answers

ANSWERS

467

Answers

CHAPTER 1

Exercise 1.1

1

P

030

,

Q

070

,

R

120

,

S

165

,

T

217

,

U

260

,

V

319

2

AB

054

,

BC

118

,

CD

193

,

DE

292

,

EF

081

,

FG

340

3 a

298

24

M

b

299

44

M

c

301

28

M

d

302

24

M

4 a

236

10

G, 231

14

M

b

080

48

G, 075

52

M

c

303

37

G, 298

41

M

d

189

09

G, 184

13

M

e

349

20

G, 344

24

M

5 a

358

52

G, 356

57

T

b

105

05

G, 103

10

T

c

064

40

G, 061

45

T

d

290

28

G, 288

33

T

e

273

23

G, 271

28

T

6 a

344

41

T, 336

04

M

b

030

12

T, 021

35

M

c

125

54

T, 117

17

M

d

238

33

T, 229

56

M

e

199

29

T, 190

52

M

Exercise 1.2

1 a

120 n.m.

b

788 n.m.

c

328 n.m.

d

471 n.m.

2 a

444 km

b

181 km

c

1304 km

d

895 km

3 a

259 n.m. b 334 n.m.c 135 n.m. d 310 n.m.

4 a 203 km b 316 kmc 1117 km d 242 km

5 a 147 n.m. b 272 km6 a i 375 n.m. ii 695 km

b i 6.5 kn ii 12 km/h7 a 3600 km b 1944 n.m.

c 2436N, 23E8 15 h 26 min9 a 84 n.m. b 112 n.m. c 76 n.m.

d 136 n.m. e 47 n.m.10 a 3 n.m. b 4.3 n.m. c 3.1 n.m.

d 3.4 n.m. e 5.7 n.m.11 a 118 n.m. b 143 n.m. c 113 n.m.

d 48 n.m. e 48 n.m.12 a 9.5 km b 7.3 km c 5 km

d 4.6 km e 5.9 km

13 a 0.75 km b 2 kmc 2.25 km d 4.25 km

14 a 213 b 049 c 338d 279 e 252

15 a 050 b 169 c 137 d 17316 a 254 b 220 c 339

d 153 e 14317 a 256 b 155 c 091

d 312 e 02018 a 229 b 116 c 305 d 080

Exercise 1.31 a Jeffreys Rocks (2155S, 15013.7E)

b Normanby Rock (2140.8S, 15012.8E)c Sail Rock (2148.2S, 15009.3E) d Howard Point (2137.4S, 15014.9E)e Hixson Islet (2144.9S, 15017.3E)

2 a Nobbys Head (3255S, 15148E)b Stockton Bight (3251S, 15158E) c Norah Head (3317S, 15134.5E)d Barrenjoey Head (3335S, 15119.5E) e Long Point (3344.8S, 15119E)

3 a Kaipara Harbour (3625S, 17407E)b Raglan Harbour (3748S, 17452E) c Three Kings Islands (3411S, 17205E)d New Plymouth (3902S, 17412E) e Manukau Harbour (3703S, 17431E)

Chapter review1 A bearing is the clockwise angle between

north and the direction of a line. It is normally stated as a three-digit number.

2 The magnetic variation is the angle between true north and magnetic north. An easterly variation means that magnetic north is east of true north.

3 Grid convergence is the angle between true north and grid north.

4 The magnetic variation changes in a fairly regular fashion. The change in one year is printed on the map, together with the variation for a particular year. The magnetic variation for another year is calculated from the change in one year and the number of years since the date printed on the map.

5 A westerly change of magnetic variation will increase the magnetic bearing of an object.

6 A nautical mile is equal to the distance travelled by moving through 1 on a great circle such as a meridian.

NQM12Ach.ans.fm Page 467 Thursday, October 3, 2002 12:55 PM

468 NEW QMATHS 12A

7 The dividers are opened to the distance between the objects on the chart. The distance is measured using the number of minutes on the vertical scale at the same latitude as the objects to obtain the distance in nautical miles.

8 The ruler is placed so that it is in the direction of the line and moved across to the compass rose printed on the chart to read the bearing.

9 Dead reckoning is the establishment of position by plotting or calculating the course of a boat using speed, time and direction from the original position of the boat, taking currents and winds into account wherever possible.

10 A x is the establishment of ones position at sea.

11 Degrees and minutes are used. Distance (in nautical miles) travelled along a meridian is equal to the change of angle, measured in minutes. Distance (in nautical miles) travelled along a line of latitude is equal to the cosine of the latitude multiplied by the change of angle, measured in minutes

12 A 020, B 060, C 085, D 133, E 176, F 220, G 265, H 300, I 338

13 a 34458 M b 34340 Mc 34210 M d 34540 M

14 a 300 n.m., 556 km b 107 n.m., 198 kmc 432 n.m., 800 km d 276 n.m., 511 km

15 a 333 n.m., 618 km b 191 n.m., 353 kmc 62 n.m., 114 km d 183 n.m., 338 km

16 a 12352 G, 11234 Mb 22618 G, 21500 Mc 00641 G, 35523 M d 07623 G, 06505 Me 358.0 G, 346.7 Mf 287.7 G, 276.4 Mg 33936 T, 32902 Mh 20001 T, 18927 Mi 04205 T, 03131 M j 35908 T, 34834 Mk 136.8 T, 126.233 Ml 001.1 T, 350.533 M

17 a 176.4 T, 185.3 Mb 014.7 T, 023.6 Mc 039.2 T, 048.1 Md 230.3 T, 239.2 Me 34510 T, 35404 Mf 10323 T, 11217 Mg 147.8 G, 147.6 Th 302.4 G, 302.2 Ti 034.6 G, 034.4 Tj 116.1 G, 115.9 Tk 22356 G, 22344 Tl 05931 G, 05919 T

18 a 8.3 kn b 15.4 km/h19 a 225 n.m. b 60S, 137 30E20 a 3.6 n.m. at 217 b 8.5 n.m. at 052

c 8 n.m. at 032 d 5.8 n.m. at 20421 a 15 n.m. at 032 b 15.4 n.m. at 131

c 9.8 n.m. at 256 d 9 n.m. at 29922 a Esmeralda Cove (3237.4S, 15219.6E)

b Little Island (3242.1S, 15214.6E)c Hawks Nest (3240.1S, 15211.4E)

CHAPTER 2For answers that depend on single measurements, a variation of about 1 mm or 1 is acceptable.For answers that depend on multiple measurements, a variation of about 3 mm or 3 is acceptable.

Exercise 2.11 a 1.05 m b 0.83 m

c 0.93 m d 0.88 m2 a 27.2 m b 16.8 m

c 70.4 m d 550.4 m3 a 875 m2 b 829 m2

c 419 m2 d 2408 m2 4 a Bearings: AB 131, BC 026, CD 175,

DE 097b Back bearings: AB 311, BC 206,

CD 355, DE 2775 a 286 b 041 c 189

d 132 e 2746 a 2.871 m b 4.7 c 1 : 12.27 5.1 cm8 a 1.904 m b 4.53 c 5.313 m

d 6.73 e 9.249 The offer isnt fair. She loses about 10.6%

of the land but the price is reduced by only 8.8%.

10 Yes, the slope is just less than 5 (4.987).11 Answers will vary. 12 1.629 m

Exercise 2.21 a 2443.46 m2 b 2743.84 m2

c 3634.74 m2 d 5225.89 m2 2 1440 m2 3 158 297 m2 4 a 19 000 m2 b 6980 m2 5 13.353 ha (133 526.7 m2)6 2000 m2

Exercise 2.31 101.1 m 2 148.6 m3 41.9 m 4 18.8 m5 59.2 m


ANSWERS 469

Exercise 2.41

2

3 a Similar to sketch in question.b

c 900 m2 d 294.6 m2 e 12 711 m2

A

B

C

D

E

Hedge

Stream

Track

RoadScale 1 : 1000Yours should betwice this size.

Sports

Boating

A B

C

complex

lake

Scale 1 : 20 000You should work at 1 : 10 000.

Fence

Stream

P

Q

RS

T

U

Factorystoragearea

Factory

Path

Roa

d

Scale 1 : 2000Yours should be twice this size.


470 NEW QMATHS 12A

4

Exercise 2.51 a

b TP = 160 m at 2602

DA 800 m (798 m)3 a No local attraction on leg AB

Dam

Creek

CrossingFence

Dairy Shed

House

A

B

C

D E

Bou

ndar

y

RoadScale 1 : 20 000 You should work at 1 : 10 000.

Scru

b ar

eaN

P

Q

RS

T

58 m

39 m

72 m55

m

22

134

101 53

Scale 1: 2000Yours should be twice this size.

N

A

B

C

D

452 m

450 m

400 m

30

80160

Scale 1: 10 000You should work at 1: 5000.

Factory

b Original Corrections Corrected

Leg Forward bearingBack

bearing Difference Forward BackForward bearing

Back bearing

AB 052 232 180 0 0 052 232BC 127 305 178 0 +2 127 307CD 084 267 183 +2 1 086 266DE 221 043 178 1 3 220 040EF 119 297 178 3 1 116 296FG 042 220 178 1 +1 041 221


ANSWERS 471

c

d GA = 147 km at 095

N

52

127

86220

116

41

30 km

28 km 42 km

46 km

53 km

62 km

A

B

C

D

E

F

G

Scale 1: 2 000 000You should work at 1: 1 000 000.

4

N

77 m

107 m

114 m

101 m

A1

B1 C1

D1A

B C

D

Scale 1: 4000Yours should be 4 times this size.

6 a

b QR = 215 m; RS = 490 m

979 m

172

m

150 m

5

168

185

347

320

14348 P

Q

R

S

Scale 1:10 000You should work at 1: 5000.

T

8

7 Leg Bearing Back bearingAB 064 244 BC 100 280CD 229 049DE 271 091EA 305 125

N

167 m 125 m

103 m181 m

124 mB

C

D

E

A

A1

B1

C1

D1

E1


Local attractions at C, E, F and A.

5 Original Corrections Corrected

Leg Forward bearingBack

bearing Difference Forward BackForward bearing

Back bearing

AB 144 324 180 0 0 144 324BC 070 253 183 0 3 070 250CD 017 194 177 3 0 014 194DE 319 138 181 0 +1 319 139EF 228 050 178 +1 1 229 049FA 278 098 180 1 1 277 097


472 NEW QMATHS 12A

Exercise 2.61

2 a 177.5 mb AQ 076, BP 298, BR 256c AR = 45 m, RB = 155 m, BQ = 52.5 m

3 a 1 : 2000 b 642 mc XB 047, XD 195, YC 099, YA 294

4 a 1 : 4000 c 1256 mb

Chapter review1 Booking a survey refers to the practice of

recording details of a survey into a eld book.2 A datum point provides a point of reference.

Distances and directions can be calculated relative to the datum point.

3 a m2 b ha or m2 depending on size of the school

4 Triangulation involves dividing a large area of land into triangles and completing the survey by chaining the lengths of the sides of the triangles.

5 Begin at a known point, then select another point (station) along the traverse path. Chain and measure the direction from the starting point to the rst station. This is the rst leg of the traverse. Repeat this procedure for as many legs as are required to reach the end point of the traverse.

6 It is most likely to be due to local attraction. It would be unlikely that the error could be

83 m 06

1

59 m 09722

m

194

24 m 273

27 m 328

N

Childrens play area

Tree

Bench

Hut

Barbecue

Swings


X

NU V

W

X

149239

191

309

228 m

264 m

300 m

464 mFarmland

11


consistently made if it were due to faulty instrumentation or poor technique.

7 It is suitable to use the radiation method in situations where a single sighting station is sufcient to view all the required points and it is convenient to chain to all required points from a single station.

8 a 20 m (20.27 m) b 225 m (224.6 m)9 a 836.9 m2 b 2476.0 m2

11 a 023 b 291 c 152 d 20312 a 4.3781 m b 5.42 c 1 : 10.5313 20.25 cm14 a 4635 m2 b 3267.43 m2 15 a

b 28 183.6 m2 16 a i ii 1216 m2

10 Leg Bearing Back bearingOA 034 214OB 109 289OC 202 022OD 255 075OE 341 161

AB

79.2 m

117 m

81.9 m123.3 m

72.9 m

26.1 m


20.7 m

Scale 1: 1000


ANSWERS 473

b i ii 50 520 m2

17 108 m18 a

b Answers will vary depending on how the outer boundary is drawn. Area is approximately 19 000 m2.

19

Scale 1: 10 000

102 m

122

m

162 m

201 mP Q

RS

Scale 1: 4000You should work at 1: 2000.

118 m

170 m

104 m

98 m

Jetty Boat

Tree

Flag

Fence

A B

CD

E


shed

pole

143 m

20

21

23


D

N

CB

A

E

F

YX

100 m

Dam

WindmillPine

B

A

N


tree

116

m


P

Q

R

S

N

5084

m

Field

A

B

b Scale 1 : 3460, perimeter about 768.1 m

22 a Station Bearing from ABearing from B

P 050 021Q 100 116R 148 195S 258 264T 339 314


474 NEW QMATHS 12A

CHAPTER 3Exercise 3.1

1 a 93.75% b 57.6% c 275%d 23.33% e 60% f 7.5%g 146% h 85% i 80%

2 a 15.79% b 18.75%3 a $1180 b 56.78% c 264.29%4 a 18.18% b 81.82% c 22.22%5 a $34.50 b $103.18 c $1341.30

d $31.20 e $12.75 f $285.20g $86.27 h $156.65 i $21.35j $50.27

6 $5760 7 $105 8 $2.249 $942.02 10 $4550 11 $400

12 $600

Exercise 3.21 4.5% 2 20% 3 5.33% 4 13% 5 4.53% 6 19%, then 22%

Exercise 3.31 a $437.50 b $480 c $414

d $412.50 e $27372 a $3600 b $1507.50 c $203

d $11.94 e $1043.013 $1775.34, $46 775.344 $507.21 5 $75 394.52 6 $94087 $1627.40, $41 627.408 $579 $279 10 $3312, $11 312

11 6.5% 12 $197 826.0913 a 6.58% b $115.50 (on $7300)14 4.78% 15 $64 285.71 16 21.29% 17 $1 248 00018 a $106.88 b $8066.04

Exercise 3.4a $6856.95, $856.95b $8509.59, $209.59c $13 349.44, $1349.44d $10 105.13, $2605.13e $11 643.97, $2243.97f $1621.49, $121.49g $9241.35, $1091.35h $5802.23, $202.23

Exercise 3.51 a $18 161.18, $3161.18

b $11 001.82, $2051.82c $5146.53, $646.53

d $4353.65, $3813.65e $15 744.67, $8204.67f $2921.72, $421.72g $9134.55, $3334.55h $7401.84, $1401.84i $4370.34, $120.34 (182 days)j $63 991.78, $38 991.78

2 $3924.56 3 $3073.58 4 8.14%5 8.16% 6 3 years 7 357 days 8 10 years 9 5.79% 10 $8263.97

11 14 years and 18 fortnights

Exercise 3.61 7.76% 2 6.93% 3 7.82%, 7.76%, 7.68%; 7.6% compounded

quarterly is best.4 8.77%, 8.62%, 8.44%; 8.5% compounded

quarterly is best.5 a 6.43% b 5.15% c 7.27% d 8.48%6 8.34%, 8.27%; 8.04% compounded monthly

is best.

Chapter review1 Divide the prot by the buying price and

multiply by 100%.2 Interest is the amount of money charged,

while the interest rate is the ratio of the interest to the principal.

3 A nominal interest rate is the pro-rata rate for 1 year.

4 Simple interest is charged on the initial amount borrowed, while compound interest is charged on the accumulated amount of principal plus interest.

5 It is the period over which interest is calculated.

6 a 56% b 134% c 250% d 3.6%7 a 16% b 116%8 a $56.35 b $8056 c $4.149 $5074 10 $72.25 11 $600

12 $20 000 13 18% 14 6.96%15 I = Pin 16 $390017 a $1200 b $756 c $1534.5018 $46 294.52 19 $175.23 20 3.6%21 A = P = P

22 $12 328.75, $4528.7523 a $6553.98, $1553.98

b $13 854.63, $1854.63c $11 875.73, $4375.73

24 a 7.76% b 14.34% c 8.67%25 4.3% 26 $3499.29 27 2 years 37 weeks 28 6.79%29 $31 327.88

1 ik--+ kt 1 ik--+

n


ANSWERS 475


1 a Mean 16.6, mode = 19, median = 17, range = 8, interquartile range = 5, SD 2.64

b Mean 51.8, mode = 47, median = 49, range = 18, interquartile range = 10.5, SD 5.89

c Mean 5.8, mode = 6, median = 6, range = 7, interquartile range = 3.5, SD 2.15

2 a Mean 11.2, mode = 11, median = 11, range = 9, interquartile range = 3, SD 2.26

b Mean 4.2, modes = 4 and 5, median = 4, range = 10, interquartile range = 3, SD 2.26

c Mean 30.2, mode = 31, median = 30, range = 8, interquartile range = 3, SD 1.99

3 a Mean 22.7, mode = 23, median = 23, range = 8, interquartile range = 3.5, SD 2.14

b Mean 44.1, mode = 44, median = 44, range = 8, interquartile range = 4, SD 2.32

c Mean 26.4, mode = 26, median = 26, range = 10, interquartile range = 3, SD 2.38

d Mean 8.23, mode = 8, median = 8, range = 8, interquartile range = 2, SD 1.74

4 a Mean 22.5, modal class = 2529, median 23.6, range = 25, interquartile range 7.9, SD 5.06

b Mean 51.3, modal class = 5059, median 52.4, range = 60, interquartile range 17.6, SD 12.97

c Mean 114.8, modal class = 110114, median 113.3, range = 25, interquartile range 10.3, SD 6.52

d Mean 145.2, modal class = 150169, median 152.7, range = 120, interquartile range 38.6, SD 27.86

5 a

Median 170, interquartile range 9.5

Cum

ulat

ive

freq

uenc

y

Height (cm)

100

80

60

40

20

0140 150 160 170 180 190

b

Median 69.5, interquartile range 106 a 52.4 g b From 49.4 g to 55.1 g7 a 2 b 3.248 17.7%

Exercise 4.21 a Supermarket shoppers; smallest, average

and largest amounts spent; smallest = $4.30, average = $87.64 and largest = $198.75

b People who watch TV; audience of each channel; Channel 9 = 40%, Channel 7 = 30%, Channel 10 = 16%, Channel 2 = 14%

c Cars; colour of cars; white = 38%, yellow = 19%, blue = 11%, red = 15%, others = 17%

d Tea-time sales of fast-foods; amount of each type; hamburgers = 31%, shnchips = 12%, salad rolls = 6%, mini pizzas = 17%, sausage rolls = 22%, pies = 11%

e Bikes ridden to schools; type of frame, type of bike; chromolly frames = 67%, mountain bikes are the most popular.

2 Choose 17, 29, 41, 3.3 a 9, 5, 14, 48, 43, 18, 26, 41, 37, 49

b 1545, 1808, 1040, 1393, 1148, 1340, 1068, 1602, 1729, 1535

c 417, 301, 251, 490, 668, 314, 349, 561, 632, 286

d 62 739, 99 482, 55 380, 51 038, 55 694,88 904, 98 147, 99 823, 57 536, 65 237

4 a 2 Year 8s, 3 Year 9s, 2 Year 10s, 1 Year 11, 2 Year 12s

b 5 sales reps, 1 manager, 6 clerical staff, 3 stores staff

c 3 surfers, 3 boogie-boarders, 2 sailboarders5 Difcult to contact all members of a

population, costly, time-consuming, 6 B 7 A8 A

Cum

ulat

ive

freq

uenc

y

Pulse (beats/min)

100

80

60

40

20

040 50 60 70 80 90


476 NEW QMATHS 12A

9 Samples are as in the table below. Note that when sex is disregarded, the number is not necessarily the total of the samples when sex is taken into account. Rounding errors give samples of 499.

10 Biasedonly those with telephones involved; poll conducted after emotive story; people can vote more than once.

11 a Student body at Grifth University in a particular year

b The number of students who have part-time jobs

12 Items 127, 135, 90, 149, 94, 105, 122, 97, 142, 87

13 a Examples include:i Use a random number table to select 10

page numbers. ii Determine the number of names on a

page and use a random number table to select 10 numbers up to that number of names; start at the top left corner of the page and work down the column.

iii Select 10 pages as in i, then select one name on each of the selected pages as in ii.

iv Count how many Wilsons there are in the book and select 10 of them using a method as in ii.

b Answers will vary.14 a Fair and unbiased, but people can be

selected more than once.b Fair and unbiased, but fairly inefcient as

many selections would probably have no person seated in that position.

Exercise 4.31 a 25 b 20 c None d 37, 512 a 4.9 b 197 c 22.1 d None3 a 5, 9

b Mean 1.9 , median = 2c Mean = 1.5 , median = 1.5d Both the mean and the median are affected

by the outliers.

State or territory

a By sex b Withoutregard to sex

(persons)Females Males

NSW 84 85 169Vic. 62 63 124Qld 46 46 93SA 19 20 39WA 25 24 49Tas. 6 6 12NT 3 2 5ACT 4 4 8

4 a 124 kgb Mean 71.4 kg, median = 71 kgc Mean 68.9 kg , median = 70 kgd Both the mean and the median are affected

by the outliers.

Exercise 4.41 The results in Chemistry are a little higher,

but more spread out. Most of these students nd Chemistry easier than Physics.

2 The humidity is generally higher in Town B, so it probably gets more storms.

3 The rst is harder, but has greater variation in sentence length.

4 Their sales performances are about the same, but Martina is more consistent.

5 Kathy generally scores lower than Jessica.6 Route B is better, because it has the same

mean but is more consistent, so there is less chance of being very late.

7 She did better in Film and TV as she was 1.13 standard deviations above the mean, whereas in Speech and Drama she was only 0.65 standard deviations above the mean.

8 David is fatter than the population he is compared with, because, in standard deviation units, his weight is further from the mean than his height.

Chapter review1 The median score item is calculated as the

(n + 1)th item. The score group that containsthis item is found. The item is calculated as a fraction of the number of items in the group. This fraction is then applied to the width of the group to obtain a score.

2 It measures the spread of the middle 50% of the data.

3 25% of the number of items is calculated. The position of this item on the graph is found and the corresponding score located.

4 A parameter is a measure of the population while a statistic is a measure from a sample.

5 A biased sample favours one part of the population but a fair sample does not.

6 Systematic sampling of households in the area would be cheap and give a reasonable result. Otherwise the council could use ABS data to perform a stratied random sample.

7 An outlier is a score that is markedly different from the majority of scores in a set of data.

8 By comparing the means and standard deviations (or other measures)

9 Mean = 15, mode = 14, median = 14.5, range = 10, interquartile range = 4, SD 2.81

12---


ANSWERS 477

10 Mean 21.3, modal class = 1519, median 21.3, range = 30, interquartile range 10.1, SD 6.48

11 a Mean 58.3, mode = 65, median = 59, range = 32, interquartile range = 12.5, SD 7.56

b Mean 70.7, SD 25.24, other statistics not reliable

12

Median 25.5, interquartile range 913 Population is people who buy takeaway food.

Parameter and statistics are the number of times they buy takeaways each week.

14 a 38, 58, 74, 67, 55b 262, 229, 346, 204, 215, 248, 343c 3328, 3905, 3983, 4249, 6902, 5777, 3117,

3307, 486515 a TV viewers in general

b TV viewers in Mackay who watched that news program, have their numbers in the book, and were interviewed by the market research company

16 The people on one page could all be of a particular group. For example, they might all have the family name Singh.

17 B 18 4819 He did slightly better in Maths.20 The mens results are more spread and lower

than the womens results.


1 $12 900 2 $1 124 0003 9.68% 4 10.79%5 a 8.78% b $48706 a 6 years b $20 000 c $33 000

d $28 000 e $5000

Scrap value is $1500.

7 Year Starting value

Depreciation during year

Written-down value

1 $7500 $1500 $60002 $6000 $1500 $45003 $4500 $1500 $30004 $3000 $1500 $1500

Cum

ulat

ive

freq

uenc

y

Score

50

40

30

20

10

05 15 20 30 3510 25 40

Useful life is 5 years.

Exercise 5.21 a $740 b 16.4%

2 a $10 000 b 28.57%

3 a $3920



Written-down value

1 $6000 $1200 $48002 $4800 $1200 $36003 $3600 $1200 $24004 $2400 $1200 $12005 $1200 $1200 $0

c Year Starting value


Written-down value

1 $4500 $740 $37602 $3760 $740 $30203 $3020 $740 $22804 $2280 $740 $15405 $1540 $740 $800

c Year Starting value


Written-down value

1 $35 000 $10 000 $25 0002 $25 000 $10 000 $15 0003 $15 000 $10 000 $5 000

b Year Starting value


Written-down value

1 $28 000 $3920 $24 0802 $24 080 $3920 $20 1603 $20 160 $3920 $16 2404 $16 240 $3920 $12 3205 $12 320 $3920 $8 4006 $8 400 $3920 $4 480

Boo

k va

lue

5000

4000

3000

2000

1000

0

Time (years)1 2 3 4 5

Book value of fryerdB

ook

valu

e

40 000

30 000

20 000

10 000

0

Time (years)1 2 3

Book value of card


478 NEW QMATHS 12A

c $4480

e Straight-line depreciation because the graph of the written-down value is a straight line.

4 a $5500

c $5500 d $3000

Value after 4 years is $5337.29.6 $3213.26 7 22.64%, $12 961.488 14.34% 9 a 34.13% b $806.23

Exercise 5.31 $531.47 2 $714.093 a 4846

b Answers will vary, depending on the year.4 a 72 m2 b 116 m2 c 80 days5 $15 463.68 6 $28 056.22 7 $14 527.288 $10 344.52 9 $15 362.65 10 $4495.69

11 $5291.43 12 $11 255.03 13 $7747.1014 $3577.30 15 $4906.76

Exercise 5.4

b Year Starting value


Written-down value

1 $27 500 $5500 $22 0002 $22 000 $5500 $16 5003 $16 500 $5500 $11 0004 $11 000 $5500 $5 500



Written-down value

1 $8900 $1068 $78322 $7832 $939.84 $6892.163 $6892.16 $827.06 $6065.104 $6065.10 $727.81 $5337.29

1 Year CPI Ination Year CPI Ination1986 73.5 1994 110.4 1.8%1987 80.4 9.4% 1995 113.9 3.2%1988 86.3 7.3% 1996 118.7 4.2%1989 92.6 7.3% 1997 120.3 1.3%1990 100.0 8.0% 1998 120.3 0%1991 105.3 5.3% 1999 121.8 1.2%1992 107.3 1.9% 2000 124.7 2.4%1993 108.4 1.0% 2001 132.2 6.0%

d

1 2 3 4 5 6

Boo

k va

lue

30 000

20 000

10 000

0

Time (years)

Book value of lathe2 a 45c in 2002

b i 10.36% ii 3.79%iii Answers will vary.

3 $95 034.72 4 $183.255 $5624.46, so a decrease in real terms6 4.64%

In real terms, greatest expenditure was in 2002.

8 252 14s 7d (or 252.729 788)9 a Prices will vary depending on the year,

which brand is chosen, etc.b and c Answers will vary for each item.d and e Answers will vary depending on

the year.f The statement does not appear to be true

because the increase in average male weekly earnings is greater than the increase in most items.

Exercise 5.51 $99 292.18 2 $18 842.783 a $52 500 b $22 611.34

c $14 695.824 a $36 000 b $9566.445 a $1632 b $8659.826 a $436 750 b $45 000

c $32 918.34d $2 164 576, $88 320, $68 914.87

respectively

Chapter review1 Appreciation is the increase in value of an

item over time.2 A business can claim the value of

depreciation as a tax deduction.3 For straight-line depreciation, the value of an

item is reduced by the same dollar amount each year. For the diminishing-value method, the value of an item is reduced by the same percentage of its value at the start of each year.

4 The value of money changes due to the effect of ination.

5 The rate of ination is calculated by nding the percentage change in the CPI.

6 Comparing values in real dollars means comparing values taking into account the effects of ination.

7 $14 200

7 Year 1998 1999 2000 2001 2002Amount(millions)

$16.71 $16.25 $18.43 $21.47 $22.94

2000 $s(millions)

$17.80 $16.77 $18.43 $20.80 $21.54


ANSWERS 479

8 a 4.12% b $23 700

10 20.83%

Scrap value is $1772.12 238 m2 13 $56 74114 $11 390.81 15 $12 731.0416 $17 961.06 17 $10 208.7518 $11 214.05 19 $3025.8620 $8210.73 21 $321 849.3022 $350 637.93 23 24.21%24 1970: $618 612.33, 1974: $660 856.71,

1979: $712 080, so the 1979 sale was the best real price.

25 a $27 000 b $63 000 c $40 430.0426 a $44 057.90 b $13 888.89 c $94 864.40


1 a 0.433 b 0.072 c 40d At least 22 e 9

2 a White 0.581, wholemeal 0.202, rye 0.079, specialty 0.138

b 0.202 c 40 d 116 e 163 a 0.204 b 0.170 c 0.2314 3575 a Taxi 0.06, train 0.337, bus 0.26, car 0.287,

bike 0.02, other 0.037b 0.287 c 7 d 8

6 a 200



Written-down value

1 $65 000 $7500 $57 5002 $57 500 $7500 $50 0003 $50 000 $7500 $42 5004 $42 500 $7500 $35 0005 $35 000 $7500 $27 5006 $27 500 $7500 $20 000



Written-down value

1 $5600 $1400 $42002 $4200 $1050 $31503 $3150 $787.50 $2362.504 $2362.50 $590.62 $1771.88

b Score 2 3 4 5Rel. freq. 0.12 0.1 0.075 0.08

Score 6 7 8 9Rel. freq. 0.065 0.105 0.085 0.09

Score 10 11 12Rel. freq. 0.11 0.11 0.06

c 40 d 4 or 5 e 16

b 29 c 2

b 6 (185 cm or more)

b 32 (less than 2 kg) c 22 (4 kg or more)

Exercise 6.21 a {1, 2, 3, 4, 5, 6}

b c d e 02 a {5R, 8G, 7B}

b c 0 d e Green

7 a Amount spent ($) Frequency Rel. freq.

0.001.99 10 14.3%

2.003.99 25 35.7%4.005.99 15 21.4%6.007.99 9 12.9%

8.009.99 5 7.1%10.0015.99 6 8.6%

Total 70

8 Rent per person($/week) Number Rel. freq.

6569 6 3%7074 10 5%7579 36 18%8084 62 31%8589 34 17%9094 25 12.5%9599 16 8%

100104 11 5.5%Total 200

9 a Height (cm) Frequency Rel. freq160169 7 10%

170174 13 18.6%

175179 23 32.9%180184 14 20%

185199 13 18.6%Total 70

10 a Mass (kg) Number Rel. freq.1.01.9 13 13%

2.02.4 18 18%

2.52.9 31 31%3.03.4 17 17%

3.53.9 12 12%4.04.4 9 9%

Total 100

16---

12---

12---

720------

25---


480 NEW QMATHS 12A

3 a {HH, HT, TH, TT}b c d e

4 a {AA, AB, AC, AD, AE, BA, , ED, EE}b c d e

5 a {6V, 5H}b c d e 1

6

a b c d

e f g

78

P(3 girls) = 9 a b c

10 a b11 a

b c

14---

14---

14---

12---

925------

125------

825------

1625------

611------

511------

611------

S = SSH = SHD = SDC = SCS = HSH = HHD = HDC = HCS = DSH = DHD = DDC = DCS = CSH = CHD = CDC = CC

S

H

D

C

116------

14---

18---

116------

14---

34---

916------

56---

G = GGGB = GGBG = GBGB = GBBG = BGGB = BGBG = BBGB = BBB

G

B

G

B

G

B

18---

320------

310------

920------

1140------

38---

eit

eit

eit

eit

eit

eit

eit

eit

v

am

h

m

v

am

h

m

r

f

14---

116------

12

13 Answers will varycheck with your teacher.14

The placement of b will depend on how the Broncos are doing.

15 a {red, green, pink, orange, blue}b c

16 P(A) = , P(B) = , P(C) = 17 a i ii

b

Exercise 6.31 a

b 0.12 = 12% c 72 a

b 0.673 a

b 0.174 a i 2.3% ii 18.9%

b i About 51 (21.2%)ii About 134 (55.8%)

5 a i 0.25 ii 0.14b i 17 or 18 ii 13

0 10.5

c d e a b

0 10.5

d f e a c

15---

25---

47---

27---

17---

38---

58---

A B

x

z

y

Prob

abili

ty 0.250.20.150.10.05

00 4

Number of siblings7

0.30.35

1 2 3 5 6 8

Prob

abili

ty 0.250.20.150.10.05

00 4

Heads in 6 tosses5

0.30.35

1 2 3 6

Prob

abili

ty 0.250.20.150.10.05

025 29

Maximum temperature (C)32

0.3

26 27 28 30 31 33


ANSWERS 481

b

b

c 0.13 d 8

b About 11% (over 185 cm)c 6 (under 156 cm)

Chapter review1 Relative frequency is the experimental

probability of an outcome, calculated as

2 A certain outcome always occurs, so has a probability of 1.

3 The expected frequency of an outcome is calculated by multiplying the probability by the number of trials.

4 Experimental probabilities derived from frequencies less than 5 are regarded as unreliable.

6 a Result 1 2 3 4 5 6Frequency 7 12 9 12 12 8

7 a Result 35 68 911 1214 1519Number 9 8 8 17 8

8 a Height (cm) 141155 156160 161165Frequency 10 8 7

Height (cm) 166170 171175 176180Frequency 8 14 13

Height (cm) 181185 186195Frequency 11 9

Prob

abili

ty

0.250.20.150.10.05

01 5

Result of rolling a die2 3 4 6

Prob

abili

ty 0.10.080.060.040.02

03

Test result

0.12

5 7 9 1911 13 15 17

Prob

abili

ty

0.0250.020.0150.010.005

0

Height (cm)

0.030.035

141

0.04

146

151

156

161

166

171

176

181

186

191

196

frequency of outcomenumber of trials---------------------------------------------------- .

5 A one-way table is a relative frequency table.6 Experimental probability is based on the

frequencies of actual occurrences but theoretical probability is based on a list of possible occurrences.

7 A fair die is one that has an equal chance of landing on each face.

8 The sample space for a probability situation is the list of possible outcomes.

9 The probability of an event is calculated using the number of outcomes in the sample space. Writing n(event) as the number of outcomes for the event, the probability of an event is given by P(event) =

10 Random selection is the choice of items from a group such that each item has an equal chance of being chosen.

11 A tree diagram shows each stage of a probability situation as branches from a common point, giving rise to a tree-like structure.

12 The probability of event A and its complement add up to 1, so the probability that A does not occur is

13 If the coin is fair, the probability that the next toss will give a head is It is not affected by previous results.

14 A probability distribution is a histogram (column graph) that shows the probabilities of individual scores.

15 The heights of the columns of a frequency histogram are divided by the total frequency and the column widths to obtain the heights of the corresponding columns of the resulting probability histogram.

16 The total area is 1, reecting the fact that one of the possible outcomes must occur.

17 D 18 A 19 D

b 13321 {3P, 5G, 2O} 22 E 23 C24 D 25 B 26

20 a Amount spent ($) 2024 2529 3034Number 24 56 68Rel. freq. 6.3% 14.6% 17.7%

Amount spent ($) 3539 4044 4549Number 66 60 48Rel. freq. 17.2% 15.6% 12.5%

Amount spent ($) 5054 5559 6064Number 30 20 12Rel. freq. 7.8% 5.2% 3.1%

n event( )n sample space( )---------------------------------------- .

23--- .

12--- .

13---


482 NEW QMATHS 12A

27 a

b 30 c d28 a

b c d

b 0.23 c 23

29 a No. computers 01 2 3 46No. families 15 15 10 12Rel. freq./unit 0.144 0.288 0.192 0.077

30 a Mass (g) 5059 6069 7079 8089 9099Number 9 11 13 9 6

Rel. freq./unit 0.019 0.023 0.027 0.019 0.013

BlueBlueBlackRedRed

BlueBlueBlackRedRed

BlueBlueBlackRedRed

BlueBlueBlueRedRed

BlueBlueBlueBlackRed

BlueBlueBlueBlackRed

Blue

Blue

Blue

Black

Red

Red

15---

1115------

H

T

H

T

T

H

HTHTHTHT

18---

38---

12---

Prob

abili

ty 0.250.20.150.10.05

00 4

Number of computers5

0.30.35

1 2 3 6

b 0.431 a 1060 b 207 c 519 d 0.5132 a b c d 1

33

34 a 0.0114 b 0.000 12

c 0.117 d 81


1 a b

35 a Fall (m) 00.4 0.50.9 11.4 1.51.9Frequency 2 5 15 18

Fall (m) 22.4 2.52.9 33.4 3.53.9Frequency 8 4 2 5

Fall (m) 44.4 4.54.9Frequency 0 1

b Fall across site (m) 00.9 11.4 1.51.9Frequency 7 15 18Rel. freq./unit 0.012 0.05 0.06

Fall across site (m) 22.4 2.54.9Frequency 8 12

Rel. freq./unit 0.027 0.008

Prob

abili

ty 0.0250.020.0150.010.005

080

Mass of apple (g)90

0.03

50 60 70 100

413------

513------

413------

2732------

188------

18184------------

Prob

abili

ty 0.050.040.030.020.01

0

Fall across site (m)

0.060.07

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x

y

22

2

4

2y = 3x 2

4

f

d

21

4

2

4

d = 2f + 32


ANSWERS 483

c d

e

2 a b

c d

e

3 a b

c d

h

m

22

6

8

4

2

m = 6 2h

k

X

22

6

8

4

2

X = k + 4

C8

4

2

C = 5 2x6

k22y6

4

22x + 4y = 10

x62 4

q2

2

p 2q = 6

p62 4

4

y6

4

x + 3y = 6

x26 4

2

v

2

4

6

3m 2v = 12

m62 4

c8

4

2

2a + c = 86

a62 4y6

4

24x + 5y = 20

x62 4

q8

4

2

p + q = 8

p82 6

6

4d

4

8

3c + 2d = 15

c26 42

6

e

6

2

d24

43d 2e = 9

e

4 a y = 3x + 4 b y = 2x + 3

c 5x + 2y = 10 d 3x + 4y = 12

e 2x + 4y = 7

Exercise 7.21 a Rooms occupied = n, prot = P,

initial cost = $300, prot/room = $25,number of rooms = 30

b P = 25n 300c

d $325 e 20 f 12 rooms2 a Resistance = R, temperature = T,

resistance at 10C = 25 , temperature rise = 2C/

b T = 2R + 5c

d 43C e 75C f 17.5

y

2

x812

43y x = 12

4

Hotel roomsP ($)500400300200100

0100200300

20 30 n

P = 25n 300

10

10

Resistance wireT (C)80

60

40

20

020 30 R ()

T = 2R + 5


484 NEW QMATHS 12A

3 a Days hired = d, cost of hire = C, initial cost = $500, cost/day = $200

b C = 500 + 200dc $1200 d 5 days

4 a Weeks without rain = W, water level = L, current level = 6.5 m, drop/week = 0.2 m, maximum level = 8 m

b L = 6.5 0.2Wc

d 4.1 m e 7 weeks ago f 32 weeks5 a Amount produced by Agora mine = A,

amount produced by Finbury mine = F, Agora concentration = 2%, Finbury concentration = 3.5%, amount to be produced = 20 tonnes/week

b 0.02A + 0.035F = 20c

d 300 tonnes6 a Number of concession tickets = c, number

of adult tickets = a, concession cost = $8, adult cost = $12, total amount = $2400

b 12a + 8c = 2400 d 100c

7 a Number of service calls = s, number of rells = r, cost of service call = $80, cost of rell = $20, total amount = $500

b 80s + 20r = 500 d 9c

10

Level of damL (m)8

6

4

2

20 30W (weeks)

L = 6.5 0.2W

0Present

10

12---

12---

400

F (t)600

400

200

600 1000 A (t)

0.02A + 0.035F = 20

0

Copper production

200 800

571.4

200

c

300

200

100

a

12a + 8c = 2400

0

Cinema tickets

100

10

r

25

15

5

s

80s + 20r = 500

05

20

106.25

Exercise 7.31 2

3 4

5 6

7 8

9 10

Exercise 7.41 10c + 25t 900 2 0.8h + 1.5s 2003 20m + 35a 1800 4 b + t 60005 30c + 40t 400

Exercise 7.51 a Number of high-quality = h,

number of medium-quality = m, high-quality materials = $300/set, medium-quality materials = $200/set,maximum cost = $2000, minimum high-quality = 2

b 300h + 200m 2000, h 2

x

y

22

2

4

2

y < 3x + 2

x

y

2

2

4

2 3x y 94

4

x

y

4x + 3y 24

4

8

12

48 x

y

2 4

y 2x 6

2

4

6

2

x

y

2

x + y > 4

2

4

4

x

y

2

y > 4 x4

4 6

2

6

x4 6 1082

y

x + 2y 10

4

2

6

x22

2

4

4

yy > 3 4x

2

x

24

4

y

22y 5x < 10

x

y

4

6x 3y 15

2

4

6

2

2

2


ANSWERS 485

c

d m = 0 7, h = 6 22 a Number of large mugs = b, number of

small mugs = s, large mug time = 15 min, small mug time = 10 min, maximum time = 240 min, minimum large mugs = 5

b 15b + 10s 240, b 5c

d s = 0 16, b = 16 53 a Cost of an ice-cream = i, cost of a

chocolate = c, number of ice-creams = 14, number of chocolates = 9, minimum cost of a chocolate = $1.50

b 14i + 9c 30, c 1.5c

d i = 0 1.17, c = 3.33 1.504 a Number of large ferries = b, number of

small ferries = s, capacity of large ferry = 120 people, capacity of small ferry = 80 people, minimum passengers = 600, minimum small ferries = 2, number of runs each day = 3, present maximum passengers = 1200 (possibly growing)

b 360b + 240s 600, 360b + 240s 1200, s 2

c

Chair upholstery

3

m

10

6

2

h0

2

8

4

1 4 5 6 7

6.67

Pottery mugss

25

15

5

b0

5

20

10

10 15

16

24

Chocolates and ice-creamsc ($)

2

1

i ($)0

1

4

2 3

3

1.5

3.33

2.14

Ferriess

5

3

1

b0

1

4

2

2 3

2.5

3.331.67

5 a Number of complete kits = c, number of emergency kits = e, complete kit price = $60, emergency kit price = $25, minimum complete kits = 10, minimum sales = $900, maximum sales = $1500

b 60c + 25e 900, 60c + 25e 1500, c 10

c

d c = 10 25, e = 36 0

Chapter review1 A linear equation has no powers. Its graph

is a straight line.2 The intercepts of a linear equation are the

points where it cuts the axes.3 A variable is a quantity that can have

different values in a problem.4 A parameter is a quantity that has a xed

value for a problem.5 A solid line is used for or to show that

the points on the line are included. A dashed line is used for or to show that points on the line are not included.

6 A constraint is an inequality that restricts the solutions to a problem by restricting the variables.

7 The region dened by the constraints of a problem is the area of the graph where the constraints all overlap.

8 a b

d No excess capacity ( 1200)Large, b 2 1 1 0 0Small, s 2 2 3 4 5

Excess capacity (> 1200)Large, b 2 1 1Small, s 3 4 5

Automotive tool kitse

60

30

10

c0

10

40

20

20 30

50

2515

36

x22

2

4

4

y

y = 4x 52

v

4v = 7 + 2f

f24

2

8

6


486 NEW QMATHS 12A

c

9 a b

c

10 a b

c

11 a y = 3x + 5 b y = 3 4x

c y = 3x + 1

t

2

4

z

z = 3 2t

24

4

x

y

2

3x + 5y 18 = 04

4 6

2

6

24 f

g

2f 3g + 9 = 04

2 4

2

6

4 2

x

y

5x 2y = 7

6

3

3

12

3

9

x

y

2

3x + 5y = 204

4 6

26

23---

c

d

d 4c = 126

2

2

12

4

8

4

10

2

g2 4

h

3g 2h = 94

2

6

412---

12 a b

c

13 a Number of donuts sold = n, prot = P, site rental = $300, prot/donut = $0.25

b P = 0.25n 300c

d 200014 a Hire time = h, cost of hire = c,

initial cost = $200, hourly cost = $50b c = 200 + 50hc

d 6 h15 a Number of children = c, number of

adults = a, number of children = c, child cost = $10, adult cost = $16, total takings = $5400

b 10c + 16a = 5400 c

d 46016 2b + 3s 50

x

y

12

3x 4y < 24

2

810

2

4 8

46

x

y

4

5x + 2y 12

4

6

2 2

2

2.4

y > 3x 6

y

4

2

6

x2 2

Donut standP ($)400

200

0

200

400

2000 n

P = 0.25n 300

10001200

Backhoe hirec ($)800

600

400

200

08 h (h)

c = 200 + 50h

2 4 6

Circus takingsa

400

300

200

100

0600 c

10c + 16a = 5400

100 200 300 400 500

540

337.5


ANSWERS 487

17 a Number of photocopiers = p,number of workstations = w,cost of photocopiers = $2500, cost of workstations = $3500, maximum cost = $12 000, minimum photocopiers = 2

b 2500p + 3500w 12 000, p 2c

d For p = 2, w = 0 2; for p = 3, w = 0 1; for p = 4, w = 0

18 a Number of cowboys = c, number of thrillers = t, cost of cowboys = $200, cost of thrillers = $400, maximum cost = $2000, minimum thrillers = 2

b 200c + 400t 2000, t 2c

d For t = 2, c = 0 6; for t = 3, c = 0 4; for t = 4, c = 0 2; for t = 5, c = 0

19 a Number of Fair Isle = f, number of plain = p, time for Fair Isle = 5 h, time for plain = 3 h, maximum time = 60 h, maximum plain = 5

b 5f + 3p 60, p 5c

d f = 0 12, p = 5 0


1 a Simple path b Circuitc Simple circuit d Pathe Simple cycle f Simple pathg Simple circuit h Chaini Simple path j Cycle

Office equipmentw

4

3

2

1

05 p1 3 42

4.8

3.43

Old movie serialst6

4

2

010 c2 6 84

Knitted jumpersp20

15

10

5

015 f5 10

12

2 a Connected undirected circuitb Connected directed mixedc Disconnected (directed trees)d Connected undirected treee Connected directed treef Connected directed cycle

3 a b

4 a

b Luke Fellowes, Kerry Stubbs, David Alsopc Felix Goldschmeid (known to LF and KS)

5

6 a and b

c MHMGBSFFBSHMH = 53 mind MHBSFFMH = 28 min

Exercise 8.21 a AEFGH = 30 b JEFC = 23

c DGFI = 26 d ABDGF = 43e ABCF = 25

2 a KIEDN = 43 b LHEF = 38c IKLM = 47 d KLMLHD = 88e KIED = 26

3 a

FK/M

DLo

E

B1

ES SB2B4

La

WC

B3B G

B2

BB3

F LaWC

KG

LoD

E

B1ES

B4

JK

LS

JCFG

KS

LF

ASJF

DA

PP

Ph

TaV

CaL

Th

Ma

S

Indo

ET

SL

BaMy

I

BhN

Ch

SH

BMG

BS

FF

MH6

14210

1512 10

15

6

Ca

Tow

G

E

MI

Ch

Too

B

$33$21

$6$18

$24

$12$15

$18

$21$15

$10.5

0$15


488 NEW QMATHS 12A

b $36 c $51d $39 e $25.50

4 a

b 8 months c $200 000d Do fast theoretical research and

government approval, normal laboratory experiments and slow marketing to complete in 11 months at $290 000.

5

RedcliffeCabooltureEsk = 30 min

Exercise 8.31 a b

2

3 a

b PortR4R5R2R1, R4R6, R5R7 and R5R3 = 25 km

2 m

$50

$70

$100

5 m

1 m

$60

$80

$120

3 m

2 m

$10

$10

$30

6 m

3 m

$80

$105

$150

5 m

4 m 2 m 4 m 4 m

Costs 1000

C

R

Br

E

TI

L

Be

20

K

35

45

1525

2015

5

10

15

50

20

10

25

Times in minutes

a bc

d e

g f hi

867

6 44 5

5

A B C D

G H I J

E F

9 5 776

7

6 65

c

b ed

a l g fm

jik h

56

4

7 5

7 76

6

8

8

8

R1

R2

1045

8

R3

R4

R5

R6

R7

Port Distance, km

6

105 6 8 2 6 9

10

8

10

10 2

11 5

5936

7

8

49

10

c

d Same connections as part b, cost = $53 0004 a

b Connections marked in purple = 140 mc

d Connections marked in purple = $22 5005 a AEHIGCBDF or ABDCGIHEF = 18

b AEFHIGCBDF = 206 Go along Front St to Johnston Rd, up

Johnston Rd to the hospital, back and along Owen St, up Dan Hart Lane and Mullavey St, back down Grogan St, up William St and Ingles St, back round Thomas St and Mill St, down Foxton Ave to Park St, along Park St, detouring up Hart St and down to the river. Then left along Mossman St, up Front St to Bow St, detour up Bow St and back to the bit of Front St between Bow St and William St. This minimises double walking of some streets.

Chapter review1 Both have lines (edges or arcs) connecting

nodes, but a network has a distance for each edge.

2 A path joins one node to another through connecting edges or arcs (in the same direction). A circuit connects edges or arcs (same direction) back to the original node.

R1

R2

148 912

R3

R4

R5

R6

R7

Port Cost $1000

10

14910

12 10 1314

12

14

14 6

15 9

9

137

1011

12813

14

6

F D

C

E A

B

25

20

20

30

20

6015

15 15

15

25 35

60

40

Distance, m

F D

E A

$3250

$8050$2650

$5650

$2650

$2650

$3250

$385

0$4450

$2650 $3250 $8050$5

050$3850

BC


ANSWERS 489

3 A tree is a network (or part of one) that has no cycles or circuits.

4 Dijkstras Algorithm grows shortest trees from the starting node until the nishing node is reached.

5 A minimum spanning tree connects every node of a network together with the smallest possible total length.

6 Starting from the smallest edge, Kruskals Algorithm adds the next smallest edges that do not form a cycle or circuit progressively until a minimum spanning tree is formed.

7 a Simple path b Simple circuitc Circuit d Pathe Simple cycle f Simple pathg Simple circuit h Chain

8 a Connected undirected circuitb Connected undirected treec Connected directed mixedd Disconnected directed (mixed)e Connected undirected mixedf Connected directed cycle

9 a AGMNO = 12b KLMIJ = 10c EIMGF = 11

10 a b

11

12 Repeat the following sections: Fairlawn between Toohey and Caneld and between Dellwood and Edgehill; Bankside between Arkindale and Fairlawn and between Glade and Edgehill; Dellwood between Edgehill and the turning circle; and Ivymount between Fairlawn and Bankside and between Glade and the turning circle.

13 From Toohey Rd, go around Caneld and Arkindale (2), then around Bankside (1) to Glade. Do the 2 in Ivymount. Come back around the rest of Bankside (2) and nally do the ones in Fairlawn (1) and Dellwood (2) before returning to Toohey Rd.

b

a

c dg

hi

e

f

2 33 2

2

23

3

BD

A

C E

GF 3

3 345

4

S1 S2 S3 S4

S5 S7S6

14

b BMoBHA = 17 hc ABHMoRTMI = $214

15 Keep CityCarindale, CityMoorooka, MoorookaAcacia Ridge, Acacia RidgeIpswich, MoorookaAnnerley, CarindaleMt Gravatt, LoganSpringwood, SpringwoodBeenleigh and either CarindaleLogan or Mt GravattLogan.


1 a b c d

2

3 a b c d

e f g

4 a b c d

e f5 a 0.93 b About 0.602 c 0.3986 About 0.075 = 7.5%7 a 0.294 b 0.6688 a b c d e9

a 0.321 b 0.143 c 0.2510

a 33 b 0.455 c 0.364d 0.545 e 0.636

Cai

T

R

B

Coo

S

Me

D

BH

Mo

MI

Can

A

10124

7

5

1

64

5

7

6

47

5

5

8

8

87

34

20

14

24

32

60

40

24

20

2020

28

50

30

34

34

Time (h)Cost ($)

40

16---

518------

518------

59---

15525------------

113------

12---

126------

713------

1213------

34---

913------

518------

1318------

14---

23---

518------

13---

13---

12---

16---

23---

13---

E B

4 5

712

928------

W A

4 8

714

1533------


490 NEW QMATHS 12A

11 49:1512 a 37:1 b 10:913 a b c

d e 61:3 f $10.67

15 a Green 24:1, red 124:1b Green $25, red $125

Exercise 9.21 a Discrete b Continuous c Discrete

d Continuous e Discrete f Discreteg Continuoush Discrete (must be to 5 cents), continuous

in bulk2 a Yes b Yes c No

b

b

c No

14 Total 2 3 4 5 6Fair odds 35:1 17:1 11:1 8:1 31:5

Total 8 9 10 11 12

Fair odds 31:5 8:1 11:1 17:1 35:1

3 a c 0 1 2 3

p(c)

4 a s 2 3 4 5 6 7

p(s)

s 8 9 10 11 12

p(s)

5 a t 11 12 13 14 15f 2 3 4 6 1p(t) 0.08 0.12 0.16 0.24 0.04t 16 17 18 19 20f 3 0 1 1 4p(t) 0.12 0 0.04 0.04 0.16

164------

164------

164------

364------

2764------

2764------

964------

164------

C10

0.15

0.250.3

0 2 3

0.40.45

0.35

0.2

0.10.05

p(c)

136------

118------

112------

19---

536------

16---

536------

19---

112------

118------

136------

p(s)

2 3 6 8 S0

0.1

0.2

4 5 7 9 10 11 12

b c

b

Neither distribution is uniform.c The data is the same, but the variable is

different, so the graphs are different.

b

8 Throwing a normal die; the number that comes up on a chocolate wheel; Gold Lotto numbers

Exercise 9.31 E(T) 5.27 2 E(X) = 13 E(X) = 1.75 4 E(X) = 3.55 76 a $1.67

b No, you should win in the long run.c One way would be to win if the total is less

than 5 or greater than 9.

b E(X) 2.18 a 0.125 002 8 b $0.53 c 47%9 3 (2.65)

10 a 0.0775 b 011 a $2400 b $100012 3 (2.875)

6 a d 0 1 2 3 4 5

p(d)

s 1 2 3 4 5 6

p(s)

7 a x 0 1 2 3 4

p(x)

7 a x 0 1 2 3

P(X = x) 0.015 625 0.281 25 0.281 25 0.421 875

325------

25---

16---

518------

29---

16---

19---

118------

1136------

14---

736------

536------

112------

136------

p(d)

0 1 3 4 D0

0.1

0.2

0.3

2 5p(s)

1 2 4 5 S0

0.10.2

0.4

3 6

0.3

116------

716------

516------

18---

116------

p(x)

0 1 3 4 X0

0.10.2

0.5

2

0.30.4


ANSWERS 491

E(T) 7.75

E(T) 8.5

Exercise 9.41 a No b Yes c Yes d No e Yes2 a 4

b p = 0.17, q = 0.83c 0.386 d 0.116

3 a n = 4, p = 0.3, q = 0.7b 0.0081 c 0.265 d 0.760

4 a 0.008 b 0.201 c 0.335 d 0.000 025 a 0.016 b 0.054 c 0.1936 a 0.125 b 0.375 c 0.8757 a 0.336 b 0.4288 0.0759 a 0.813 b 0.570

10 a 0.087 b 0.116 c 0.682 d 0.82311 a 0.322

b P(1 caused by fatigue) 0.5, so it was not highly unlikely.

12 a 0.599 b 0.401 c 0.9885

Chapter review1 The probability of two events occurring

together is the product of the probabilities of the individual events. This is written as P(A B) = P(A and B) = P(A) P(B).

2 P(A or B) = P(A) + P(B) P(A and B)3 A Venn diagram shows a sample space as an

enclosing rectangle and events within the sample space as circles, overlapping where they have sample points in common.

4 Event A either occurs or it doesnt, so P(A) + P(A) = 1. To nd the probability that event A does not occur, subtract P(A) from 1.

5 Fair odds are calculated so that the long-term return from bets is zero. They are the ratio of the probability of losing to the probability of winning.

6 A discrete variable is a numeric variable that can take only specic values with a value determined by a random event.

7 0 p(x) 1 and P(X = x) = 1.

13 a t 2 3 4 5 6 7p(t) 0.0083 0.0333 0.0583 0.0833 0.1083 0.1667

t 8 9 10 11 12

p(t) 0.1583 0.1333 0.1083 0.0833 0.0583

b t 2 3 4 5 6 7p(t) 0.0025 0.015 0.0375 0.06 0.0825 0.125

t 8 9 10 11 12

p(t) 0.1725 0.15 0.1275 0.105 0.1225

16---

56---

8 A uniform probability distribution has a constant function.

9 The expected value E(X) of a probability distribution is the mean of the distribution.

10 Bernoulli trials have only two possible outcomes, called success and failure; the probability of success ( p) and the probability of failure (q) do not change; p + q = 1; the results of one trial do not affect others.

11 A binomial probability situation consists of a xed number of Bernoulli trials where the probability of a particular number of successes is required, without reference to order.

12 B 13 0.000 003 714 0.784 15 B16 5:3 17 D18 a Yes b No c Yes d No e No

b

c E(D) 1.94420 3.5 21 E(X) = 4.4522 a 0.2061 b 0.144323 C 24 A25 a 0.0114 b 0.0004

E(D) 2.52627 a 0.1074 b 0.2684 c 0.879128 a 0.0160 b 0.0219 c 0.170229 a 0.0874 b 0.6814 c 0.8227


1 a 313 T b 119 T c 349 Td 228 T e 093 T f 023 Tg 009 T h 020 T i 191 Tj 219 T

2 a 247 M b 357 M c 161 Md 071 M e 143 M f 011 Mg 203 M h 004 M i 237 Mj 024 M

19 a d 0 1 2 3 4 5

p(d)

26 d 1 2 3 4P(D = d) 0.3823 0.2362 0.1459 0.0901d 5 6 7P(D = d) 0.0556 0.0344 0.0555

16---

518------

29---

16---

19---

118------

p(d)

0 1 3 4 D0

0.1

0.2

0.3

2 5


492 NEW QMATHS 12A

3 a 3.69 m b 0.93 mc 1.88 m d 0.15 m exposede 1.65 m f 5.15 m

4 a 147 T b 241 T c 304 Td 172 T e 044 T

5 a 128 b 015 c 091d 251 e 316 f 228g 335 h 015

6 a 223 T b 127 T c 012 Td 335 T

7 a 208 b 130 c 005d 352 e 061 f 221g 034 h 185

8 Rock: 2.44 m at 6 am, 3.44 m at 5 pm; sandbank: 1.54 m at 6 am, 2.54 m at 5 pm

9 5 h

Exercise 10.21 a 3244.6S, 15214.4E

b 3242S, 15220.8Ec 3239.5S, 15213.2Ed 3242S,15215.6E

2 a 3307.5S, 15158Eb 3347S, 15150Ec 3327S, 15215Ed 3259S, 15223E

3 a 2313.3S, 15054.6 Eb 2301.7S, 15103Ec 2307.2S, 15050.5E

Exercise 10.31 a 2249S, 15012.3E

b 2244.8S, 1509.5Ec 2252S, 15023.5Ed 2241.5S, 15027.6Ee 2231.2S, 15011.7E

2 a 2035S, 17813Eb 1740S, 1764Ec 1818S, 17942Ed 1822S, 17636Ee 1753S, 17942E

3 a 3956S, 14843Eb 4007S, 14915Ec 4038S, 14826Ed 4050S, 14926Ee 4037.5S, 14824E

4 a 2111S, 14945Eb 2038S, 14917Ec 2116S, 14947Ed 2030S, 14932Ee 2114S, 14936E with 2 n.m. error

Exercise 10.41 a 2.7 n.m. b 3.1 n.m.

c 4.3 n.m. d 4.3 n.m.

2 a 11.8 n.m. b 19.8 n.m.c 12.2 n.m. d 14.1 n.m.

3 2238S, 15010E 4 4037S, 14840E

Exercise 10.51 a 6 kn at 166 b 4.3 kn at 272

c 8.3 kn at 072 d 7.2 kn at 237e 9.7 kn at 076

2 a 290, 9.1 kn, compass heading 275b 149.5, 7.7 kn, compass heading 160.5c 146, 2.6 kn, compass heading 140d 319, 9.2 kn, compass heading 328e 050, 11.2 kn, compass heading 044

3 a 15 kn at 252, current 4 kn at 293b 9 kn at 120, current 3 kn at 162c 9.5 kn at 017, current 3.5 kn at 077

4 142 at 13.1 kn, compass heading 155 for 10 h 41 min

5 009 at 6.7 kn, compass heading 017 for 5 h 4 min

Exercise 10.61 1011S, 13430W 2 3920S, 15430E3 1215N, 113E 4 3144N, 15930W5 In navigation on land, you need to consider

the slope of the land, but are never out of sight of landmarks. There are no currents or winds to consider. In air navigation, you are travelling so much faster than in marine navigation or land navigation that you need to work out your course before taking off.

Chapter review1 Variation is the difference between magnetic

north and true north. Deviation is the difference between magnetic north and the direction shown by a ships compass. Deviation normally varies with the direction (heading) of the ship.

2 The chart datum is the level from which water depths and heights of exposed objects are measured. It is normally set at the lowest possible low tide under normal weather conditions.

3 The rule of twelfths says that in the 4th hour after a tide, the water level will change by (1 + 2 + 3 + 3) = 9 twelfths (= ) of the difference between the tide levels. So 4 hours after low tide, the water level will be of theway to high tide level. The water height over the sandbar is found by subtracting the height of the sandbar from the 4-hour water level.

4 If a bearing is taken from point A to point B, then the back bearing is from B to A. It differs by 180 from the rst bearing.

34---

34---


ANSWERS 493

5 A cocked hat uses three bearings and shows the error. A cross bearing uses only two bearings and does not show the error.

6 In a running x, two bearings are taken while the boat is moving in a known direction at a known speed, so that the distance between the bearings is known. This allows the position to be determined.

7 Doubling of the bow is a simplied running x where the boat travels a distance such that the angle between the heading and the landmark doubles.

8 For a four-point bearing, the bow angle doubles from 45 to 90 as the boat passes the landmark.

9 A transit bearing is the most accurate because it involves no measurement.

10 An object is dipping the horizon when the top of the object is just visible at the horizon, so that as the boat rocks with the waves the object seems to appear and disappear at the horizon.

11 When arrows showing the speed through the water and the current are placed head to tail, an arrow drawn from the tail of the rst to the head of the second shows the speed and course made good.

12 An accurate clock can be set to Greenwich Mean Time (GMT). Longitude can then be calculated using the difference in time between an event (such as noon) at Greenwich and the same event at the observers position.

13 The complementary angle to the altitude of the Sun at noon shows the difference in latitude between the observer and the latitude where the Sun is overhead.

14 a 208 T b 353 Tc 358 T d 120 T

15 a 008 M b 130 Mc 079 M d 222 M

16 a 0.15 m b 4.6 m17 a 3.2 n.m. b 19 n.m.18 a 11.4 kn at 149 b 7.2 kn at 256

c 12.7 kn at 03719 a Heading 205, speed made good 11 kn,

compass heading 197b Heading 349, speed made good 7.7 kn,

compass heading 33720 3920N, 17345W21 a 127 T b 051 T c 309 T22 a 086 b 297 c 212 d 26623 At 6:30 am: rock about 2.6 m, channel about

4.0 m. At 12 noon: rock about 1.6 m, channel about 3.0 m

24 a 2632S, 15330E b 2647S, 15326E

25 a 2735S, 15341E b 2748S, 15348Ec 2705S, 15316E

26 04827 27 23S, 15344E28 Speed 11 kn at 040, current 4 kn at 343


1 a 8 km b 210 km c 5 kmd 7 km e 4.5 km

2 a 197.5 km 148.75 kmb 29 378 km2 c 6

3 a 114.5 km b 42 km c 71.8 km4 a Geographic or general reference map

b 1 cm:125 kmc 875 kmd 1150 km (Answers will vary depending on

the route takenroads are not straight lines.)

e Straight line 530 km, road 650 kmf i 28S, 143.8E ii 27.6S, 151.9E

iii 22.5S, 143.1E iv 25.5S, 152.8E5 a i 82.2 km ii 124.3 km iii 213.4 km

b i 21.7S, 139.5E ii 21.8S, 139.9Eiii 22.5S, 138.8E iv 20.7S, 139.5E

6 a Cannon Hill Anglican Collegeb C5 c 800 md 280 m e 192 000 m2 (19.2 ha)

7 a Topographic map b 2.8 kmc 700 m d 285569e 257542 f Ramp

Exercise 11.21 a 25 m

b A 125 m, B 37.5 m, C 164 m, D 43.8 mc i 3.7 km ii 4.7 km iii 3.4 kmd i 0.0236 ii 0.0269 iii 0.0354

2 a 5 mb i 159 m ii 62 m

iii 77.5 m iv 64 mc i 2500 m ii 900 m iii 600 md i 0.0388 ii 0.0906 iii 0.1583e i 2.22 ii 5.17 iii 9.00

3 a 5 m b 1250 m c 0.0244 a 5 m

b i 81 m ii 84 m iii 115 miv 84 m v 138 m

c i 7.86 ii 5.49 iii 3.06iv 12.86 v 3.43


494 NEW QMATHS 12A

Exercise 11.31 a 50 m b

2 a 20 m b

3 a 1:25 000 b 20 m c 426164 fd 425166 e 433158

4 a 1:50 000 b 25 mc i 797361 ii 791360 iii 858321 iv 837372d

e

6 a 1:25 000b

QP

m500

0

400300200100

Scale 1 : 100 000 (half-width)

River

ZW

Creek

m200

0

1601208040

Scale 1 : 25 000 (half-width)

Island

Stream RoadRiver

160140120100806040200

m

X YScale 1 : 50 000 (half-width)

180

Stream

Scale 1 : 100 000 (half-width) QP

m

250

0

20015010050

Scale 1 : 100 000 (half-width) NM

m

250

0

20015010050

m

850800750700650600550500

900

P QScale 1 : 25 000

56

78

9

10

5

4

6

6

6

7

7

7

67

58

8

8

9 910

8

10

6

4

6

4

5

4

3

3 22

1

N

Heights in metresScale 1 : 800 (half-size)


ANSWERS 495

Exercise 11.41 a Lots 1, 2, 3 and 4

b Lengths 231 m and 223 m, width 45 mc 10 215 m2, 1.0215 had 35 m 20 m, 700 m2 e $13 261

2 a 3540 m2 b They are corner blocks and have been

trimmed.c 12 714 m2 (1.2714 ha)

3 a 825 m2 b 2.3 mc 44 posts, 63 rails, 800 palings; $2938.25d 6.3 m3 e $77 900

4 a 20 m and 40 m b 6 mc 760 m2 d $6800

5 a 374 m b 3848 m2

Chapter review1 A geographic or general reference map2 a Cadastral map b B3 Eastings and northings are a system of grid

lines on topographic maps used to locate the positions of features.

4 When contour lines are close together, the ground rises steeply.

5 A6 a i 19.4S, 146.8E ii 14S, 143.2E

iii 18.2S, 142.3E iv 17.7S, 141.1Eb i 438 km ii 763 km iii 900 km

7 a i Allan Border Field and Queensland Cricket HQ

ii Oriel Park iii Jettyiv Newstead House and Newstead Parkv Albion Park Raceway

b i G15 ii H20 iii M17iv H18 v L19

c 1:20 000d About 800 m from each school centree 400 m f 15 600 m2 (1.56 ha)

8 a i 311509 ii 302485iii 318496 iv 326477

b 900 m c 1 km d 3.8 km9 a 362.5 m 190 m

b i 0.05 ii 0.033 iii 0.218c

F

BA Scale 1 : 5000 (half-width)

m

76

72

68

64

60

0

40

C

DE

10 a Lawry Ave and Anna Place 12 m, Landra Road 16 m

b i 624 m2 ii 707 m2 iii 712 m2 11 a 602 m2 b 192 m2 c 69.4 m


1 a 24 b 122 a 23 b 7 c 64

d 53 e 130

b 11

b 9

3 Age of car (years)14 57 811

Holden 400 370 310 1080Ford 280 360 370 1010Toyota 180 260 430 870Mitsubishi 150 220 250 620

1010 1210 1360 3580

5 a Female MaleGarden City 11 7 18Grifth Uni 8 4 12

19 11 30

6 a Female MaleAFL 8 9 17Soccer 5 6 11Rugby 12 20 32

25 35 60

Vehicle preference

Sedan0

10

20

Stn wagon 4WD

440

30

Num

ber

Ute

CityCountry


496 NEW QMATHS 12A

b 49 c 61

b 3 (for 25 years)c In all age groups, there are not many with

more than 3 siblings.

Exercise 12.21 a The people interviewed

b The women interviewed c Those in favour of daylight savingd Teenagerse Pet-owners

2 a 30% b 22% c 15%d 5% e 15%

3 a 27% b 15% c 26%d 26% e 12%

7 a Republic

Yes Dont care

No

Labor 28 4 17 49Liberal/National 25 3 27 55

Minor parties 8 7 6 2161 14 50 125

8 a Age 25 2635 35

0 3 3 1 71 3 1 2 62 3 4 2 93 1 2 1 44 2 0 1 35 0 1 1 26 1 1 0 2

13 12 8 33

4 Oil exploration support

Yes NoNSW, Queensland 7 13 20ACT, NT, Victoria, SA 8 21 29Tasmania, WA 5 9 14

20 43 63

5 Age group 40 40

2 5 8 13

2 5 5 10

2 12 5 17

22 18 40

Sibl

ings

Take

aw

ays

Exercise 12.3

b Obese 32%, not obese 6.3%c Obese people are 5 times more likely to die

of heart attack than non-obese people.

b 98.1% c 0.0053%

a 21% b 4.5%

b 26% c 67%d Less than 5 e 27

5 a i 33 ii 49 or 50 iii Less than 5b i 17 ii 19

a Correct b No, 50% c No, 87%

Chapter review1 A contingency table shows data concerning

two different characteristics of the same population. The frequencies for one characteristic are in rows, while the other characteristic has frequencies in columns.

2 Marginal frequencies of a contingency table are the totals of rows, placed on the right, and the totals of columns, placed at the bottom.

3 1

1 a Heart attackYes No

Obese 3020 2 820 5 840Not obese 6380 41 780 48 160

9400 44 600 54 000

2 a HIVYes No

Positive 0.095% 4.995% 5.09%

Negative 0.005% 94.905% 94.91%

0.100% 99.90% 100.00%

3 PregnantYes No

Positive 27% 7% 34%

Negative 3% 63% 66%

30% 70% 100%

4 a Paid up In arrearsMarried 62% 4% 66%Single 26% 8% 34%

88% 12% 100%

6 Aust. born Overss bornAFL 27% 4% 31%Rugby 37% 5% 42%Soccer 18% 9% 27%

82% 18% 100%

Test

Test


ANSWERS 497

4 The percentages calculated by row use the row totals as the denominators, while those calculated by column use the column totals as denominators. The percentages are out of different totals.

5 The rule of 5 ensures that probabilities and percentages are not taken from too little data and that expected frequencies are not stated too accurately.

6 The difference suggests that there is some relationship between the row and column characteristics.

7 A false positive is a positive result to the pathology test although the person does not have the condition tested.

8 a 330 b 520

10 a The number attending the gameb Teenagers at Kalkadoon SHS

11 a 32% b 15% c 39% d 32% e 17%

b 60c Very active 17%, sedentary 33%d Those with sedentary lifestyles were twice

as likely to develop osteoporosis as those with very active lifestyles.

b No-one spent more than an hour.c 16% spent less than an hour.

a 84% b 0.008%

9 Caffeine contentNone Low Medium High

Herbal 3 4 1 2 10No herbs 0 4 1 4 9

3 8 2 6 19

12 a OsteoporosisYes No

Very active 10 50 60Sedentary 60 120 180

70 170 240

13 a Homework time (min)30 3059 6089 90119 119

8 7 9 0 0 0 169 2 4 9 2 0 1710 2 5 3 2 0 1211 0 5 2 2 7 1612 1 2 5 3 8 19

12 25 19 9 15 80

14 Blood disorderYes No

Positive 0.392% 1.992% 2.384%Negative 0.008% 97.608% 97.616%

0.400% 99.600% 100.000%

Year

Test

a 4.6% b 61% c 516 The contingency table with percentages

calculated by table total shows that 4% of the students were boys who used conditioner and 42% were girls who used conditioner, giving 46% overall who used conditioner.


1 $14 835, $247.25 2 $19 680, $4103 $2730, $26.25 4 $33.255 $114.426 a $8526.40 b $4604.26

c $13 130.66 d $364.747 a $5411 b $3571.26

c $8982.26 d $187.138 a $2482.32 b $1141.87 c $3624.19

d $34.85 e 89 at $35 and 15 at $349 a $1922.66 b $937.30

c $2859.96 d $44.00e 65 payments of $44

10 a $26 582.02 b $29 240.23c $55 822.25 d $930.37e 37 at $930 and 23 at $931

Exercise 13.21 A costs $9041.76 and B costs $8962.40, so

B is better.2 A costs $4158.60, B costs $4035.96 and

C costs $4004.52, so C is best.3 A costs $35 423.50 and B costs $32 473.65,

so B is better.4 A costs $2953.74 and B costs $2912.76, so

B is better.5 a $3338.40, 22.33% b $3583.84, 22.28%

c $22 302.80, 9.76% d $3590.08, 22.09%e $20 088, 18.98%

Exercise 13.3

15 Very tall Not very tallGirl 1.8% 58.2% 60%Boy 2.8% 37.2% 40%

4.6% 95.4% 100%

1 a Mth Principal($)

Interest($)

Payment($)

Balance($)

1 35 000 350 386 34 9642 34 964 349.64 386 34 927.64

3 34 927.64 349.28 386 34 890.92

4 34 890.92 348.91 386 34 853.835 34 853.83 348.54 386 34 816.37


498 NEW QMATHS 12A

b Mth Principal($)

Interest($)

Payment($)

Balance($)

1 42 000 294 520 41 7742 41 774 292.42 520 41 546.423 41 546.42 290.82 520 41 317.244 41 317.24 289.22 520 41 086.465 41 086.46 287.61 520 40 854.076 40 854.07 285.98 520 40 620.05

c Mth Principal($)

Interest($)

Payment($)

Balance($)

1 40 000 260 490 39 7702 39 770 258.51 490 39 538.513 39 538.51 257.00 490 39 305.514 39 305.51 255.49 490 39 071.005 39 071.00 253.96 490 38 834.966 38 834.96 252.43 490 38 597.39

d Fnt Principal($)

Interest($)

Payment($)

Balance($)

1 40 000 120 245 39 8752 39 875 119.63 245 39 749.633 39 749.63 119.25 245 39 623.884 39 623.88 118.87 245 39 497.755 39 497.75 118.49 245 39 371.246 39 371.24 118.11 245 39 244.357 39 244.35 117.73 245 39 117.088 39 117.08 117.35 245 38 989.439 38 989.43 116.97 245 38 861.40

10 38 861.40 116.58 245 38 732.9811 38 732.98 116.20 245 38 604.1812 38 604.18 115.81 245 38 474.9913 38 474.99 115.42 245 38 345.41

e Fnt Principal($)

Interest($)

Payment($)

Balance($)

1 25 000 125 175 24 9502 24 950 124.75 175 24 899.753 24 899.75 124.50 175 24 849.254 24 849.25 124.25 175 24 798.505 24 798.50 123.99 175 24 747.496 24 747.49 123.74 175 24 696.237 24 696.23 123.48 175 24 644.718 24 644.71 123.22 175 24 592.939 24 592.93 122.96 175 24 540.89

10 24 540.89 122.70 175 24 488.5911 24 488.59 122.44 175 24 436.0312 24 436.03 122.18 175 24 383.2113 24 383.21 121.92 175 24 330.13

2 a $622 b $343c $787 d $848e $798 f $332g $326.43 h $406.50

3 a $1349.60 b $1437.31c $663.01 d $1115.14e $839.44

4 There are 26 fortnights in a year, so this is effectively the same as making 13 monthly payments in a year. You can work out the correct fortnightly rate by multiplying the monthly payment by

Exercise 13.41 $91 000 2 $76 0003 $112 000 4 $89 0005 Yes, if they take it over 20 or more years.

Exercise 13.51 a $1680.33 b $1684.902 a $486.67 b $652.783 a $840.33 b $846.294 a $1030 b $1050.835 a $452 b $458.58 c $460.446 The debit card ($4.44 vs $22.56)

f Mth Principal($)

Interest($)

Payment($)

Balance($)

1 70 000 875 980 69 8952 69 895 873.69 980 69 788.693 69 788.69 872.36 980 69 681.054 69 681.05 871.01 980 69 572.065 69 572.06 869.65 980 69 461.716 69 461.71 868.27 980 69 349.98

g Mth Principal($)

Interest($)

Payment($)

Balance($)

1 70 000 875 1500 69 3752 69 375 867.19 1500 68 742.193 68 742.19 859.28 1500 68 101.474 68 101.47 851.27 1500 67 452.745 67 452.74 843.16 1500 66 795.906 66 795.90 834.95 1500 66 130.85

h Mth Principal($)

Interest($)

Payment($)

Balance($)

1 70 000 875 900 69 9752 69 975 874.69 900 69 949.693 69 949.69 874.37 900 69 924.064 69 924.06 874.05 900 69 898.115 69 898.11 873.73 900 69 871.846 69 871.84 873.40 900 69 845.24

613------ .


ANSWERS 499

7 a The $50 annual charge card (by $29.44)b The $25 annual charge card (by $11.38)

8 a $4.86 b $37.92c $12.12 d Keep his present card.

Chapter review1 Calculate the interest using the full amount

borrowed over the period of the loan. Add the interest to the amount borrowed and divide the result by the number of months in the period of the loan.

2 Stamp duty, agents commission and loan insurance

3 The expression on terms refers to the terms of the nance and is usually stated as a payment amount per period along with the number of periodsfor example, $10/week for 27 weeks.

4 The interest is calculated using the amount still owing at the beginning of each compounding period or rest. As the loan is paid off, the amount of interest payable in each period steadily reduces as the loan balance reduces.

5 Calculate your after-tax monthly income. Work out a quarter (25%) of this amount and subtract the monthly value of other loan repayments from this amount. The remaining amount is the monthly payment you can reasonably afford to repay on a mortgage.

6 A credit card transaction involves borrowing money and interest may need to be paid on the borrowed amount. A debit card transaction involves using your own money and therefore no interest is charged.

7 $4949, $137.478 $30.529 a $7648 b $4894.72

c $12 542.72 d $261.3110 A costs $13 725.60, B costs $13 659.50 and C

costs $13 653.36, so C is best.11 $1300, 20.93%12 $4258.80, 25.09%13 Mth Principal

($)Interest

($)Payment

($)Balance

($)1 19 000 142.50 251 18 891.502 18 891.50 141.69 251 18 782.193 18 782.19 140.87 251 18 672.064 18 672.06 140.04 251 18 561.105 18 561.10 139.21 251 18 449.316 18 449.31 138.37 251 18 336.68

15 a $461 ($460.95) b $867 ($866.12)16 $118 00017 a $917 b $921.48 c $7.4218 a $622 b 25 September

c $635.1619 a $3572 b $1535.96 c $5107.96

d $49.12 e 92 at $49 and 12 at $5020 Yes, if they take the loan out over 25 years

or more.

21 A costs $56.32, B costs $32.82 and C costs $23.95, so C is best.


1 a

b 20 c 112 a

b 25

14 Mth Principal($)

Interest($)

Payment($)

Balance($)

1 135 000 900 1050 134 8502 134 850 899 1050 134 6993 134 699 897.99 1050 134 546.994 134 546.99 896.98 1050 134 393.975 134 393.97 895.96 1050 134 239.936 134 239.93 894.93 1050 134 084.877 134 084.87 893.90 1050 133 928.778 133 928.77 892.86 1050 133 771.639 133 771.63 891.81 1050 133 613.44

10 133 613.44 890.76 1050 133 454.2011 133 454.20 889.69 1050 133 293.8912 133 293.89 888.63 1050 133 132.52

Prob

abili

ty 0.0250.020.0150.010.005

039.5 79.5

Mass (kg)89.5

0.030.035

49.5 59.5 69.5 99.5

Masses of 17-year-olds

109.5

Prob

abili

ty 0.10.080.060.040.02

010.5 18.5

Time (s)20.5

0.120.14

12.5 14.5 16.5 22.5

Times of students for 100 m

24.5


500 NEW QMATHS 12A

3 a 0.075 = 7.5% b 13 c 474 a

b 0.25 = 25% c 205 a

b 0.167 = 16.7% c 8 times

Exercise 14.21 a Yes b No c No d Yes

e Yes f Yes, but not well2 a

b About 51% c About 74%3 a i 42 kg ii 62 kg iii 76.4 kg

b i 75 cm ii 127.5 cm iii 190 cm4 a, b

c About 7 d About 21

Prob

abili

ty

10 30 400

0.01

0.02

0.03

20

Flies caught

Distance (cm)

Time waiting for bus

Prob

abili

ty

10 300

0.01

0.02

0.03

20Time (min)

0.04

40 80

60

40

20

0

Mathematics A (%)

Stud

y of

Soc

iety

(%)

Mid-semester tests

80

100

20 60 100

190 210

15

10

5

0

Height (cm)

Reb

ound

s/ga

me

Average rebounds

20

180 200

5 a, b

c About 45% d About 5%e It is too far outside the data range.

6 a a = 3.625t 1.0625b About 97 c About 122d No, it is too far outside the data range

they might cook!7 a p = 0.225y 430.913

b About 23.6 million c About 15.3 milliond No, it is too far outside the data range.

Conditions have changed too much (e.g. immigration after World War II).

Exercise 14.31 a

c

2 a About 8.6 cm b After 1.3 days3 a About 10.1 cm2 b About 6 cm2

c If the trend continued it would be negative, and this is silly for area.

4 a, b

c About 1 s d About 81 dB

40 80

90

60

300

Hours studying

Exam

mar

ks (%

)

Results of DEET exam

20 60 11030 50 70 90 100

4050

7080

b

Reflexes with background noise

7

6

5

4

3

2

1

030 40 50 60 70 80 90 100 110 120

Noise level (dB)

Ref

lex ti

me

(s)


ANSWERS 501

5 a, b

c About 53% d About 15 ppme No, the concentration would be so high that

the insecticide would probably be dangerous to humans.

Chapter review1 The column heights in a probability

histogram depend on the class width so that the probability is calculated for a width of 1.

2 The rule of 5 means that conclusions should be based on a reasonable amount of information. A frequency of 5 is considered the minimum reasonable amount.

3 To work out expected frequency, multiply the probability by the total number.

4 The area of the histogram for the range of values is worked out to nd the probability.

5 A positive relationship has an upward trend and a negative relationship a downward trend.

6 A line of prediction passes through the central point as close as possible to all points. It shows the expected values of the second variable from values of the rst variable.

7 Interpolation is the prediction of values within the range of real data.

8 Extrapolation is the prediction of values outside the range of real data.

9 The means of the points above and below the middle of the data for the rst variable are calculated. A line is drawn through these two points and its equation is worked out.

10 A curved line of prediction is drawn when the points on a scatterplot do not form a straight line, but do show a relationship.

11 a No b Yes c Yes, but not welld Yes e Yes f Yes

12 a

Cockroaches killed by insecticide

10 20

100

50

0

Insecticide concentration (ppm)

Coc

kroa

ches

left

5 15

160 170

170

160

150

0

Husbands height (cm)

Wife

s h

eigh

t (cm

)

Heights of married couples180

150 180 190

b About 157 cm13 a 4200 kg b 27 h c No14 a

b

15 a About 13.8C b About 16Cc No, the pool would be far too big.

16 a

b 5117 a, b

c m = 0.741e + 14.43d About 63% from the graph

18 a, b

c About $5.30/kg d About 6

Masses of watermelonsPr

obab

ility 0.25

0.20.150.10.05

00.45

Mass (kg)

0.30.35

1.45 2.45 3.45 4.45 5.45 6.95

Mid-semester assessment

70

Engl

ish

(%)

Modern History (%)30

0

50

60

80

40 50 60 70 80 90

40

Mullet prices and ratings

7

Who

lesa

le p

rice

($/kg

)

Rating1

0

3

5

2 3 7 9 10

1

4 5 6 8

2

4

6


502 NEW QMATHS 12A


1 a $101.25 b $374.40 c $1212.75d $275.00 e $1189.00

2 a $6.14 b $423 c $54d $300.93 e $2306.10

3 $37.454 $13.71 (November minimum = $2666.85)

b $6.02

Exercise 15.21 a 12.24% b 8.18%2 a 16.13% b 2.20%3 a 5.88% b 3.51%4 a 9.42% b 4.60%5 a 10.27% b 5.18%

Exercise 15.31 a Fosters Group b Freedom and Futuris

c Funtastic (2.67%) d Freedom (1.54%)e Futuris (96.30%)

2 a 20.0% b 4.17% c 12.03 a 12.0% b 7.06% c 11.34 a $7.75 b 62 cents

c 31% d 6.95 a Shares $13 500, brokerage $202.50, total

cost $13 702.50b $170.90

6 a Shares $77 400, brokerage $774, total cost $78 174

b $689.637 Brokerage $55, amount received $20208 a $55 b $4515 c $55

d $4645 e $130 prot9 C is best with 11.11% dividend yield.

10 The dividend yield (5.29%) is better than the credit union interest.

11 a $3210 b $5105c $1895 prot

5 a Date Deposit Withdrawal Balance1 Apr $1391.578 Apr $2245.80 $3637.37

17 Apr $550.00 $3087.3727 Apr $1278.30 $1809.076 May $2245.80 $4054.878 May $2137.20 $1917.67

25 May $1687.45 $230.223 Jun $2245.80 $2476.02

12 Jun $1109.28 $1366.7429 Jun $1287.56 $79.18

Exercise 15.41 9.2%, which is better than the bank.2 Painting is 10.3%, sculpture is 11.3%, so

sculpture is better.3 8.16%, which is reasonable.4 Land is 6.06%, which is about the same as

the shares.5 A is 13.41%, B is 8.24%, C is 3.23%, so the

painting is best.6 A is 9.86%, B is 8.05%, C is 15.26%, so the

didgeridoo is best.7 a 5.38% b 1.89%

c Yes, the bank is better.8 a Bank 3.52% (after tax), shares 2.34%

(after tax)b After capital gain, shares grow by 11.74%

including dividends, so Pauls decision was wrong.

Chapter 15 review1 Because lower return investments usually

involve lower risk as well.2 Capital growth and rental income3 A share in the prots and a say in how the

company is run4 A share of the companys prots paid for

each share held5 The par or face value is the price at which the

shares are initially released (oated) on the market. The market price is the price at which the shares are currently traded.

6 Capital growth (growth in the share price) and dividends

7 A fee paid to a stockbroker for completing a share transaction

8 Calculate the compound growth rate for each using the compound interest formula.

9 a $206.25 b $166.67 c $3984.9010 $43.89

b $12.81

11 a Date Deposit Withdrawal Balance1 Mar $2134.588 Mar $1567.80 $3702.38

11 Mar $750.00 $2952.3821 Mar $225.00 $2727.3822 Mar $1567.80 $650.00 $3645.18

3 Apr $1050.25 $2594.935 Apr $1567.80 $4162.73

14 Apr $1389.75 $2772.9819 Apr $1567.80 $4340.7830 Apr $1067.50 $3273.28


ANSWERS 503

12 a 15.69% b 12.24%13 a 5.70% b 3.24%14 a 9% b 6.62% c 11.315 Shares $13 050, brokerage $195.75,

total cost $13 245.7516 a Brokerage $55, amount received $3645

b $38.7017 Sell

answers

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