Answers

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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 b299°44′ M c 301°28′ M d302°24′ M 4 a 236°10′ G, 231°14′ M

b080°48′ G, 075°52′ M c 303°37′ G, 298°41′ M d189°09′ G, 184°13′ M e 349°20′ G, 344°24′ M 5 a 358°52′ G, 356°57′ T

b105°05′ G, 103°10′ T c 064°40′ G, 061°45′ T d290°28′ G, 288°33′ T e 273°23′ G, 271°28′ T 6 a 344°41′ T, 336°04′ M b030°12′ T, 021°35′ M c 125°54′ T, 117°17′ M d238°33′ T, 229°56′ M e 199°29′ T, 190°52′ M

Exercise 1.2

1 a 120 n.m. b788 n.m.

c 328 n.m. d471 n.m.

2 a 444 km b181 km

c 1304 km d895 km

3 a 259 n.m. b 334 n.m.

c 135 n.m. d 310 n.m.

4 a 203 km b 316 km

c 1117 km d 242 km

5 a 147 n.m. b 272 km

6 a i 375 n.m. ii 695 km

b i 6.5 kn ii 12 km/h

7 a 3600 km b 1944 n.m. c 24°36′N, 23°E

8 15 h 26 min

9 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 km

c 2.25 km d 4.25 km

14 a 213° b 049° c 338° d 279° e 252°

15 a 050° b 169° c 137° d 173° 16 a 254° b 220° c 339°

d 153° e 143°

17 a 256° b 155° c 091° d 312° e 020°

18 a 229° b 116° c 305° d 080°

Exercise 1.3

1 a Jeffreys Rocks (21°55′S, 150°13.7′E) b Normanby Rock (21°40.8′S, 150°12.8′E) c Sail Rock (21°48.2′S, 150°09.3′E) d Howard Point (21°37.4′S, 150°14.9′E) e Hixson Islet (21°44.9′S, 150°17.3′E) 2 a Nobbys Head (32°55′S, 151°48′E)

b Stockton Bight (32°51′S, 151°58′E) c Norah Head (33°17′S, 151°34.5′E) d Barrenjoey Head (33°35′S, 151°19.5′E) e Long Point (33°44.8′S, 151°19′E) 3 a Kaipara Harbour (36°25′S, 174°07′E)

b Raglan Harbour (37°48′S, 174°52′E) c Three Kings Islands (34°11′S, 172°05′E) d New Plymouth (39°02′S, 174°12′E) e Manukau Harbour (37°03′S, 174°31′E)

Chapter review

1 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.

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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 ﬁx is the establishment of one’s 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 344°58′ M b 343°40′ M

c 342°10′ M d 345°40′ M 14 a 300 n.m., 556 km b 107 n.m., 198 km

c 432 n.m., 800 km d 276 n.m., 511 km 15 a 333 n.m., 618 km b 191 n.m., 353 km c 62 n.m., 114 km d 183 n.m., 338 km 16 a 123°52′ G, 112°34′ M

b 226°18′ G, 215°00′ M c 006°41′ G, 355°23′ M d 076°23′ G, 065°05′ M e 358.0° G, 346.7° M f 287.7° G, 276.4° M g 339°36′ T, 329°02′ M h 200°01′ T, 189°27′ M i 042°05′ T, 031°31′ M j 359°08′ T, 348°34′ M k 136.8° T, 126.233° M l 001.1° T, 350.533° M 17 a 176.4° T, 185.3° M

b 014.7° T, 023.6° M c 039.2° T, 048.1° M d 230.3° T, 239.2° M e 345°10′ T, 354°04′ M f 103°23′ T, 112°17′ M g 147.8° G, 147.6° T h 302.4° G, 302.2° T i 034.6° G, 034.4° T j 116.1° G, 115.9° T k 223°56′ G, 223°44′ T l 059°31′ G, 059°19′ T

18 a 8.3 kn b 15.4 km/h

19 a 225 n.m. b 60°S, 137° 30′E 20 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 204° 21 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 299° 22 a Esmeralda Cove (32°37.4′S, 152°19.6′E)

b Little Island (32°42.1′S, 152°14.6′E) c Hawks Nest (32°40.1′S, 152°11.4′E)

CHAPTER 2

For 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 2.1

1 a 1.05 m b 0.83 m

c 0.93 m d 0.88 m

2 a 27.2 m b 16.8 m

c 70.4 m d 550.4 m

3 a 875 m2 _{b 829 m}2

c 419 m2 _{d 2408 m}2

4 a Bearings: AB 131°, BC 026°, CD 175°,

DE 097°

b Back bearings: AB 311°, BC 206°,

CD 355°, DE 277°

5 a 286° b 041° c 189°

d 132° e 274°

6 a 2.871 m b 4.7° c 1 : 12.2 7 5.1 cm

8 a 1.904 m b 4.53° c 5.313 m d 6.73° e 9.24°

9 The offer isn’t 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 2.2

1 a 2443.46 m2 _{b 2743.84 m}2

c 3634.74 m2 _{d 5225.89 m}2

2 1440 m2 _{3 158 297 m}2

4 a 19 000 m2 _{b 6980 m}2

5 13.353 ha (133 526.7 m2₎

6 2000 m2

Exercise 2.3

1 101.1 m 2 148.6 m

3 41.9 m 4 18.8 m

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Exercise 2.4

1

2

3 a Similar to sketch in question. b

c 900 m2 _{d 294.6 m}2 _{e 12 711 m}2 A

B

C

D E

Hedge

Stream Track

Road

Scale 1 : 1000 Yours should be twice this size.

Sports Boating

A B

C

complex lake

Scale 1 : 20 000

You should work at 1 : 10 000.

Fence

Stream

P

Q

R S

T U

Factory storage area

Factory

Path Road

Scale 1 : 2000

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4

Exercise 2.5

1 a

b TP = 160 m at 260° 2

DA ≈ 800 m (798 m)

3 a No local attraction on leg AB

Dam

Creek

Crossing Fence

Dairy Shed

House

A B

C

D E

Boundary

Road

Scale 1 : 20 000

You should work at 1 : 10 000.

Scrub area

N

P

Q

R

S

T

58 m

39 m 72 m

55 m

22°

134°

101°

53°

Scale 1: 2000

Yours should be twice this size.

N

A

B

C

D 452 m 450 m

400 m

30°

80° 160°

Scale 1: 10 000

You should work at 1: 5000. Factory

b Original Corrections Corrected

Leg Forward bearing

Back

bearing Difference Forward Back

Forward bearing

Back bearing

AB 052° 232° 180° 0° 0° 052° 232°

BC 127° 305° 178° 0° +2° 127° 307°

CD 084° 267° 183° +2° −1° 086° 266°

DE 221° 043° 178° −1° −3° 220° 040°

EF 119° 297° 178° −3° −1° 116° 296°

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c

d GA = 147 km at 095° N

52°

127°

86°

220°

116°

41°

30 km 28 km

42 km

46 km

53 km

62 km A

B

C

D

E

F

G

Scale 1: 2 000 000

You should work at 1: 1 000 000.

4

N

77 m

107 m

114 m 101 m

A1

B1 _C

1

D1

A

B _C

D

Scale 1: 4000

Yours 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°

14° 348° _P

Q R

S

Scale 1:10 000

You should work at 1: 5000. T

8

7 Leg Bearing Back bearing

AB 064° 244°

BC 100° 280°

CD 229° 049°

DE 271° 091°

EA 305° 125°

N

167 m _{125 m}

103 m 181 m

124 m B

C

D

E A

A1

B1

C1

D1

E1

Scale 1: 4000

Local attractions at C, E, F and A.

5 Original Corrections Corrected

Leg Forward

bearing

Back

bearing Difference Forward Back

Forward bearing

Back bearing

AB 144° 324° 180° 0° 0° 144° 324°

BC 070° 253° 183° 0° −3° 070° 250°

CD 017° 194° 177° −3° 0° 014° 194°

DE 319° 138° 181° 0° +1° 319° 139°

EF 228° 050° 178° +1° −1° 229° 049°

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Exercise 2.6

1

2 a 177.5 m

b AQ 076°, BP 298°, BR 256°

c AR = 45 m, RB = 155 m, BQ = 52.5 m

3 a 1 : 2000 b 642 m

c XB 047°, XD 195°, YC 099°, YA 294°

4 a 1 : 4000 c 1256 m

b

Chapter review

1 Booking a survey refers to the practice of recording details of a survey into a ﬁeld 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 ﬁrst station. This is the ﬁrst 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 061°

59 m 097°

22 m 194°

24 m 273° 27 m 328°

N

Children’s play area

Tree

Bench

Hut Barbecue

Swings

Scale 1: 2000

Yours should be twice this size. X

N

U

V

W X

149° 239°

191° 309°

228 m

264 m 300 m

464 m

Farmland

11°

Scale 1: 8000

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 sufﬁcient 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 m}2

11 a 023° b 291° c 152° d 203° 12 a 4.3781 m b 5.42° c 1 : 10.53 13 20.25 cm

14 a 4635 m2 _{b 3267.43 m}2

15 a

b 28 183.6 m2

16 a i ii 1216 m2

10 Leg Bearing Back bearing

OA 034° 214°

OB 109° 289°

OC 202° 022°

OD 255° 075°

OE 341° 161°

A B

79.2 m

117 m 81.9 m 123.3 m

72.9 m 26.1 m

Scale 1: 4000

Yours should be twice this size. 20.7 m

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b i ii 50 520 m2

17 108 m 18 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 m

P Q

R S

Scale 1: 4000

You should work at 1: 2000.

118 m

170 m 104 m

98 m

Jetty Boat

Tree Flag

Fence

A B

C D

E

Scale 1: 4000

You should work at 1: 2000.

shed

pole

143 m

20

21

23

Scale 1: 5000

Yours should be twice this size. D

N

C

B A

E

F

Y X

100 m

Dam

Windmill Pine

B

A

N

Scale 1: 2000

You should work at 1: 1000. tree

116 m

Scale 1: 4000

You should work at 1: 2000. P

Q

R S

N

50° 84 m

Field

A

B

b Scale 1 : 3460, perimeter about 768.1 m 22 a Station Bearing

from A

Bearing from B

P 050° 021°

Q 100° 116°

R 148° 195°

S 258° 264°

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CHAPTER 3

Exercise 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 $285.20 g $86.27 h $156.65 i $21.35 j $50.27

6 $5760 7 $105 8 $2.24

9 $942.02 10 $4550 11 $400 12 $600

Exercise 3.2

1 4.5% 2 20%

3 5.33% 4 13%

5 4.53% 6 19%, then 22%

Exercise 3.3

1 a $437.50 b $480 c $414 d $412.50 e $2737

2 a $3600 b $1507.50 c $203 d $11.94 e $1043.01

3 $1775.34, $46 775.34

4 $507.21 5 $75 394.52

6 $9408

7 $1627.40, $41 627.40 8 $57

9 $279 10 $3312, $11 312

11 6.5% 12 $197 826.09

13 a 6.58% b $115.50 (on $7300)

14 4.78% 15 $64 285.71

16 21.29% 17 $1 248 000

18 a $106.88 b $8066.04

Exercise 3.4

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

Exercise 3.5

1 a $18 161.18, $3161.18

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

d $4353.65, $3813.65 e $15 744.67, $8204.67 f $2921.72, $421.72 g $9134.55, $3334.55 h $7401.84, $1401.84

i $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 3.6

1 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 review

1 Divide the proﬁt 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 $4.14 9 $5074 10 $72.25 11 $600 12 $20 000 13 18% 14 6.96% 15 I = Pin 16 $3900

17 a $1200 b $756 c $1534.50 18 $46 294.52 19 $175.23 20 3.6%

21 A = P = P

22 $12 328.75, $4528.75 23 a $6553.98, $1553.98

b $13 854.63, $1854.63 c $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 i

k

--+

 

 k t ₁ i

k

--+

 

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CHAPTER 4

Exercise 4.1

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 = 25–29,

median ≈ 23.6, range = 25, interquartile range ≈ 7.9, SD ≈ 5.06 b Mean ≈ 51.3, modal class = 50–59,

median ≈ 52.4, range = 60,

interquartile range ≈ 17.6, SD ≈ 12.97 c Mean ≈ 114.8, modal class = 110–114,

median ≈ 113.3, range = 25, interquartile range ≈ 10.3, SD ≈ 6.52 d Mean ≈ 145.2, modal class = 150–169,

median ≈ 152.7, range = 120, interquartile range ≈ 38.6, SD ≈ 27.86 5 a

Median ≈ 170, interquartile range ≈ 9.5

Cumulati

v

e

frequenc

y

Height (cm)

100

80

60

40

20

0

140 150 160 170 180 190

b

Median ≈ 69.5, interquartile range ≈ 10 6 a 52.4 g b From 49.4 g to 55.1 g

7 a 2 b 3.24

8 17.7%

Exercise 4.2

1 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%, ﬁsh’n’chips = 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 sailboarders 5 Difﬁcult to contact all members of a

population, costly, time-consuming, …

6 B 7 A

8 A

Cumulati

v

e

frequenc

y

Pulse (beats/min)

100

80

60

40

20

0

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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 Biased—only those with telephones involved; poll conducted after emotive story; people can ‘vote’ more than once.

11 a Student body at Grifﬁth 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 inefﬁcient as many selections would probably have no person seated in that position.

Exercise 4.3

1 a 25 b 20 c None d 37, 51

2 a 4.9 b 197 c 22.1 d None

3 a 5, 9

b Mean ≈ 1.9 , median = 2 c Mean = 1.5 , median = 1.5

d Both the mean and the median are affected by the outliers.

State or territory

a By sex b Without

regard to sex (persons) Females Males

NSW 84 85 169

Vic. 62 63 124

Qld 46 46 93

SA 19 20 39

WA 25 24 49

Tas. 6 6 12

NT 3 2 5

ACT 4 4 8

4 a 124 kg

b Mean ≈ 71.4 kg, median = 71 kg c Mean ≈ 68.9 kg , median = 70 kg

d Both the mean and the median are affected by the outliers.

Exercise 4.4

1 The results in Chemistry are a little higher, but more spread out. Most of these students ﬁnd Chemistry easier than Physics.

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

3 The ﬁrst 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 review

1 The median score item is calculated as the (n + 1)th item. The score group that contains this 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 stratiﬁed 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

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---10 Mean ≈ 21.3, modal class = 15–19, 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 ≈ 9 13 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, 55

b 262, 229, 346, 204, 215, 248, 343

c 3328, 3905, 3983, 4249, 6902, 5777, 3117, 3307, 4865

15 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 48

19 He did slightly better in Maths.

20 The men’s results are more spread and lower than the women’s results.

CHAPTER 5

Exercise 5.1

1 $12 900 2 $1 124 000

3 9.68% 4 10.79%

5 a 8.78% b $4870

6 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 $6000

2 $6000 $1500 $4500

3 $4500 $1500 $3000

4 $3000 $1500 $1500

Cumulati

v

e

frequenc

y

Score

50

40

30

20

10

0

5 10 15 20 25 30 35 40

Useful life is 5 years.

Exercise 5.2

1 a $740 b 16.4%

2 a $10 000 b 28.57%

3 a $3920 8 Year Starting

value

1 $6000 $1200 $4800

2 $4800 $1200 $3600

3 $3600 $1200 $2400

4 $2400 $1200 $1200

5 $1200 $1200 $0

c _Year Starting value

1 $4500 $740 $3760

2 $3760 $740 $3020

3 $3020 $740 $2280

4 $2280 $740 $1540

5 $1540 $740 $800

c _Year Starting value

1 $35 000 $10 000 $25 000

2 $25 000 $10 000 $15 000

3 $15 000 $10 000 $5 000

b _Year Starting value

Written-down value

1 $28 000 $3920 $24 080

2 $24 080 $3920 $20 160

3 $20 160 $3920 $16 240

4 $16 240 $3920 $12 320

5 $12 320 $3920 $8 400

6 $8 400 $3920 $4 480

Book v

alue

5000

4000

3000

2000

1000

0

Time (years)

1 2 3 4 5

Book value of fryer d

Book v

alue

40 000

30 000

20 000

10 000

0

Time (years)

1 2 3

(12)

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%, $12 961.48 8 14.34% 9 a 34.13% b $806.23

Exercise 5.3

1 $531.47 2 $714.09 3 a 4846

b Answers will vary, depending on the year. 4 a 72 m2 b 116 m2 c 80 days

5 $15 463.68 6 $28 056.22 7 $14 527.28 8 $10 344.52 9 $15 362.65 10 $4495.69 11 $5291.43 12 $11 255.03 13 $7747.10 14 $3577.30 15 $4906.76

Exercise 5.4

b Year Starting

value

1 $27 500 $5500 $22 000

2 $22 000 $5500 $16 500

3 $16 500 $5500 $11 000

4 $11 000 $5500 $5 500

5 _Year Starting value

1 $8900 $1068 $7832

2 $7832 $939.84 $6892.16

3 $6892.16 $827.06 $6065.10

4 $6065.10 $727.81 $5337.29

1 Year CPI Inﬂation Year CPI Inﬂation

1986 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

Book v

alue

30 000

20 000

10 000

0

Time (years) Book value of lathe

2 a 45c in 2002

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

3 $95 034.72 4 $183.25

5 $5624.46, so a decrease in real terms 6 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 5.5

1 $99 292.18 2 $18 842.78 3 a $52 500 b $22 611.34

c $14 695.82

4 a $36 000 b $9566.44

5 a $1632 b $8659.82

6 a $436 750 b $45 000 c $32 918.34

d $2 164 576, $88 320, $68 914.87 respectively

Chapter review

1 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 inﬂation.

5 The rate of inﬂation is calculated by ﬁnding the percentage change in the CPI.

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

7 $14 200

7 Year 1998 1999 2000 2001 2002

Amount (millions)

$16.71 $16.25 $18.43 $21.47 $22.94

2000 $s (millions)

(13)

8 a 4.12% b $23 700

10 20.83%

Scrap value is $1772.

12 238 m2 _{13 $56 741}

14 $11 390.81 15 $12 731.04 16 $17 961.06 17 $10 208.75 18 $11 214.05 19 $3025.86 20 $8210.73 21 $321 849.30 22 $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 $40 430.04 26 a $44 057.90 b $13 888.89 c $94 864.40

CHAPTER 6

Exercise 6.1

1 a 0.433 b 0.072 c 40

d 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 16

3 a 0.204 b 0.170 c 0.231 4 357

5 a Taxi 0.06, train 0.337, bus 0.26, car 0.287, bike 0.02, other 0.037

b 0.287 c 7 d 8

6 a 200

1 $65 000 $7500 $57 500

2 $57 500 $7500 $50 000

3 $50 000 $7500 $42 500

4 $42 500 $7500 $35 000

5 $35 000 $7500 $27 500

6 $27 500 $7500 $20 000

1 $5600 $1400 $4200

2 $4200 $1050 $3150

3 $3150 $787.50 $2362.50

4 $2362.50 $590.62 $1771.88

b Score 2 3 4 5

Rel. freq. 0.12 0.1 0.075 0.08

Score 6 7 8 9

Rel. freq. 0.065 0.105 0.085 0.09

Score 10 11 12

Rel. 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 6.2

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

b c d e 0

2 a {5R, 8G, 7B}

b c 0 d e Green

7 a Amount

spent ($) Frequency Rel. freq.

0.00–1.99 10 14.3%

2.00–3.99 25 35.7%

4.00–5.99 15 21.4%

6.00–7.99 9 12.9%

8.00–9.99 5 7.1%

10.00–15.99 6 8.6%

Total 70

8 Rent per person

($/week) Number Rel. freq.

65–69 6 3%

70–74 10 5%

75–79 36 18%

80–84 62 31%

85–89 34 17%

90–94 25 12.5%

95–99 16 8%

100–104 11 5.5%

Total 200

9 a Height (cm) Frequency Rel. freq

160–169 7 10%

170–174 13 18.6%

175–179 23 32.9%

180–184 14 20%

185–199 13 18.6%

Total 70

10 a Mass (kg) Number Rel. freq.

1.0–1.9 13 13%

2.0–2.4 18 18%

2.5–2.9 31 31%

3.0–3.4 17 17%

3.5–3.9 12 12%

4.0–4.4 9 9%

Total 100

1 6

--- 1

2

--- 1

2

---7 20

--- 2

(14)

---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

7 8

P(3 girls) =

9 a b c

10 a b

11 a b c 1 4 --- 1 4 --- 1 4 --- 1 2 ---9 25 --- 1 25 --- 8 25 --- 16 25 ---6 11 --- 5 11 --- 6 11

---S = SS H = SH D= SD C = SC S = HS H = HH D= HD C = HC S = DS H = DH D= DD C = DC S = CS H = CH D= CD

C = CC S H D C 1 16 --- 1 4 --- 1 8 --- 1 16 ---1 4 --- 3 4 --- 9 16 ---5 6

---G = GGG

B= GGB G= GBG

B = GBB G = BGG

B = BGB G= BBG

B= BBB G B G B G B 1 8 ---3 20 --- 3 10 --- 9 20 ---11 40 --- 3 8 ---ei t ei t ei t ei t ei t ei t ei t ei t v am h m v am h m r f 1 4 --- 1 16 ---12

13 Answers will vary—check 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 6.3

1 a

b 0.12 = 12% c 7

2 a

b 0.67 3 a

b 0.17

4 a i 2.3% ii 18.9%

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

5 a i 0.25 ii 0.14

b i 17 or 18 ii 13

0 0.5 1

c d e a b

0 0.5 1

d f e a c

1 5 --- 2 5 ---4 7 --- 2 7 --- 1 7 ---3 8 --- 5 8 ---A B x z y Probability 0.25 0.2 0.15 0.1 0.05 0 0 4

Number of siblings

7 0.3

0.35

1 2 3 5 6 8

Probability 0.25 0.2 0.15 0.1 0.05 0 0 4

Heads in 6 tosses

5 0.3

0.35

1 2 3 6

Probability 0.25 0.2 0.15 0.1 0.05 0 25 29

Maximum temperature (°C)

32 0.3

(15)

b

c 0.13 d 8

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

Chapter review

1 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 6

Frequency 7 12 9 12 12 8

7 a Result 3–5 6–8 9–11 12–14 15–19

Number 9 8 8 17 8

8 a Height (cm) 141–155 156–160 161–165

Frequency 10 8 7

Height (cm) 166–170 171–175 176–180

Height (cm) 181–185 186–195

Frequency 11 9

Probability

0.25

0.2

0.15

0.1

0.05

0

1 5

Result of rolling a die

2 3 4 6

Probability

0.1 0.08 0.06 0.04 0.02 0

3

Test result

0.12

5 7 9 11 13 15 17 19

Probability

0.025 0.02 0.015 0.01 0.005

0

Height (cm)

0.03 0.035

141

0.04

146 151 156 161 166 171 176 181 186 191 196

frequency of outcome number 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, reﬂecting the fact that one of the possible outcomes must occur.

17 D 18 A 19 D

b 133

21 {3P, 5G, 2O} 22 E 23 C

24 D 25 B 26

20 a Amount spent ($) 20–24 25–29 30–34

Number 24 56 68

Rel. freq. 6.3% 14.6% 17.7%

Amount spent ($) 35–39 40–44 45–49

Number 66 60 48

Rel. freq. 17.2% 15.6% 12.5%

Amount spent ($) 50–54 55–59 60–64

Number 30 20 12

Rel. freq. 7.8% 5.2% 3.1%

n event( )

n sample space( ) --- .

2 3 ---.

1 2 ---

.

(16)

---27 a

b 30 c d

28 a

b c d

b 0.23 c 23

29 a No. computers 0–1 2 3 4–6

No. families 15 15 10 12

Rel. freq./unit 0.144 0.288 0.192 0.077

30 a Mass (g) 50–59 60–69 70–79 80–89 90–99

Number 9 11 13 9 6

Rel. freq./unit 0.019 0.023 0.027 0.019 0.013 Blue Blue Black Red Red

Blue Blue Black Red Red

Blue Blue Blue Red Red

Blue Blue Blue Black Red

Blue Blue Blue Black Red Blue

Blue

Black

Red

1 5

--- 11

15

---H

T

H

T

T H

H T H T H T H T 1

8

--- 3

8

--- 1

2

---Probability

0.25 0.2 0.15 0.1 0.05

0

0 4

Number of computers

5 0.3

0.35

1 2 3 6

b 0.4

31 a 1060 b 207 c 519 d 0.51

32 a b c d 1

33

34 a ≈ 0.0114 b ≈ 0.000 12

c 0.117 d 81

CHAPTER 7

Exercise 7.1

1 a b

35 a Fall (m) 0–0.4 0.5–0.9 1–1.4 1.5–1.9

Frequency 2 5 15 18

Fall (m) 2–2.4 2.5–2.9 3–3.4 3.5–3.9

Frequency 8 4 2 5

Fall (m) 4–4.4 4.5–4.9

Frequency 0 1

b Fall across site (m) 0–0.9 1–1.4 1.5–1.9

Rel. freq./unit 0.012 0.05 0.06

Fall across site (m) 2–2.4 2.5–4.9

Frequency 8 12

Rel. freq./unit 0.027 0.008

Probability

0.025 0.02 0.015 0.01 0.005

0

80

Mass of apple (g)

90 0.03

50 60 70 100

4 13

--- 5 13

--- 4 13

---27 32

---1 88

--- 1

8184

---Probability

0.05 0.04 0.03 0.02 0.01 0

Fall across site (m)

0.06 0.07

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x y

2 −2

2 4

−2

y = 3x − 2

−4

f d

2 −1

4

−2

−4

d = 2f + 3

(17)

c d

e

2 a b

c d

e

3 a b

c d

h m

2 −2

6 8

4

2

m = 6 − 2h

k X

2 −2

6 8

4

2

X = −k + 4

C

8

4

2

C = 5 − 2x

6

k

2 −2

y

6

4

2

2x + 4y = 10

x

6

2 4

q

2

−2

p − 2q = 6

p

6

2 4

−4

y

6

4

−x + 3y = 6

x

−2 −6 −4

2

v

−2

−4

−6

3m − 2v = 12 m

6

2 4

c

8

4

2

2a + c = 8

6

a

6

2 4

y

6

4

2

4x + 5y = 20

x

6

2 4

q

8

4

2

p + q = 8

p

8

2 6

6

4

d

−4

−8

3c + 2d = −15 c

−2 −6 −4

−2

−6

e

6

2

d

−2 −4

4

3d − 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 7.2

1 a Rooms occupied = n, proﬁt = P, initial cost = $300, proﬁt/room = $25, number of rooms = 30

b P = 25n − 300 c

d $325 e 20 f 12 rooms

2 a Resistance = R, temperature = T, resistance at 10°C = 25 Ω, temperature rise = 2°C/Ω b T = 2R + 5

c

d 43°C e 75°C f 17.5 Ω

y

2

x

−8 −12

4

3y − x = 12

−4

Hotel rooms P ($)

500 400 300 200 100 0 −100 −200 −300

20 30 n

P = 25n − 300

10

Resistance wire T (°C)

80

60

40

20

0

20 30 R (Ω)

(18)

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

c $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 − 0.2W c

d 4.1 m e 7 weeks ago f 32 weeks 5 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 = 20

c

d 300 tonnes

6 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 100

c

7 a Number of service calls = s, number of reﬁlls = r, cost of service call = $80, cost of reﬁll = $20, total amount = $500 b 80s + 20r = 500 d 9

c

10

Level of dam L (m)

8

6

4

2

20 30

W (weeks) L = 6.5 − 0.2W

0 Present −10

1 2

--- 1

2

---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

0 5 20

10

6.25

Exercise 7.3

1 2

3 4

5 6

7 8

9 10

Exercise 7.4

1 10c + 25t 900 2 0.8h + 1.5s 200 3 20m + 35a 1800 4 b + t 6000 5 30c + 40t 400

Exercise 7.5

1 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

2 −2

2 4

−2

y < 3x + 2

x y

2 2 4

−2

3x − y 9

−4

4

x y

4x + 3y −24

−4

−8

−12 −4

−8 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 −x 4

4 6 2

6

x

4 6 8 10 2

y

x + 2y 10

4

2 6

x

2 −2

2 4

−4

y

y > 3 − 4x

−2

x

2 4

−4

y

−2 −2

y − 5x < 10

x y

4

6x − 3y 15

−2

−4

−6 −2

2

(19)

c

d m = 0 → 7, h = 6 → 2

2 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 5

c

d s = 0 → 16, b = 16 → 5

3 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 1.5 c

d i = 0 → 1.17, c = 3.33 → 1.50

4 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

h

0

2 8

4

1 4 5 6 7

6.67

Pottery mugs s

25

15

5

b

0

5 20

10

10 15

16 24

Chocolates and ice-creams c ($)

2

1

i ($)

0

1 4

2 3

3

1.5 3.33

2.14

Ferries s

5

3

1

b

0

1 4

2

2 3

2.5

3.33 1.67

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 review

1 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 ﬁxed 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 deﬁned 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 0

Small, s 2 2 3 4 5

Excess capacity (> 1200)

Large, b 2 1 1

Small, s 3 4 5

Automotive tool kits e

60

30

10

c

0

10 40

20

20 30

50

25 15 36

x

2 −2

2 4

−4

y

y = 4x − 5

−2

v

4 v = 7 + 2f

f

−2 −4

2 8

(20)

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

−2

4 4

x y

2

3x + 5y − 18 = 0

4

4 6 2

6

−2

−4 f

g

2f − 3g + 9 = 0

4

2 4 2 6

−4 −2

x y

5x − 2y = −7

6

3 3

12

−3 9

x y

2

3x + 5y = 20

4

4 6 2

623

---c d

d − 4c = 12

6

2 2 12

−4 8

4 10

−2

g

2 4

h

3g − 2h = 9

4

2 6

412

---12 a b

c

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

c

d 2000

14 a Hire time = h, cost of hire = c, initial cost = $200, hourly cost = $50 b c = 200 + 50h

c

d 6 h

15 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 460 16 2b + 3s 50

x y

12

3x − 4y < 24

−2

−8

−10 2

4 8

−4

−6

x y

4

5x + 2y 12

4 6

−2 2

2

2.4

y > 3x − 6 y

−4 −2

−6

x

−2 2

Donut stand P ($)

400

200

0

−200

−400

2000 n P = 0.25n − 300

1000

1200

Backhoe hire c ($)

800

600

400

200

0

8 h (h) c = 200 + 50h

2 4 6

Circus takings a

400

300

200

100

0

600 c 10c + 16a = 5400

100 200 300 400 500

(21)

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 2 c

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 2

c

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 5 c

d f = 0 → 12, p = 5 → 0

CHAPTER 8

Exercise 8.1

1 a Simple path b Circuit c Simple circuit d Path e Simple cycle f Simple path g Simple circuit h Chain i Simple path j Cycle Office equipment w

4

3

2

1

0

5 p

1 2 3 4

4.8 3.43

Old movie serials t

6

4

2

0

10 c

2 4 6 8

Knitted jumpers p

20

15

10

5

0

15 f

5 10

12

2 a Connected undirected circuit b Connected directed mixed c Disconnected (directed trees) d Connected undirected tree e Connected directed tree f Connected directed cycle

3 a b

4 a

b Luke Fellowes, Kerry Stubbs, David Alsop c Felix Goldschmeid (known to LF and KS) 5

6 a and b

c MH–MG–BS–FF–B–SH–MH = 53 min d MH–BS–FF–MH = 28 min

Exercise 8.2

1 a AEFGH = 30 b JEFC = 23 c DGFI = 26 d ABDGF = 43 e ABCF = 25

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

3 a

F K/M

D Lo E

B1

ES S

B2 B4

La

WC B3

B G

B2

B B3

F La

WC K G

Lo D

E

B1 ES

B4

JK LS

JC FG

KS LF

AS JF

DA PP

Ph Ta V

Ca L

Th

Ma

S

Indo

ET SL

Ba My

I Bh N

Ch

SH

B MG

BS

FF

MH

6

14 2 10

15 12 10

15 6

Ca

Tow

G E

MI

Ch

Too

B

$33 $21

$6 $18

$24

$12 $15

$18

$21

$15

$10.50

(22)

b $36 c $51

d $39 e $25.50

4 a

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

government approval, normal laboratory experiments and slow marketing to complete in 11 months at $290 000. 5

Redcliffe–Caboolture–Esk = 30 min

Exercise 8.3

1 a b

2

3 a

b Port–R4–R5–R2–R1, R4–R6, R5–R7 and R5–R3 = 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 T I L Be 20 K 35 45 15 25 20 15 5 10 15 50 20 10 25

Times in minutes

a b

c

d e

g

f h i

8 6 7 6 4 4 5 5

A B C D

G H I J

E F

9 5 7

7 6 7 6 6 5 c

b _d e

a l m _g f

j i k h 5 6 4 7 5 7 7 6 6 8 8 8 R1 R2 10 4 5 8 R3 R4 R5 R6 R7 Port Distance, km

6 10

5 ₆

8 2 69 10 8 10 10 2 11 5 5 9 3 6 7 8 4 9 10 c

d Same connections as part b, cost = $53 000 4 a

b Connections marked in purple = 140 m c

d Connections marked in purple = $22 500 5 a AEHIGCBDF or ABDCGIHEF = 18

b AEFHIGCBDF = 20

6 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 review

1 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 14 8 9 12 R3 R4 R5 R6 R7 Port Cost × $1000

10 14

9 10

12 1013 14 12 14 14 6 15 9 9 13 7 10 11 12 8 13 14 6 F D C E A B 25 20 20 30 20 60 15 15 15 15 25 35 60 40 Distance, m F D E A $3250 $8050 $2650 $5650 $2650 $2650 $3250 $3850 $4450

$2650 $3250 $8050 $5050

$3850

(23)

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

4 Dijkstra’s Algorithm ‘grows’ shortest trees from the starting node until the ﬁnishing 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, Kruskal’s 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 circuit

c Circuit d Path

e Simple cycle f Simple path g Simple circuit h Chain 8 a Connected undirected circuit

b Connected undirected tree c Connected directed mixed d Disconnected directed (mixed) e Connected undirected mixed f Connected directed cycle 9 a AGMNO = 12

b KLMIJ = 10 c EIMGF = 11

10 a b

11

12 Repeat the following sections: Fairlawn between Toohey and Canﬁeld 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 Canﬁeld and Arkindale (2), then around Bankside (1) to Glade. Do the 2 in Ivymount. Come back around the rest of Bankside (2) and ﬁnally do the ones in Fairlawn (1) and Dellwood (2) before returning to Toohey Rd.

b a

c d g

h i e f 2 3 3 2 2 2 3 3 B D A C E G F 3 3 3 4 5 4

S1 S2 S3 S4

S5 S6 S7

14

b B–Mo–BH–A = 17 h c A–BH–Mo–R–T–MI = $214 15 Keep City–Carindale, City–Moorooka,

Moorooka–Acacia Ridge, Acacia Ridge– Ipswich, Moorooka–Annerley, Carindale– Mt Gravatt, Logan–Springwood,

Springwood–Beenleigh and either Carindale– Logan or Mt Gravatt–Logan.

CHAPTER 9

Exercise 9.1

1 a b c d

2

3 a b c d

e f g

4 a b c d

e f

5 a 0.93 b About 0.602 c 0.398 6 About 0.075 = 7.5%

7 a 0.294 b 0.668

8 a b c d e

9

a ≈ 0.321 b 0.143 c 0.25

10

a 33 b ≈ 0.455 c 0.364

(24)

---11 49:15

12 a 37:1 b 10:9

13 a b c

d e 61:3 f $10.67

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

Exercise 9.2

1 a Discrete b Continuous c Discrete d Continuous e Discrete f Discrete g Continuous

h Discrete (must be to 5 cents), continuous in bulk

2 a Yes b Yes c No

b

c No

14 Total 2 3 4 5 6

Fair 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 15

f 2 3 4 6 1

p(t) 0.08 0.12 0.16 0.24 0.04

t 16 17 18 19 20

f 3 0 1 1 4

p(t) 0.12 0 0.04 0.04 0.16 1 64 --- 1 64 --- 1 64 ---3 64 ---27 64 --- 27 64 --- 9 64 --- 1 64 ---C 1 0 0.15 0.25 0.3

0 2 3

0.4 0.45 0.35 0.2 0.1 0.05 p(c) 1 36 --- 1 18 --- 1 12 --- 1 9 --- 5 36 --- 1 6 ---5 36 --- 1 9 --- 1 12 --- 1 18 --- 1 36 ---p(s)

2 3 6 8 S

0 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 9.3

1 E(T) ≈ 5.27 2 E(X) = 1 3 E(X) = 1.75 4 E(X) = 3.5 5 7

6 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) ≈ 2.1

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

10 a 0.0775 b 0

11 a $2400 b $−1000

12 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

3 25 --- 2 5 ---1 6 --- 5 18 --- 2 9 --- 1 6 --- 1 9 --- 1 18 ---11 36 --- 1 4 --- 7 36 --- 5 36 --- 1 12 --- 1 36 ---p(d)

0 1 3 4 D

0 0.1 0.2 0.3 2 5 p(s)

1 2 4 5 S

0 0.1 0.2 0.4 3 6 0.3 1 16 --- 7 16 --- 5 16 --- 1 8 --- 1 16 ---p(x)

0 1 3 4 X

(25)

E(T) ≈ 7.75

E(T) ≈ 8.5

Exercise 9.4

1 a No b Yes c Yes d No e Yes 2 a 4

b p = ≈ 0.17, q = ≈ 0.83

c 0.386 d 0.116

3 a n = 4, p = 0.3, q = 0.7

b 0.0081 c 0.265 d 0.760 4 a 0.008 b 0.201 c 0.335 d 0.000 02 5 a 0.016 b 0.054 c 0.193

6 a 0.125 b 0.375 c 0.875 7 a 0.336 b 0.428

8 0.075

9 a 0.813 b 0.570

10 a 0.087 b 0.116 c 0.682 d 0.823 11 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 review

1 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 doesn’t, so

P(A) + P(A′) = 1. To ﬁnd 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 speciﬁc values with a value determined by a random event.

7 0 p(x) 1 and ΣP(X = x) = 1.

13 a t 2 3 4 5 6 7

p(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 7

p(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

1 6

--- 5

6

---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 ﬁxed 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 7

14 0.784 15 B

16 5:3 17 D

18 a Yes b No c Yes d No e No

b

c E(D) ≈ 1.944

20 3.5 21 E(X) = 4.45

22 a 0.2061 b 0.1443

23 C 24 A

25 a 0.0114 b 0.0004

E(D) ≈ 2.526

27 a 0.1074 b 0.2684 c 0.8791 28 a 0.0160 b 0.0219 c 0.1702 29 a 0.0874 b 0.6814 c 0.8227

CHAPTER 10

Exercise 10.1

1 a 313° T b 119° T c 349° T d 228° T e 093° T f 023° T g 009° T h 020° T i 191° T j 219° T

2 a 247° M b 357° M c 161° M d 071° M e 143° M f 011° M g 203° M h 004° M i 237° M j 024° M

19 a d 0 1 2 3 4 5

p(d)

26 d 1 2 3 4

P(D = d) 0.3823 0.2362 0.1459 0.0901

d 5 6 7

P(D = d) 0.0556 0.0344 0.0555 1

6 --- 5

18 --- 2

9 --- 1

6 --- 1

9 --- 1

18

---p(d)

0 1 3 4 D

0 0.1 0.2