn 8 = X M in tn 8 8.*. x * 6.77 i.e., length * 6.77 mm x m m x m x 4 sin 41 = km x sin 41 = x x km 1.8 i.e., length * 1.8 km 11. os 6 = - x x os 6 = x m os 60 x * 3.86 i.e., length * 3.86 m 5 tn 51 = m 6 x x tn 51 = 9 x m tn m x * 7.9 i.e., length * 7.9 m 35 sin 61 = - 35 m 6 os 3 = - x 1 1 x os 3 = x i.e., length * m iiil r, 18.4 sin = - x x x sin 6 = sin 6 x * 0.84 i.e., length * 0.84 m 1 m x m 18.4 m x m 9 sin 0 = os 0 = 0.1 ;. 0 = sin-1 (0.654).'. 0 = os-1 (0.1) * * 84.6 tn 0 = 1 ;. 0 = tn-1 (1) 0 = tn 0 = 5.3 sin 0 = i = tn-1(5.3).*. 0 = sin'().6 mm.*. 0 * , 0 = 30 > os 0 = M (M).'. 0 * os 0 = 0 tn 0 = A 7 0 = COS-1(0).'. 0 = tn-1 (1) 0 = 90 sin 0 = 0.95 = sin (0.95) 0 * tn 0 = 1 7 m 6 0 = tn-1 (i) 6 m 04' sin 0 = 11 m 0 = sin-1 (*) 0 4'7.0 5m 34 os 0 = 13 0 = os-' (i) sin 0 = = sin-1 ( : 4 ).'. 0# m 13 mm 0 8 mm 3.m x x sin 61 = x m sin 610 x * 40.0 i.e., length * 40.0 m 35 tn 0 = = tn-1 (N) ;. * 55. os 0 = = os ' (113) 0# m 8 - m 13.4 m 11.5 m

14 36 sin0 = 8..'. 0 = sin-1 (H).* m 148. m 45 h m x m m m m x 37 sin 13.6 = x sin13.6 = x x x * tn 0 = 16 0 = tn-1 (t) tnx = 7..diellik 7 5 ;. X = tn-1() 35.5 x m 58 0 m 0 tn 58 = - x x x tn 58 = 0 0 x tn58 x * 1.5 os q 11. =_ os-1(11.6\ k 17.9 / x4: sin x = 5.1 X = sin-1 ( 1. ) 5.1 x4: 49.9 tn x = 4.6 x tn44.. x4: sin 53 = - x tnx = x mm sin xo=t,1 ) ll., x. 10. mtk...4.._ 11.1 m m tn 63 = x tn 63 = h h 18.9 i.e., the flgpole is 18.9 m high. 46 tn 37 = 10 d x tn37 = d - tn 37 d 159. i.e., the nl is 159. m wide m= 4 km 10 x r ground 4 tn 10 = - x x x tn10 = 4 4 tn 10 x *.685 km distne trvelled speed = time tken hour * km/h 48 B A m C 185m EE 49 =-:41 { mm = to hr} 80 mr-r ngle of elevtion is BC = m = 55 m Now tn = = tn-1(*) * i.e., the ngle of elevtion is sin (g) d m B 10m C A sin-1 = i.e., the ngle t the entre is 77.36

15 5 15 m 37 m m 'm os 37 0 = 15 sin 37 = 10 = 15 x os 37 = 15 x sin 37 * * Now re = se x height = x x = * x 9.07 re * m 57 X 1461) ering of X from Y is ering of X from Y is m A AB In OAB, tn 36 = x tn 36 = AB AB * ering of X from Y is 155 In AOAC, tn x tn 38 = AC AC* Now, BC.= AC AB.. BC * * i.e., the flgpole is 10.9 m in height T Ah m 0 B, A-4 gorn s-4 xm 1.-C h In ABTC, tn 3 = x... h=xxtn3 h In AATC, tn 0 = x h = (x + 80) x tn 0... x x tn3 = (x + 80) x tn 0... x tn 3 = x tn tn 0... x tn 3 x tn 0 = 80 tn 0... x(tn 3 tn 0 ) = 80 tn 0 80 tn 0 tn 3 tn 0 80 tn 0 h= x tn 3 tn 3 tn 0 h = 04.8 i.e., the uilding is 04.3 m high. ering of R from Q is ering of R from Q is ering of R from Q is 187

16 59 ] A -,--..- ering of B from A is = 180 {o-interior ngles}... 0 = = 360 {ngles t point} ) = ) = ering of B from C= = ) = 180 {ngles in Al = = 33 Thus 58 + = ering of C from A is N LSXF is right ngle S 13 km N 1 km - x lt F x = {Pythgors} :. x = :. x = x = \ 1/ {x > 0}... x , distne from strting point is km. Now tn 0 = 1,... 0 = tn'().'. 0 * Thus ering of finl position from strting point is N NI l' e x km N 3 km km 3 &Lit WV J LMSJ is right ngle x = {Pythgors} :. x = x = x =. /1754 {x > 0}... x Mry nd Jne re km prt. tn0 = N... 0 = tn-1() But = 180 {o-interior ngles}.' = Jne is on ering of from Mry.

17 1 Convert the following lengths: 41.6 km to m 90 m to m mm to m 3 4 Convert the following lengths: 5 mm to m 0.53 m to mm 1145 m to km Convert the following lengths: 310 mm to m lm to m.3 lm to mm Convert the following lengths: 46.7 m to mm 500 m to km 4 km to mm 5 Jne runs 6.4 km while Jenny wlks 600 m. How muh further did Jne run thn Jenny wlk? 6 Due lives.7 km from shool. He runs the first 800 m on his wy home, then wlks the rest of the wy. How fr does he wlk? 9 Find the perimeter of: tringle with sides 1. m, 11.4 m nd 10.8 m rhomus with one side 7.9 m irle with dimeter 8 m. 10 Find the perimeter of: n equilterl tringle with sides.7 m retngle with djent sides m nd 3. m irle with rdius 3 m. 11 Find the perimeter of: n isoseles tringle with equl sides 13.4 m nd third side 11. m rhomus with sides 7.1 m irle with rdius 10 m. 13 Find the perimeter of: 14 Find the perimeter of 5 m 10 7 m 16 Find the perimeter of setor of rdius 8 m nd ngle Find the perimeter of the given shpe: 10 m 19 Find the perimeter of the following shpe:, m

18 0 Find the perimeter of the following shpe: 3 m (You my need to use Pythgors' theorem.) 9 m _I. 5 Determine the length of fening round 60 m y 150 m retngulr plying field if the fene is to e 15 m outside the edge of the plying field. 6 A new house is to hve eight luminium windows, identil in shpe to tht shown in the digrm. The outer frming osts $9.50 per metre nd the inner slts ost $4.0 per metre. Find the totl ost for the frming nd slts. 1.5 in 1. In 7 A frmer wishes to sudivide his property into six setions s shown. If he wishes to fene his entire property, s well s eh setion, with fening whih osts $0.34 per metre, find the totl ost of the fening m 700 m 8 A frmer wnts to fene his house grden to keep out wndering ttle. The grden mesures 6 in y 40 m. The fene is to hve five strnds of wire, nd posts re to e pled every two metres. Clulte the perimeter of the grden. Wht length of wire is needed? Now mny posts re needed? d Find the totl ost of the fene if wire is $0.34 per metre nd eh post osts $ Find the rdius of irulr fish pond with irumferene 5 m. (Answer to the nerest m.) 41 Find the shded res: l4kmfl10 11m km 4 Find the re of the following figures: 4 m Atm 1 m 43 Find the re of the following figures: 13 m m

19 44 Find the shded re of the following figures: d 5 m 3 m 1.14 mi, 45 Find the re of the following figure: 1 m 0 m 10 m 47 Find the re of the following figure: 9m 48 Find the re of the following figure: T1) 6.4 m 3 m 49 Find the re of: trpezium with prllel sides 10 m nd 6 m, nd height 1 m setor of rdius 10 m nd ngle m ' m Find: the rdius of the semi-irle the perimeter of the figure the enlosed re. 63 Find formul for the re A, of the following region: 69 Find the surfe re of the following solids: 00 4 m Alo;m 10 m m 15 m

20 71 Find the surfe re of the following solids: 5 m naidhl if4r4(040; 3 6 m 5 m 73 Find the totl surfe re of the following solids 8 m 74 Find the totl surfe re of the following solids: m 80 Find the rdius of sphere of surfe re 500 m. 91 Find the volume of: sphere of rdius 5. m one of rdius 1 m nd slnt height 15 m. 9 Find the volume of: ylinder of height 16 m nd se dimeter 40 m n equilterl tringulr prism with tringles of side 4 m nd length 10 m. 93 A swimming pool hs the dimensions shown. Find the re of the trpezium-shped side. Find the volume of wter needed to fill the pool. Find the pity of the pool in: i kilolitres ii litres. d If wter runs into the pool t 80 L per minute, how long will it tke to fill?.5 m 0 in P1P-- 1 in A glss vse is hemispheril, s shown, with hemispheril hollow for wter. Find the pproximte volume of glss, if the vse is slightly flttened on the ottom for stility. I-I 1 in 0.8 m pipes with dimensions s shown re to e mde of onrete. Find: the re of the onentri irulr end the volume of eh pipe the totl weight of onrete required to mke the pipes, given tht eh ui metre of onrete weighs 3.17 tonnes. IIII 1.4m I* 1. m PI m end km = (41.6 x 1000) m = m mm = ( ) m = 11.5 m 5 mm = (5 ± 10) m = 5. m 1145 m = (1145 ± ) Iin = km SOLUTIONS 90 m = (90 ± 100) m = 9. in 0.53m = (0.53 x 1000) mm = 530 mm mm = (310± 1000) m = 3.1 m km = (0.076 x 1000 x 100) m = 7600 m.3 km = (.3 x 1000 x 1000) mm = mm

21 m 500m = (46.7 x 10) mm = ( ) km = 467 mm =.5 km 4 km = (4 x 1000 x 1000) mm = mm 5 Jne runs 6.4 km Jenny wlks 600 m = ( ) km =.6 km Differene = = 3.8 km i.e., Jne runs 3.8 km further. 6 Due wlks.7 km m =.7 km - ( ) km = (.7-0.8) km = 1.9 km i.e., Du wlks 1.9 km 9 Perimeter Perimeter = m = 4 x 7.9 m = 34.4 m = 31.6 m Perimeter =7r x 8 m *5.13 m 10 Perimeter Perimeter = 3 x.7 m = x ( + 3.) m = 8.1 m = x 5. m Perimeter = 10.4 m = x7rx 3m * m 11 Perimeter Perimeter = m = 4 x 7.1 m = 38 m = 8.4 m Perimeter = x7rx 10 m * 6.83 m 13 Perimeter = 5 -F 5 -F (t) x 7r x 5 = 10 + (t) x 107r * m 14 Perimeter = (*) x 7r x 7 * 8.66 m 147r 16 Perimeter =8+8±( 131-g)x 7r x 8 g m = 16 + (*) x 167r m Perimeter = semi-irle dimeter 1 m + semi-irles dimeter 6 m =ix7rx 1 ± xix7rx6 = 67r + 67r = 17r * m 0 x m 3 m =4 m -4-3 m-p-4-3 m-p-4-3 m m perimeter = = m 150 m 90m 60 m m Length of fening = x ( ) = 540 m 6 Length of outer frming for 1 window = x (1.5 ± 1.) = 5.4 m 7 Length of inner slts for 1 window = x x 1. = 5.4 m A By Pythgors x = = x x = 5 x = {x > x = 5 Totl ost for 8 windows = 8 x (5.4 x $ x $4.0) = $ m Length of outer fenes = x (1100 ± 700) = 3600 m Length of inner fenes: Now AB ± CD = 700 EF + GH = 700 IJ + GK = 1100 length of inner fenes = = 500 m totl length of fening = = 6100 m totl ost of fening = 6100 x $0.34 = $ m P = rd = 0 ± ir x 10 * 51.4 m

22 8 d Length Totl 31 C 7rr 5 = 7rr 5 r = 40 m P = x (6 -F 40) P = 13 m Numer r * m r * 80 m 41 A = ( ± x h (14 - F 1(:) A = x 1. A = 1 x 1 A= 144 lm 6 m of wire required = 13 x 5 {5 strnds} = 660 m of posts = 13 ± = 66 ost = 660 x $ x $3.95 = $ nin 10 km 1 km t AilW 1-4=0.4. 5m 10 m 0-1 h + 5 = 13 {Pythgors}... h + 5 = 169.*. h = h = 1 {h > 0} re = se x height = (5 + 10) x 1.=. 180 m A = (35) x 7r x 8 x 647r A = 30.7 m 44 By Pythgors h + 3 = 5 {Pyth}.*. h + 9 = 5 h m :. h = 16 5 m 3 m Are shded = x 5 x 4 = 10 m A = {h > 0}... h = 4 mdi 4 m A = re lrge irle re smll irle A = 7r x 4 ir x A = m 4 4 m h em \tri 45 Af% D] 4 mk- Are = re lrge semi-irle re smll semi-irle = x (7r x 8 ) 1 X (7r X 4 ) * m A 1 m B E0m 10 m /4 m h 4 gm d 4 m 1 m Are = re of A + re of B h 4 = 5 {Pythgors} = ( ) x x 0 h + 16 = 5 = h = 9 = 540 m h = 3 {h > m 47 re = ( + ) x h /4 + 1 \ = ) = 8 x 3 = 4 m A= () xirx6 A = x 367r 360 A * 4.41 m 9m 1/1 Are = re of A + re of B + re of C = (9 x ) + (4 x 5) +( x x 4) = = 4 m

23 48 3.4m m 3.4m 3 m Are = re squre 6.4 m x 6.4 m re tringle se 3.4 m, height 3.4 m = 6.4 x 6.4 x 3.4 x 3.4 = m 6 1 m 10 m Are = (3 460o) X ir X 10. * 09.4 m 5 8m 10 m r m = ( + Are r m x h /6+ 10) = x 1 = 96 m to m The figure hs fes of 10 m x 8 m fes of 10 m x 4 m fes of 8 m x 4 m totl surfe re = x 10 x 8 + x 10 x 4 + x 8 x 4 = = 304 m The figure hs 1 squre se 4 tringulr fes 15 m 15 m totl surfe re =15x xx 15 x 10 = = 55 m 71 The figure hs 4 fes of 8 m x 5 m fes of 5 m x 5 m.-. totl surfe re =4 x8x5 + x5 x5 = 10 m The figure hs 1 fe of 6 m x 4 m 1 fe of 6 m x 3 m /ON 1 fe of 6 m x 5 m 3 m 4 m fes of i x 3 m x 4 m (r) +8 = 10 {Pythgors} The tringle with sides 3 m, 4 m, 5 m is right ngled s = 5 {Pythgors} 4r + 64 = 100 4r = totl surfe re r = 9 =6 x4+6 x 3+6 x 5+x x 3 x 4 r = 3 {r > 0} = i.e., rdius = 3 m = 84 m 63 P = (7rr) P=18+7rx 3 P = 7.4 m A = re of tringle + re of semi-irle = x 8 x r + 1 (7rr ) =8 x 3 + xirx 3 * m 3r. 5 m m 73 Totl surfe re = 7rrh 7rr {r =, h 8} = 7r x x 8+ 7r x 15.7 m 74 Totl surfe re = 47rr {r = 5} = 47r x m 80 A = 4R-r 500 = 4n-r.. r= 47r Are = re of squre re of irle = 3r x 3r T-7. =9r 7rr = (9 7r) r r= / {r > 0} r i.e., rdius is 6.31 m

24 91 V = n-r3 = x ir x * m3 15 m h + 1 = 15 m {Pythgors}... h = m N-.'. h 81 h = 9 {h > 0} V = iirr h =ix7rx 1 x9 * i.e., volume is m3 9 V = re of end x height = r x 0 x 16 * m 411k1114 m 10 m h + _4 {Pythgors} Akm h + 4 = 16 h = 1 m h = N/T {h > 0} V = re of end x height = 1 x4 xvf x 10 * 69.8 i.e., volume is 69.8 m3 97 outer rdius = 6 m, inner rdius = 5. m Approximte volume of glss = outer volume inner volume =x 7rx63 1 xi x 7rx * m3 100 end 1.4m 1.7 m re of end = re of lrge irle re of smll irle =7rx (0.7) 7rx(0.6) * m volume = re of end x length * x * m3 totl weight * x 000 x 3.17 tonnes * 5179 tonnes 93 0 m.5 m 1m Are of trpezium (.5 + 1) x 0 = 35 m V = re of end x length = 35 x 6 = 10 m3 i.e., volume of wter is 10 m 3 i 1 ms = 1 kl pity = 10 kl ii 10 kl = 10 x 1000 L = L d time tken = 80 = 65 minutes = 43i hours

25 9 7 Construt vertil olumn grph for the following dt: Numer of lollies in pket Frequeny STATISTICS The following mrks were sored for test where the mximum mrk ws 40: Construt n ordered stem-nd-lef plot for the dt. 8 Fmilies t shool were surveyed nd the numer of hildren in eh fmily ws reorded. The results of the survey re shown longside: Construt vertil olumn grph of the dt. Numer of hildren Frequeny Totl The weights (in kg) of pumpkins grown y mrket grdener re reorded elow: Construt n ordered stem-nd-lef plot for the dt using stems, 3, 4, et. Copy nd omplete: "The modl weight ws etween... nd... kg." 11 The mximum dily temperture (in C) ws reorded (orret to 1 deiml ple) in Queenstown in April s follows: Construt n ordered stemplot for the dt using sterns 14, 15, 16, et. Desrie the distriution of the dt. Wht frtion of April dys hd mximum tempertures greter thn 18 C? d Wht frtion of April dys hd mximum tempertures etween 16 C nd 0 C? 1 A frequeny tle for the heights of footll squd Height (m) Frequeny is given longside < Explin why height is ontinuous vrile < Construt histogrm for the dt. The xes < should e refully lelled nd inlude heding < for the grph < Wht is the modl lss? < < A frequeny tle for the ges of students t ountry high shool is given longside. Explin why ge is ontinuous vrile. Construt histogrm for the dt. The xes should e refully lelled nd inlude heding for the grph. Wht is the modl lss? Age (yers) Frequeny Josef reorded the msses of the pumpkins tht he grew on histogrm s shown: How mny pumpkins did Josef grow? Wht perentge of the pumpkins weighed more thn 4 kg? Wht perentge of the pumpkins weighed etween kg nd 8 kg? 0 ITN Josef's pumpkins I. 0 0 I 3 I mss (kg)

26 Stff slries p.. The slries of employees of Better Helth Phrmeutils re shown in the histogrm longside. How mny employees were there? Wht perentge of the employees erned more thn $50 000? Wht perentge of the employees erned etween $ nd $ per yer? 16 A group of Yer 10 students orgnised golf ompetition to find out who ould hit golf ll the furthest. The results of the ompetition re displyed using the histogrm shown: How mny students took prt in the ompetition? Wht perentge of the students hit the ll from 00 m up to 10 m? Wht perentge of the students hit the ll 00 m or more? Find the men nd medin of the set of sores: 18 Find the men nd medin of the set of sores: 0, 1, 1, 1,, 3, 3, 3, 5, 8 3, 4, 5, 5, 4, 5, 19 The following tle shows the verge monthly rinfll for ity. Month S 10 NID Av. rinfll (mm) 14 3 F 36 r 43 A 104 I M 168 T 1657 A 1104 I I 19 Clulte the men verge monthly rinfll for this ity. 0 The selling pries of the lst 9 houses sold in prtiulr distrit were: $186000, $193000, $10000, $19000, $185000, $179000, $194000, $03000, $ Clulte the men nd medin selling pries of these houses. Comment on your results. The next house sold rought prie of $ Clulte the men nd medin selling pries of the 10 houses. Whih of your results in etter represents the typil selling prie? Give reson for your nswer. 1 A mnufturer mintins tht pkets ontin 50 mthes. A qulity ontrol inspetor tested 50 pkets nd found the following distriution: Find the: i men ii medin numer of mthes in pket Comment on these results in reltion to the mnufturer's lim. Numer of mthes Frequeny A mnufturer mintins tht pkets ontin 7 lollies. To test this qulity ontrol inspetor tested 58 pkets nd found the following distriution: Find the: i men ii medin numer of lollies in pket Comment on these results in reltion the mnufturer's lim. Numer of lollies in pket Frequeny

27 3 Niko grew fine rop of pumpkins in his kyrd. He reorded their msses in the tle shown. Find the: i pproximte men ii modl lss iii pproximte medin mss of the pumpkins. Comment on the sttement tht `Niko's pumpkins usully weigh 4 kg or more'. 4 The frequeny tle longside shows the numer of gols sored y squd of hokey plyers over the ourse of seson. Construt olumn grph for this dt. Desrie the distriution of the dt. Find the i men ii medin of the dt. d Whih mesure of entre is the most suitle for this dt set? Mss (kg) Frequeny 1 - < 4 - < < < < < < < 9 No. of gols Frequeny For the dt set 11, 113, 114, 117, 118, 10, 11, 13, 14, 15, 15 find the: medin lower qurtile upper qurtile d interqurtile rnge 30 For the dt set 3.5, 4.1, 4.3, 4.8, 4.8, 4.9, 5.0, 5.1, 5., 5., 5., 5.6 find the: medin lower qurtile upper qurtile d interqurtile rnge 31 The time spent (in minutes) y 4 people in queue t hrdwre store, witing to e ttended, hs een reorded s: Find the medin witing time nd the upper nd lower qurtiles. Find the rnge nd interqurtile rnge of the witing time. Supply the missing numers: i "50% of the witing times were greter thn minutes." ii "75% of the witing times were less thn minutes." iii "The minimum witing time ws minutes nd the mximum witing time ws minutes. The witing times were spred over minutes." 33 Stem Lef Stem Sle: 7 I 1 mens 7.1 Lef Sle: 4 I 7 mens 4.7 For the dt set given, find the: minimum vlue medin e upper qurtile g interqurtile rnge For the dt set given, find the: minimum vlue medin e upper qurtile g interqurtile rnge mximum vlue lower qurtile rnge mximum vlue lower qurtile rnge

28 35 Stem Lef For the dt set given, find the: 0 9 minimum vlue mximum vlue 1 56 medin d lower qurtile e upper qurtile f rnge g interqurtile rnge 5 0 Sle: 4 I 0 mens The oxplot given summrises the points sored y sketll tem for the --I I 1 mthes plyed in seson Wht ws the: points sored y sketll tem i highest sore ii lowest sore? Wht ws the medin sore for the seson? Wht ws the rnge of sores for the seson? In wht perentge of the mthes did the tem sore 43 points or more? Wht ws the interqurtile rnge for the seson? The lowest 5% of the sores were etween nd Is sore of 44 points in the upper 50% of the sores? 38 A oxplot hs een drwn to show the distriution of mrks (out of 50) in siene Mrks in Siene test test for lss. 1 F- Wht ws the highest mrk? Explin the signifine of the ster (1 1(1 isk on the oxplot. The middle 50% of the lss sored mrk etween nd d Find the medin mrk. e 50% of the students sored mrk greter thn or equl to f Find the rnge of mrks sored A ox-nd-whisker plot hs een drwn Mrks in Mthemtis test to show the distriution of mrks (out of 50) in mthemtis test for prtiulr 1 1 I * lss. Wht ws the: i highest mrk ii lowest mrk sored? Wht ws the medin mrk for this test? Wht ws the rnge of mrks for this test? Wht perentge of students sored 30 or more for this test? Wht ws the interqurtile rnge for this test? The top 5% of students sored mrk etween nd If you sored 35 for this test, would you e in the top 50% of students in this lss? 41 An English teher reorded the numer of spelling mistkes mde y 30 students in test. The results were: 4, 7, 9, 3, 1, 11, 5, 4, 6, 6, 4, 4, 4, 1, 8,, 3, 3, 6, 7, 8, 4, 4, 3, 5, 5, 5, 8, 7, 0 Find the medin, lower qurtile nd upper qurtile of the dt set. Find the interqurtile rnge of the dt set. Drw ox-nd-whisker plot for the dt set. 4 Donn ounted the numer of hooltes in some oxes of hooltes nd tulted the dt s shown elow: Numer of hooltes Frequeny Find the five-numer summry for this dt set. 1) Find the i rnge ii IQR for the dt set. Construt ox-nd-whisker plot for the dt set.

29 43 Chinwe worked in the sles nd mrketing deprtment for pulisher. She reorded the numer of interntionl telephone lls she mde eh work dy for month, s follows: Numer of lls Frequeny Find the five-numer summry for this dt set. Find the i rnge ii IQR for the dt set. Construt ox-nd-whisker plot for the dt set. 47 A survey ws rried out on the witing time in supermrket queue on Fridy fternoon t 4.00 pm. The results re s shown in the tle longside. Cn we urtely give the longest witing time? Drw histogrm of the dt. Wht is the length of eh lss intervl? d Wht is the modl lss? e f Comment on the shpe of the distriution. Add further olumns to help find the pproximte men. 48 The speeds of 100 rs pssing point in n 80 km/h limit zone were reorded s follows: Cn we urtely give the highest speed? Drw histogrm of the dt. Wht is the length of eh lss intervl? d Wht is the modl lss? e Add further olumns to help find the pproximte men. f Comment on the shpe of the distriution. Time (seonds) Frequeny 0 - < < < < < Speed (km/h) 55 - < < < < < < < < 95 Frequeny The following tle gives the height groups of sketll plyers in legue. Drw umultive frequeny grph of the dt nd use it to estimte: the medin height of the plyers the numer of plyers shorter thn 183 m the height elow whih lie the 4 shortest plyers. 50 The following tle gives the lp times (in seonds) of 40 ring r drivers. Drw umultive frequeny grph of the dt nd use it to estimte: the medin lp time the numer of drivers whose time ws less thn 5 seonds the time in whih the fstest 10 drivers ompleted the lp. 51 The following tle gives the sores of n exmintion hieved y group of students: Drw umultive frequeny grph of the dt nd use it to estimte: the medin exmintion mrk the numer of students who sored less thn 63 mrks the perentge of students who sored distintion, given tht the distintion mrk ws 75. Frequeny 165 x < x < x < x < x < x < x < 00 7 Time (seonds) Frequeny 30 t < t < < t < t < t < 60 1 Sore 0 x < 30 x < 40 x < 50 x < 60 x < 70 x < 80 x < 90 x < Frequeny

30 STATISTICS SOLUTIONS 7 Contents of nket 1 Height my tke ny vlue, not just the integers 170 to 05 m. 15 Heights of footll squd Numer of hildren in fmilies 30 frequeny height (m) < 190 m 13 Age n tke ny vlue, not just the integers 1 to 0 yers I I I I numer of hildren Stem Lef Stem Lef 318 mens mens 38 Stem Lef Stem Lef I 8 mens 7.8 The modl weight ws etween 5.0 kg nd 6.0 kg. 11 Stem Lef mens ge (yers) 14 - < 15 yers =80 i.e., Josef grew 80 pumpkins Perentge weighing more thn 4 kg x100% 80 = x 100% 80 =70% Perentge weighing etween kg nd 8 kg x100% 80 = 80 x100% =90% =5 i.e., there were 5 employees Perentge erning more thn $50 000/yer = 5 x 100% = x100% 5 = 0% Perentge erning etween $30 000/yer nd $40 000/yer = A x 100% =3% =116 i.e., there were 116 students Perentge who hit from 00 m up to 10 m = x 100% 116 *.4% Perentge who hit the ll 00 m or more x100% 116 x 100% 116 * 81.0%

31 sum of sores 17 men = 1 numer of sores = medin 5th i of nd 6th sores} =.5 sum of sores 18 men = numer of sores = medin = 4 {middle sore} No. of mthes Totl Freq Produt sum of ll dt vlues i men = numer of dt vlues = * 49.7 ii medin is verge of 5th nd 6th sores =50 These results tend to support the mnufturer's lim. 19 men verge monthly rinfll * 79.6 mm per month 0 $ $ men = 9 $ * $ The ordered dt set is: , , , 19000, , , 03000, $ , medin = $ {middle vlue} The men nd medin re lmost the sme whih mens tht oth re relile mesures of the entre of the distriution of this dt set. $ $ $ men = 10 = $ $ $ medin = = $ The new dt vlue of $ is n outlier nd ffets the men only. The medin gives etter representtion of the selling prie in this se s the medin is not ffeted y outliers. No. of lollies Freq. Produt Totl sum of ll dt vlues i men = numer of dt vlues = * 6. ii medin is verge of 9th nd 30th sores 6 +6 =6 These results suggest tht the mnufturer's lim is inorret. The pkets ontin 6 lollies would e more urte.

32 3 Mss (kg) 1 - < - < < < < < < < 9 Totl Frequeny Midpoint of intervl sum of ll dt vlues i men * numer of dt vlues kg ii modl lss is 5 - < 6 kg Produt th sore + 31st sore iii medin = {7 vlues re <5 16 nd = 30.5} * 5. kg Niko's pumpkins usully weigh 4 kg or more, sine the men * 5 kg (from i) nd the medin is * 5. kg (from In ft, 43 out of 60 weih 4 k or more No. of gols Freq. Produt Totl no. of gols The dt is positively skewed. sum of ll dt vlues C i men = numer of dt vlues = =, ii medin is the 15th dt vlue = 1 (the 8th through 16th dt vlues re ll 1) d The men is the most suitle mesure of entre in this se euse the men tkes into ount the skewness of the dt. 9 11, 113,114; 117, 118,401 11, 13,441 15, 15 Qi medin Q3 medin = 6th sore = 10 lower qurtile, Q i = 114 upper qurtile, Q 3 = 14 d interqurtile rnge = Q3 - = = , , 4.3, 4.8, 1.4.8;( , 5.1, 5., 5., 5., The medin is the verge of the 6th nd 7th sores medin = 9.9 = 4.95 As the medin is not dt vlue we split the originl dt into two equl groups of , 4.1,14.3, 4.8; 4.8, Qi = = =4.55 C 5.0, 5.1, 5., 5.; 5., 5.6 Q3 = 5. d interqurtile rnge = Q3 - Qi = = The ordered dt set is: 0, 0, 0, 0, 0, 0.8, 1., 1.3, 1.3, 1.5, 1.5,11.6,.5,.7,.8,.9, 3.0, 3.1, 3.1, 3.1, 3.6, 4.7, 5.5 (4 sores) n As n = 4 =--= 1.5 1th sore + 13th sore medin = = 1.95 minutes As the medin is not dt vlue we split the originl dt into two equl groups of 1. 0, 0, 0, 0, 0,10.8, 1.; 1.3, 1.3, 1.5, 1.5, 1.6.3, 4.7, Qi=.5,.7,.8, 5.5 t = = 1.0 minutes , 3.1, 3.6, 6.1 = = 3.05 minutes Rnge of the witing times = = 5.5 minutes Interqurtile rnge = =.05 minutes i 50% of the witing times were greter thn 1.95 minutes. ii 75% of the witing times were less thn 3.05 minutes. iii The minimum witing time ws 0 minutes nd the mximum witing time ws 5.5 minutes. The witing times were spred over 5.5 min.

33 n Sine n = zz, - = - = th sore + 1th sore medin = = 5.15 d As the medin is not dt vlue, we split the originl dt into two equl groups of 11. So, Qi is the 6th sore = 4.5 e Q3 is the 17th sore = 5.9 I rnge = mx. vlue - min. vlue = =4.1 g interqurtile rnge = Q3 - = = The or ered stemplot is: d e f g Stem Lef mens n Sine n = 3, = 1 medin = 1th sore = 3.3 Qi is the 6th sore =.6 Q3 is the 18th sore = 4. rnge = mx. vlue - min. vlue = =4. interqurtile rnge = Q - Qi = = n Sine n = 36 = = 18.5 ' 18th sore + 19th sore medin = d e I g 38 d e I 9th sore + 10th sore d Qi= 7+ 7 =7 e Q3 = = th sore + 8th sore = 39.5 f rnge = mx. vlue - min. vlue = 50-9 = 41 g interqurtile rnge = Q3 - Q1 = = 1.5 i 66 ii rnge = = 53 Perentge soring 43 points or more = 50% Interqurtile rnge = Q -Q = 55-5 = 30 The lowest 5% of the sores were etween 13 nd 5. Yes, sine the medin sore is The sterisk represents n outlier, mrk of 5 out of 50. The middle 50% of the lss sored mrk etween 30 nd % of the students sored mrk greter thn or equl to 38. rnge = 50-5 = i 49 ii rnge = mx. mrk - min. mrk = =31 d e 75% of students sored 30 or more IQR = Q3 - Q1 = = 11 f The top 5% of students sored mrk etween 41 nd 49. g Yes, sine the medin mrk is 34.

34 41 The ordered dt set is 0, 1,, 3, 3, 3, 3,1,4; 4, 4, 4, 4, 4, 4,( , 5, 6, 6, 6, 7,Ct 7, 8, 8, 8, 9, 11, 1 (30 sores) medin = = 5, Qi = 4, Q3 = 7 IQR = Q3 = 7 4 = 3 I I.i spelling mistkes 47 No, s we do not hve ext dt vlues, only dt intervls. Witing time in supermrket queue r)-' 30 A 1=1 5,11= ii 4 min. vlue = 3, mx. vlue = 30 Totl numer of sores is 49 medin is the 5th sore i.e., medin = 6 {there re 7 sores from 3 to 6} 1th + 13th sore Qi = = = 5 37th + 38th sore Q3 =7 = i rnge = mx. vlue min. vlue = 30 3 =7 ii IQR = Q3 Qt = 7 5 = numer of hooltes 43 min. vlue = 0, mx. vlue = 15 Totl numer of sores is 11th sore + 1th sore medin = =4 Qi = 6th sore = 3, Q3 = 17th sore = 5 i rnge = mx. vlue min. vlue = 15-0 =15 ii IQR Q3 Q1 = 5 3 = I' 1. 11' P- r) numer of lls 0 seonds d 40 - < 60 seonds Time (s) Freq. Midpoint of in Produt 0 - < < < < < Totl GO estimted men * 60 * 49 seonds f The distriution is pproximtely symmetril. 48 No, s we do not hve ext dt vlues, only dt intervls. >, 60. i Fl Speed of rs speed (km/h) 5 km/h d 75 - < 80 km/h Speed (km/h) Freq. Midpt. of int. Produt 55 - < < < < < < < < Totl estimted men 7810= 78.1 km/h 100 f The distriution is slightly negtively skewed.

35 umultive frequeny umultive frequeny I () Fd! M height (m) There re 91 plyers in totl. the medin is the 46th height. Reding from the grph, the medin * 185 m From the grph, pproximtely 37 plyers re shorter thn 183 m. From the grph, the shortest 4 plyers re elow pproximtely 169 m A umuh five frequeny mrks There re 17 students in totl. the medin is the 64th sore. Reding from the grph, the medin * 67 mrks. From the grph, pproximtely 5 students sored less thn 63 mrks. From the grph, 87 students sored less thn 75 mrks = 40 students sored more thn 75 mrks. the perentge of students who sored distintion* 14 7 x 100%.* 31.5%. 70 umultive frelueny v I I time (seonds) The medin is the verge of the 0th nd the 1st times. Reding from the grph, the medin * 4.5 seonds. From the grph, pproximtely 36 drivers hd times less thn 5 seonds. From the grph, the fstest 10 drivers ompleted the lp in pproximtely 38 seonds or less speed (km/h) There re 59 rs in totl the medin is the 30th dt vlue. Reding from the grph, the medin * 63 km/h From the grph, pproximtely 7 rs were driving slower thn 55 km/h. From the grph, pproximtely 40 rs were driving slower thn 65 km/h = 19 rs were driving fster thn 65 km/h.