year 12 maths a textbook - answers

A n s w e r s

515

answ

ers

CHAPTER 1 Simple and compound interest

Skills check

1 a

$1

b

$8.75

c

$1.50

d

$0.25

e

$4

f

$0.25

2 a

7.25

b

0.0725

c

0.2

d

0.002

e

0.125

f

0.001

3 a

years

b

years

c

years

d

2 years

e

4 years

f

2 years

4 a

450

b

525

c

21 000

d

1.157 625

e

1.083

f

1.877

Exercise 1A — Simple interest

1

2

3

16

$584.50

17 a

The Big-4 Bank offers the best rates.

b

The Big-4 Bank charges 11 % p.a. for a loan while The Friendly Building Society charges 12% (

=

12

×

1% per month).

18 a

$627.13

b

$12 542.50

19 a i

$1540.63

ii

$6162.50

b

Yes

20 a

$2247

b

$15 729

c

7 years

Exercise 1B — Finding

P

,

R

and

T

1

2

3

14 a

Yes ($1112.50)

b

No

c

Yes ($1600 in 23 months)

d

Yes ($1281.60)

Exercise 1C — Graphing simple interest functions

1 a

b

2 a

b

c

1600

d

$16 000

3 a

b

c

Answers

a

$136.00

b

$56.70

c

$145.25

d

$110.40

e

$255

f

$336.89

g

$178.57

h

$43.88

i

$11.76

j

$229.68

k

$544.05

a

$103.50

b

$2700

c

$325

d

$131.25

a

$360

b

$1020

c

$27 700

d

$17.70

e

$13.67

4

C

5

A

6

B

7

B

8

B

9

A

10

D

11

B

12

A

13

$465.50

14

$25.50

15

$2418.75

a

$3070

b

$4400

c

$5425

d

$236.36

e

$2500

a

10%

b

6.25%

c

80%

d

2.125% or 2 %

e

3.36%

a

1 year

b

18 months

c

3 months

d

7 years

e

1 month

4

$1515.79

5

$2133.33

6

$352

7

24 months

8

3 years

9

C

10

B

11

B

12

D

13

A

14--- 1

6--- 2

3---

112------ 5

12------ 1

2---

13---

12---

18---

No. of years

1 2 3 4 5

Interest

$400 $800 $1200 $1600 $2000

No. of years

1 2 3 4 5

Interest

$1600 $3200 $4300 $6400 $8000

01 2 3 4 5

500

Inte

rest

($)

Years

1000

1500

2000

0

01 2 3 4 5 6 7 8 9 10

2000

Inte

rest

($)

Years

400060008000

10 00012 00014 00016 000

0

01 2 3 4 5

2000Inte

rest

($)

Years

4000

6000

0

1 2 3 4 5

200

Inte

rest

($)

Years

400

600

800

1000

00

01 2 3 4 5

1000

Inte

rest

($)

Years

2000

3000

4000

0

1A

➔

1C

Maths A Yr 12 - Answers Page 515 Wednesday, September 11, 2002 3:41 PM

516 A n s w e r san

swer

sd

4 288

5 a

b

6 a

b

7 a

b

8 a $4500, $5000, $5200

b $875

9

10 Quick Questions 11 $800 2 $1260 3 $2700 4 $1.905 $7000 6 $138 7 $215.44 8 $1998.759 54c 10 $173.58

Exercise 1D — Calculation of compound interest1 $6655 2 $17 173.50 3 $2938.664 a $4630.50 b $9274.19 c $24 488.80

d $13 503.78 e $12 588.725 $70 555.26 6 $5027 $14 059.91 8 $31 850.339 a $1003.69 b $8985.76 c $181 402.12

d $20 039.67 e $1 264 568.9510 B 11 B 12 B 13 C14 a $15 746.40 b $15 793.09

c $15 817.40 d $15 833.9915 a 0.0219% b $108 320.71

c $8320.71 d $320.7116 a $4720 b $4726.24

c Compounding interest

Exercise 1E — Graphing compound interest functions

1 a

b

2 a

b c $25 900

3 a

b

No. of years 1 2 3 4 5

Interest (5%) $1250 $2500 $3750 $5000 $6 250

Interest (6%) $1500 $3000 $4500 $6000 $7 500

Interest (8%) $2000 $4000 $6000 $8000 $10 000


Interest (3%) $165.00 $330.00 $495.00 $660.00 $825.00

Interest (3.5%) $192.50 $385.00 $577.50 $770.00 $962.50

Interest (3.75%) $206.25 $412.50 $618.75 $825.00 $1031.25

01 2 3 4 5

5000

Inte

rest

($)

Years

10 000

15 000

20 000

0

01 2 3 4 5

100

Inte

rest

($)

Years

200

300

400

0

1 2 3 4 5

Inve

stm

ent

($)

Years

3000

0

3500

4000

0

01 2 3 4 5

5000

Inte

rest

($)

Years

10 000

0

8%6%5%

01 2 3 4 5

500

Inte

rest

($)

Years

1000

1500

0

3.75%3.5%3%

1 2 3 4 5

4000

Inte

rest

($)

Years

4500

5000

00

6.2%5.8%5.2%


Future value $8400 $8820 $9261 $9724 $10 210


Future value $12 960 $13 997 $15 117 $16 326 $17 632

1 2 3 4 5

Fut

ure

valu

e ($

)

Years

8000850090009500

0

10 000

0

1 2 3 4 5 6 7 8 9 10

Fut

ure

valu

e ($

)

Years

10 000

15 000

20 000

25 000

30 000

00

1 2 3 4 5

Fut

ure

valu

e ($

)

Years

15 000

20 000

25 000

00

1 2 3 4 5

Fut

ure

valu

e ($

)

Years

2000

3000

4000

00


A n s w e r s 517

answ

ers

4 a

b

5 a b

6 a

b

7 a

b

10 Quick Questions 21 $2051.28 2 $17 253 3 $3437.504 $429.69 5 $2315.25 6 $315.257 $6792.58 8 $6805.66 9 $6812.41

10 $14 700.68

Exercise 1F — Nominal and effective interest rates1 4.06% p.a. 2 4.08% p.a. 3 3.79% p.a.4 a 4.27% p.a. effective rate

b 4.26% p.a. effective rateSo 4.2% p.a. compounding quarterly is better.

5 5.095% effective rate, 5.12% effective rate. Choose 5% p.a. compounding monthly.

Chapter review1 $10002 a $1296 b $2820 c $42

d $4.05 e $7617.583 a $7280 b $39 780 c $4554 D 5 B 6 C7 6.5% 8 15 months 9 B

10 B 11 C

12 a

b

c 450 d $450013 $2778.30 14 $5700.4715 a $3932.39 b $12 596.90 c $14 457.96

d $23 031.76 e $5334.6716 $756.9417 a $12 024.02 b $12 052

c Compounding interest18 a

b

c $29 60019 3.94% p.a.20 4.18% p.a., 4.08% p.a.

Choose 4.1% p.a. compounding monthly.21 a 3.95% p.a. effective

b 3.97% p.a. effectivec 3.96% p.a. effective

Choose 3.895% p.a. compounding monthly.

CHAPTER 2 Appreciation and depreciationSkills check1 a 1.1 b 1.01 c 1.0175

d 0.9625 e 0.9775 f 0.99752 a $4 b $5 c $3.95

d 41c e $1.10 f $4.373 a $44 b $3 c $7.64

d $18 e $135.94 f $39404 a 40% b 87.5% c 64.5%

d 4.5% e 51.4% f 1.56%5 a 1.06 b 0.94 c 1.04

d 0.97 e 1.03 f 0.97

Years 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

FV $2060 $2122 $2186 $2252 $2318 $2388 $2460 $2534 $2610 $2688


Interest (4%) $320 $653 $999 $1359 $1733

Interest (6%) $480 $989 $1528 $2100 $2706

Interest (8%) $640 $1331 $2078 $2884 $3755


Annually $4240 $4496 $4764 $5048 $5352

Six-monthly $4244 $4504 $4776 $5068 $5376

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Fut

ure

valu

e ($

)

Years

20002100220023002400250026002700

00

1 2 3 4 5 6 7 8 9 10

Fut

ure

valu

e ($

)

Years

1400

1600

1800

1200

00

01 2 3 4 5 6 7 8 9 10

100

Com

poun

d in

tere

stea

rned

($)

Years

200300400500600

0

1 2 3 4 5

Inte

rest

($)

Years

8000

10 000

12 000

00

Interest (8%)Interest (6%)Interest (4%)

1 2 3 4 5

4000Fut

ure

valu

e ($

)

Years

4500

5000

5500

00

AnnuallySix-monthly


Interest $450 $900 $1350 $1800 $2250


Future value $20 800 $21 632 $22 497 $23 397 $24 333

01 2 3 4 5 6 7 8 910

500

Inte

rest

($)

Years

1000

350040004500

1500200025003000

0

00

1 2 3 4 5 6 7 8 9 10

Fut

ure

valu

e ($

)

Years

20 00021 00022 00023 00024 00025 00026 00027 00028 000

1D➔

1F


518 A n s w e r san

swer

sExercise 2A — Inflation and appreciation1 $20 8002 a $618 b $48.15 c $1.91

d $579.60 e $932.403 a $878.05 b $901.764 $117.90 5 $619 6 $2.527 $1.20 8 $122.80 9 D

10 $500 11 $2350 12 $2460

Exercise 2B — Modelling depreciation1 a

b V = 100 000 − 10 000A2 V = 50 000 − 8000A

3 a

b $20 000 c 9 years4 a V = 6400 − 2000A

b c 4

5 a b $2000

6 a b $17 000 c 7

7 a i $160 000 ii $128 000 iii $102 400iv $81 920 b

8 B

b See part d.

d

e 6 years

Exercise 2C — Straight line depreciation1 $20 0002 a $1000 b $10 300 c $270 000

d $145 e $32 0003 a $7 125 000 b $3 750 0004 $10 6005 8 years6 a 6 years b 5 years

c 8 years d 7 years7 $2500/year8 a $4000/year b $12 500/year c $14 500/year9 $900/year

10 $25 00011 a $110 000 b $26 500 c $145012 $78 000

0

20 000

40 000

60 000

80 000

100 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

0

10 000

20 000

30 000

40 000

50 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

0

10 000

20 000

30 000

40 000

50 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

01000200030004000500060007000

10 2 3 4

Val

ue (

$)

Age (years)

0

4 000

8 000

12 000

16 000

20 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

0

120 000

240 000

360 000

480 000

620 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

9 aAge (years) Value ($)

New (0) 30 000

1 26 000

2 22 000

3 18 000

4 14 000

5 10 000

cAge (years) Value ($)

New (0) 30 000

1 24 000

2 19 200

3 15 360

4 12 228

5 9 830

0

40 000

80 000

120 000

160 000

200 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

0

6 000

12 000

18 000

24 000

30 000 Straight line valueDeclining balance

value

0 2 4 6 8 10

Val

ue (

$)

Age (years)


A n s w e r s 519

answ

ers

Exercise 2D — Declining balance or diminishing value method of depreciation1 $20 4802 a $2220 b i $750 ii $3903 7 years4 $383 0005 a $5900 b $68 100 c $1200

d $62 100 e $39006 $61747 $676 0008 a $14 600 b $20 4009 A

10 C11 a $5360 b $2640

c $3591 d $176912 5 years

10 Quick Questions 11 $650 2 $2350/year3 7 years 4 $21 000

5 S = Vo 6 $7250

7 $11 000 8 $389 0009 $37 500 10 16 years

Exercise 2E — Depreciation tables1 a $1638.50 b $9537.50 c $34 8702 a

b c After 6 years

b

Age of car (years)

Straight line value ($)

Diminishing value ($)

New (0) 40 000 40 000

1 35 000 32 000

2 30 000 25 600

3 25 000 20 500

4 20 000 16 400

5 15 000 13 100

6 10 000 10 500

7 5 000 8 400

8 0 6 700

1 R100---------–

T

0

8 000

16 000

24 000

32 000

40 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

Straight line valueDiminishing value

3 aAge of equpment(years)

Straight line value ($)

Diminishing value ($)

New (0) 100 000 100 000

1 90 000 85 000

2 80 000 72 250

3 70 000 61 400

4 60 000 52 200

5 50 000 44 350

6 40 000 37 700

7 30 000 32 050

8 20 000 27 250

9 10 000 23 150

10 0 19 700

4Age of computer (years)

Salvage value at 20% ($)

Salvage value at 35% ($)

1 3520.00 2860.00

2 2816.00 2288.00

3 2252.80 1830.40

4 1802.24 1464.32

5 1441.79 1171.46

6 1153.43 937.16

7 922.75 749.73

8 738.20 599.79

9 590.56 479.83

10 472.45 383.86

5

YearsSalvage value

($)Tax deduction

($)

1 4355.00 2145.00

2 2917.85 1437.15

3 1954.96 962.89

4 1309.82 654.14

5 877.58 432.24

6 0 877.58

0

20 000

40 000

60 000

80 000

100 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

Straight linevalueDiminishingvalue

2A➔

2E

Maths A Yr 12 - Answers 19 Wednesday, September 11, 2002 3:41 PM

520 A n s w e r san

swer

s

b

8 a $10 000 b c $25009 a $3000 b $75 c $1600 d $750

Exercise 2F — Future and present value of an annuity1 $7049.372 a $6691.13 b $16 859.14

c $6158.56 d $3974.56e $17 713.21 f $3530.21

3 $4472.93 4 $90 237.495 a $20 326.23 b $24 297.37

c $45 881.32 d $69 770.036 C 7 $4067.238 a $4524.37 b $7068.59 c $1930.089 $4787.76

10 a $1324.00 b $23 932.35c $7503.81 d $62 953.50

11 a 4% b 10 c $6003.0512 a $4103.92 b $5535.38 c $7546.7413 5% for 6 years. $1 will grow to $6.8019 but at 6%

for 5 years it will grow to $5.6371.14 D 15 $6918.5016 a $1845.09 b $12 289.20

c $4455.79 d $16 604.40

10 Quick Questions 21 $15 937.42 2 $15 937.423 $13 537.79 4 $1435.915 $5084.04 6 $19 277.167 $22 094.93 8 $8513.569 $10.63 10 $13 295.75

Investigation — A growing investment

1 A = 20 000

2

6

YearsSalvage value

($)Tax deduction

($)

1 33 750 11 250

2 25 313 8 438

3 18 984 6 328

4 14 238 4 746

5 10 679 3 560

6 8 009 2 670

7 6 007 2 002

8 4 505 1 502

7 aAge of truck (years)

Straight line value

($)

Diminishing value

($)

New (0) 250 000 250 000

1 225 000 200 000

2 200 000 160 000

3 175 000 128 000

4 150 000 102 400

5 125 000 81 920

6 100 000 65 536

7 75 000 52 429

8 50 000 41 943

9 25 000 33 554

10 0 26 844

c

Age of truck (years)

Salvage value —

straight line ($)

Tax deduction

($)

1 225 000 25 000

2 200 000 25 000

3 175 000 25 000

4 150 000 25 000

5 125 000 25 000

6 100 000 25 000

7 75 000 25 000

8 50 000 25 000

9 25 000 25 000

10 0 25 000

0

50 000

100 000

150 000

200 000

250 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

Straight line value

Diminishingvalue

Age of truck (years)

Salvage value — declining

balance ($)

Tax deduction

($)

1 200 000 50 000

2 160 000 40 000

3 128 000 32 000

4 102 400 25 600

5 81 920 20 480

6 65 536 16 384

7 52 429 13 107

8 41 943 10 486

9 33 554 8 389

10 26 844 6 71014---

1 6100---------+

T


A n s w e r s 521

answ

ers

3

4

The investment grows to $30 000 in 6 years; that is, a year earlier.The graph shows that $30 000 is actually reached in 5.3 years.

5

Investing $15 000 at 9.1% p.a. will reach $30 000 in 8 years.

Chapter review1 $83.44 2 $149.90 3 $7900

4 a b V = 200 000 − 20 000A

5 a

b V = 3500 − 250A c $12506 a

b $10 500 c 17 years7 a

b $6500 c 9 years

8

9 $650010 a $1300 b $15 000 c $235 00011 12 years 12 $250/year 13 After 6 years14 $20 88015 $474 00016 a $23 620 b $1000 c $24 290

d $27 210 e $49 38017 a $167 100 b $432 900

b

20 $41 039.2021 a $4399.95 b $34 641.25

c $1842.84 d $51 014.2522 $2242.9523 a $1516.32 b $14 047.20

c $4055.45 d $11 177.64

0

40 000

80 000

120 000

160 000

200 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

0

1 000

2 000

3 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

0

4 000

8 000

12 000

16 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

0

10 000

20 000

30 000

40 000

50 000

60 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

18 a

Age (years)

Salvage value — straight line

($)

Salvage value —

15% p.a. ($)

New (0) 100 000 100 000

1 90 000 85 000

2 80 000 72 300

3 70 000 61 400

4 60 000 52 200

5 50 000 44 400

6 40 000 37 700

7 30 000 32 100

8 20 000 27 200

9 10 000 23 100

10 0 19 700

19

YearSalvage value

($)Tax deduction

($)

1 3015 1485

2 2020 995

3 1353 667

4 907 446

5 0 907

0

2000

4000

6000

8000

10 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

0

20 000

40 000

60 000

80 000

100 000

0 2 4 6 8 10

Val

ue (

$)

Age (years)

Salvage value — straight line

Salvage value — 15% p.a.

The investment will grow to $30 000 in 7 years.

2F➔

2F


522 A n s w e r san

swer

sCHAPTER 3 Consumer credit and investmentsSkills check1 a b 1 c

d 2 e f

2 a 0.1 b 0.01 c 0.025d 0.0375 e 0.005 f 0.0125

3 a 18 b 33 c 8d 2 e 69 f 15

4 a i 30 ii 30 iii 31b i 45 ii 33 iii 56c i 25 Feb. ii 24 Aug. iii 13 Apr.

5 a $440 b $20 c 37.5%

Exercise 3A — Flat rate interest1 a $700 b $1200

c $7500 d $2850 e $3902 $15843 $50004 a $4060 b $21 330

c $1803.75 d $308.25e $275 000

5 a $1650 b $3850c $693 d $6193

6 a $1600 b $6600c $137.50

7 a $800 b $2800c $53.85

8 a $2000 b $6000c $2160 d $8160e $226.67

9 $43.3310 B 11 C 12 8% p.a.13 a $2400 b $9600

c $16 319.88 d 15% p.a.14 15% p.a.

Exercise 3B — Home loans1 a $800 b $79 9502 a $312.50, $49 848.99 b $151.01

c $311.56, $49 697.04 d $151.95

c $440.665 $243 1236 a $302 308.80 b $241 500 c $60 808.807 A8 B9 a $112 034 b $86 072

c $61 789.40 d $39 329.6010 a Smith – $6000, Jones – $9000

3

MonthPrincipal

($) Interest ($)Balance

owing ($)

1 150 000.00 1200.00 149 791.99

2 149 791.99 1198.34 149 582.37

3 149 582.32 1196.66 149 370.93

4 149 370.97 1194.97 149 157.98

5 149 157.93 1193.26 148 943.12

6 148 943.18 1191.55 148 726.72

7 148 726.72 1189.81 148 508.52

8 148 508.52 1188.07 148 288.58

9 148 288.58 1186.31 148 066.88

10 148 066.88 1184.54 147 843.41

112------ 1

2--- 1

365---------

12--- 1

52------ 1

26------

4 a

MonthPrincipal

($)Interest

($)Balance

owing ($)1 85 000.00 637.50 84 872.732 84 872.73 636.55 84 744.513 84 744.51 635.58 84 615.324 84 615.32 634.61 84 485.165 84 485.16 633.64 84 354.036 84 354.03 632.66 84 221.927 84 221.92 631.66 84 088.818 84 088.81 630.67 83 954.719 83 954.71 629.66 83 819.60

10 83 819.60 628.65 83 683.4811 83 683.48 627.63 83 546.3412 83 546.34 626.60 83 408.17

b

MonthPrincipal

($)Interest

($)Balance

owing ($)1 85 000.00 637.50 84 837.502 84 837.50 636.28 84 673.783 84 673.78 635.05 84 508.834 84 508.83 633.82 84 342.655 84 342.65 632.57 84 175.226 84 175.22 631.31 84 006.537 84 006.53 630.05 83 836.588 83 836.58 628.77 83 665.359 83 665.35 627.49 83 492.84

10 83 492.84 626.20 83 319.0411 83 319.04 624.89 83 143.9312 83 143.93 623.58 82 967.51

b i Smith family

MonthPrincipal

($)Interest

($)Balance owing

($)1 50 000.00 395.83 49 895.83 2 49 895.83 395.01 49 790.84 3 49 790.84 394.18 49 685.02 4 49 685.02 393.34 49 578.36 5 49 578.36 392.50 49 470.86 6 49 470.86 391.64 49 362.49 7 49 362.49 390.79 49 253.28 8 49 253.29 389.92 49 143.21 9 49 143.21 389.05 49 032.26

10 49 032.26 388.17 48 920.43 11 48 920.43 387.29 48 807.72 12 48 807.72 386.39 48 694.11


A n s w e r s 523

answ

ers

c $3134.16

10 Quick Questions 11 $420 2 $1135.203 $7025 4 $9755 $5525 6 $2817.757 $8342.75 8 $231.749 $1640 10 $265 577

Exercise 3C — The cost of a loan1 11.6%2 a 11.6% b 8.32% c 15.2%

d 10.6% e 12.2%3 a 8.32% b 8.66% c 9.01%

d 9.39% e 11.6% f 18.3%4 a $213 996 b $128 996 c 6.0704%5 9.01%6 Loan 17 a $231 546 b $200 745.60

c $145 593.608 Loan 2 – they will save $60419 C

10 a $341 376 b $337 57811 D12 a $562 279.20 b 6.25% c 5.8%

Exercise 3D — Loan repayments1 $674.252 a $90.46 b $341.25 c $819.84

d $1101.00 e $1515.543 a $400 b $3600 c $123.054 They will not need to increase their repayments.5 a $1510.20 b $1620.146 Yes. The repayment is $744 and the most he can

afford is $750.7 a $7000 b $1750 c $178 0008 a $733.40 b $174.80

Exercise 3E — Bonds, debentures and term deposits1 $3200 2 $315 3 $472.50 4 $15005 $1800 6 $612.50 7 B 8 A9 C 10 D 11 C 12 B

1314

Exercise 3F — Bank savings accounts123 9%4

5

6

7

10 Quick Questions 21 $2835 2 160% p.a.3 $299 191.20 4 10.3%5 6.0% 6 $45.237 $264.40 8 $3759 $5825 10 $2.23

b ii Jones family

Principal ($)

Interest ($)

Balance owing ($)

1 50 000.00 395.83 49 645.83

2 49 645.83 393.03 49 288.86

3 49 288.86 390.20 48 929.06

4 48 929.06 387.36 48 566.42

5 48 566.42 384.48 48 200.90

6 48 200.90 381.59 47 832.49

7 47 832.49 378.67 47 461.16

8 47 461.16 375.73 47 086.89

9 47 086.89 372.77 46 709.66

10 46 709.66 369.78 46 329.44

11 46 329.44 366.77 45 946.21

12 45 946.21 363.74 45 559.95

a $22.15 b $84.99 c $297a i $1406.25 ii $1350 iii $1321.88b No difference

a $2066.10 b $9.47a 8 cents b 12 cents

1998 Transaction Debit Credit Balance1 May Balance B/F 2132.203 May Cheq 4217 460.27 1671.937 May Deposit 230.16 1902.0917 May Cheq 4218 891.20 1010.8926 May Wages 1740.60 2751.4931 May Interest 5.69 2757.182 June Deposit 415.10 3172.288 June Cheq 4220 2217.00 955.2819 June Cheq 4219 428.50 526.7821 June Cheq 4222 16.80 509.9823 June Wages 1740.60 2250.5830 June Interest 2.87 2253.451 July Deposit 22.80 2276.254 July Cheq 4221 36.72 2239.5318 July Cheq 4223 280.96 1958.5726 July Wages 1740.60 3699.1731 July Interest 11.02 3710.19

a $34.03 b $14.45

1998 Transaction Debit Credit Balance

4 Aug5 Aug10 Aug15 Aug18 Aug20 Aug30 Aug31 Aug1 Sept2 Sept5 Sept5 Sept15 Sept20 Sept29 Sept30 Sept

SalaryHealth fundHealth fundElectricity a/cSalaryRentVisaInterestSalaryTelephone a/cTax refundHealth fundSalaryRentSalaryInterest

327.60

150.26

620.80180.00

180.64

327.60

620.80

1410.20

68.20

1410.20

6.251410.20

461.27

1410.20

1410.2017.78

1410.201082.601150.801000.542410.741789.741609.741615.993026.192845.553306.822979.224389.423768.625178.825196.60

a i $6.25 ii $13.35 iii $7.10b i $4.79 ii $4.76 iii –$0.03c i $10.94 ii $16.86 iii $5.92

3A➔

3F


524 A n s w e r san

swer

sExercise 3G — Investing in real estate1 a $2448.75 b $3656.25 c $12 0002 a $244.88 b $365.63 c $12003 a $77 256.37 b $124 228.12 c $448 8004 $80 7505 a $1873 b $3175.50 c $12 832.506 a $189 123 b $272 415.50 c $554 952.507 a $169 692.50 b $278 375 c $8682.508 $87459 $127 500

10 $146 45011 $289 500

Exercise 3H — Investing in the stock market1 $19 131.25 2 $2511.253 $5071 4 $15405 25 c/share 6 $1.50/share7 6 c/share 8 29.27 c/share9 a $1.224 million b $2.176 million

c 43.52 c/share10 a $5.22 million b $9.28 million

c $1.66/share11 $3.276 million12 4.57%

13

14 2.91% 15 D 16 $36417 a 6.6% b $1.06/share18 a $1.14 b $5.928 million19 8.5%20 a 0.59% b $10.64

c 6.44 c/share d 0.61%21 a $77.50 b 1.2% c 82.622 a $60 b 1.2% c 83.323 a 5000 b 0.75% c 133.324 23.3

Exercise 3I — Graphing share performance1 a

b $7.002 a

b $1.75

3 a

b $1.20

4 a

b $16.00

5 a

b $14.50

History of mathematics — The Wall Street Crash1 Soaring share prices were suddenly reversed.

2 Share prices declined rapidly.

3 People stopped investing, banks and businesses collapsed, unemployment rose and Hitler came to power.

History of mathematics — The Dow Jones Industrial Average1 Wall Street Journal journalists Charles Dow and

Eddie Jones.

2 30.

3 Sum of 30 stock prices divided by 0.2252.

4 Technology, telecommunications.

Chapter review1 a $1120 b $7187.50 c $1281.60

d $39.60 e $12 285.00

2 $6760

3 $191.02

4 6.15%

5 a $1250 b $124 873.64

Dividend Share price Dividend yield

$0.56 $8.40 6.7%

$0.78 $7.40 10.5%

$1.20 $23.40 5.1%

$1.09 $15.76 6.9%

$0.04 $0.76 5.3%

6.506.706.907.107.30

6.306.105.905.70

1–M

ay1–

Jun 1–Jul

1–Aug

1–Se

pt1–

Oct

Shar

e pr

ice

($)

Month

4.504.003.503.002.50Sh

are

pric

e ($

)

1–Ju

n1–

Jul

1–Aug

1–Sep

t

1–Oct

1–Nov

1–Dec

1–Ja

n

1–Feb

1–M

ar

1–Apr

1–M

ay

1.101.121.14

1.081.061.041.021.00

2.022.04

2.001.181.16

1–Jan

1–Fe

b1–

Mar1–

Apr

1–M

ay1–

Jun1–Jul

1–Aug

1–Se

pt1–

Oct

1–Nov

1–Dec

Shar

e pr

ice

($)

Month1–

Jan1–

Feb1–

Mar1–

Apr

1–M

ay1–

Jun1–Jul

1–Aug

1–Se

pt1–

Oct

1–Jan

1–Feb

1–M

ar

1–Apr

1–M

ay1–

Jun1–Jul

1–Aug

1–Sep

1–Oct

1–Nov

1–Dec

Shar

e pr

ice

($)

Month

14.00

16.00

10.00

12.00

1–Jan

1–Feb

1–M

ar

1–Apr

1–M

ay1–

Jun

1–Jan

1–Feb

1–M

ar

1–Apr

1–M

ay1–

Jun1–

Jul

1–Aug

1–Sep

t

1–Oct

1–Nov

1–Dec

Shar

e pr

ice

($)

Month

12.50

12.00

13.00

13.50

14.00

14.50


A n s w e r s 525

answ

ers

c $5137.217 a $596 844 b $18 8848 a 7.25% b 13.70%

c 25.65% d 14.11%9 a $18 223.20 b $4723.20 c 7%

10 Loan 211 $21.1512 a $316.75 b $599.40

c $2369.11 d $510013 a $2453.49 b $2618.0614 B 15 A 16 D17 $2700 18 $694.17 19 $252020 $5000 21 D22 a $1.08 b $1.1523 a $3.33 b $3.6424 $3075 25 $401.63 26 $236 425.4527 $270 662.50 28 $15 832.50 29 $46 687.5030 $1.93/share 31 $14.74/share 32 5.22%

33 1.6% 34 81.8 c/share35 a $260 b 2.5% c 40.436 a

b $18.00

CHAPTER 4 Populations, samples, statistics and probabilitySkills check1 a 0.375 b 0.083 c 0.813 d 0.5902 a 75% b 12.5% c 42.5% d 4%3 Answers will vary.4 a 4 b 4 c 3 d 7 e 15 a a = 8 b b = 9 c c = 22.5

d d = 17.5 e e = 10.56 Scale on axes, omitting certain values, giving a 3D

visual impression, using a non-linear scale on the axes.

7 a 73 b 7.3 c 7 d 6e 6 f 8 g 6 h 2

Investigation — Australia’s population and housing census1 This is a statistical collection of data to determine the

number of people in Australia on Census Night, the characteristics of these people and the dwellings in which they live.

2 All people in Australia on Census Night take part.3 It is compulsory.4 Questions asked include: age, marital status,

birthplace, income, type of dwelling, type of job… The questions have changed over the years to take into account changing social conditions of the population; such as language spoken at home, computer usage…

5 A census can provide information necessary for future planning.

6 The ABS has access to the information and details of individuals are protected by the Privacy Act.

7 All dwellings are issued with census booklets, which are delivered and collected by ABS workers. The booklets are completed by all individuals on the same night.

Exercise 4A — Populations and samples1 Census, sample2 Census — every member of the population

participates.3 Survey4 a Survey b Survey c Census

d Census e Survey 5 a Survey b Census c Census

d Survey6 Survey

6 a

MonthPrincipal

($)Interest

($)Balance

owing ($)

1 130 000.00 866.67 129 779.30

2 129 779.30 865.20 129 557.12

3 129 557.12 863.71 129 333.47

4 129 333.47 862.22 129 108.32

5 129 108.32 860.72 128 881.67

6 128 881.67 859.21 128 653.51

7 128 653.51 857.69 128 423.83

8 128 423.83 856.16 128 192.62

9 128 192.62 854.62 127 959.87

10 127 959.87 853.07 127 725.56

11 127 725.56 851.50 127 489.70

12 127 489.70 849.93 127 252.26

b

MonthPrincipal

($)Interest

($)Balance

owing ($)

1 130 000.00 866.67 129 366.67

2 129 366.67 862.44 128 729.11

3 128 729.11 858.19 128 087.31

4 128 087.31 853.92 127 441.22

5 127 441.22 849.61 126 790.83

6 126 790.83 845.27 126 136.10

7 126 136.10 840.91 125 477.01

8 125 477.01 836.51 124 813.52

9 124 813.52 832.09 124 145.61

10 124 145.61 827.64 123 473.25

11 123 473.25 823.15 122 796.40

12 122 796.40 818.64 122 115.05

15.50Shar

e pr

ice

($)

Month

16.00

16.5017.00

1–Jan

1–Feb

1–M

ar

1–Apr

1–M

ay1–

Jun1–

Jul

1–Aug

1–Sep

t

1–Oct

1–Nov

1–Dec

3G➔

4A


526 A n s w e r san

swer

sExercise 4B — Samples and sampling1 80, 84, 70, 85, 79, 54, 56, 51, 81, 672 Range of answers3 Range of answers4 Should be three different sets of numbers.5 a Random sampling

b Judgemental samplingc Accessibility samplingd Systematic samplinge Stratified sampling

6 a Systematic b Stratified c Systematicd Random e Stratified

7 A8 C9 Year 7— nine, Year 8 — eleven, Year 9 — nine, Year

10 — eight, Year 11 — seven, Year 12 — six10 36 men and 24 women

11

12 4000 13 40014 a 10 000 b 25 000 c 366315 No — estimated population 20 00016 a 625 b 500 c 625 17 a 833 b 1000 c 882

d 905

10 Quick Questions 11 Census 2 Survey3 Census 4 Random sample5 Systematic sample 6 Stratified sample7 Cluster sample 8 Judgemental sample9 Quota sample 10 Random sample

Exercise 4C — Bias1 Check with your teacher.2 Check with your teacher.3 a Sample does not represent characteristics of

population.b No control over responsesc Unrepresentative sampled Abnormal conditionse Only extreme groups in sample

4 The decrease in the value of the Australian dollar compared with the American dollar is accentuated by the large scale on the y-axis. The decrease is actually only 2 cents. The scale on the x-axis is not uniform (9 May, 11 May, 12 May).

5 What type of university tests? What do the terms ‘consistently’, ‘majority’, ‘more effective’, ‘most other’ mean? No hard evidence has been provided to support the claim.

6 a There would be many more student drivers in Year 12 than in Year 11 — perhaps also some in Year 10.

b Students with part-time jobs are in lower year levels as well.

c Residents not at the neighbourhood watch meeting have been ignored.

d Other music students who play instruments and don’t belong to the choir have been excluded.

e The composition of cars in a shopping centre car park is not representative of the cars on the road.

f Females have been excluded.g Users of the local library would not reflect the

views of teenagers.

Investigation — Contingency tables from census data1 a 37.8% b 41.9%

c Part (a) compares the number of males in the retail trade with the total number of male workers, while in part (b) the comparison is with the total number in the retail trade.

d It would be easier to survey those in the retail trade rather than surveying the male population, as the former number is smaller than the second.

2 Choose another category to survey.3 Percentage of persons in agriculture, forestry and

fishing

= × 100%

= 4.2%4 Analyse data from the 2001 census.

Exercise 4D — Contingency tables

3 a 1000 b 75 c 96.7% d 60% 4 a 200 b 44 c 90.9% d 5.1%

e 94% f Check with your teacher.5 B 6 D 7 A

b i 96% ii 3.3% iii 4% iv 96.5%

Age Male Female

20–29 10 7

30–39 7 8

40–49 12 3

50–59 1 2

1 Test results

TotalAccurate Not accurate

With virus 98 2 100

Without virus 388 12 400

Total 486 14 500

2 Test results


Telling truth 777 23 800

Telling lies 156 44 200

Total 933 67 1000

8 a Test results

TotalAccurateNot

accurate

Bags with prohibited items

48 2 50

Bags with no prohibited items

145 5 150

Total 193 7 200

324 3307 636 319-----------------------


A n s w e r s 527

answ

ers

9 a ii b ii c ii10 a

b

c

d

e 1.7%f 52.5%g Survey a sample of those in education rather than

conducting a survey on a sample of females as the total education group is fewer in number than the total female group.

11 a

b

c

d

e 10.6%f 65.9%g Same as g in question 10. Comments should note

increases in percentage and the reasons for this.

12 a

b i 35.8% ii 39.9%c No — more than 39% of the labour force are

female.13 a

b No — 33% of the males in the community were born overseas, while 49% of the people born overseas were male.

Exercise 4E — Applications of statistics and probability1 a Yes b 3 c Yes, both equal 3 d 32 a No b 5–9 and 20–24 c No

d 5–9 and 20–24 e 25–293 a b Yes

c 0 and 5 d Yes, both equal 2.5e 0

4 a 2 b 2 c 55 a b 0.73

6 A7 a b No

c 151–200 and 201–250d 0.67e 0.17

8 a Chemistry is symmetrical.Maths is not symmetrical.

b Chemistry: mode = 41–50 and 81–90,Maths: mode = 71–80

c Maths, because there are more scores further away from the centre of the distribution.

Male Female

Education 3 685 4 071

Other 239 389 240 029

Male Female Total

Education 3 685 4 071 7 756

Other 239 389 240 029 479 418

Total 243 074 244 100 487 174

Male Female

Education 1.5% 1.7%

Other 98.5% 98.3%

Total 100% 100%

Male Female Total

Education 47.5% 52.5% 100%

Other 49.9% 50.1% 100%

Male Female

Education 184 287 355 776

Other 4 087 764 3 008 492

Male Female Total

Education 184 287 355 776 540 063

Other 4 087 764 3 008 492 7 096 256

Total 4 272 051 3 364 268 7 636 319

Male Female

Education 4.3% 10.6%

Other 95.7% 89.4%

Total 100% 100%

Male Female Total

Education 34.1% 65.9% 100%

Other 57.6% 42.4% 100%

Male Female Total

In 2641 1752 4393

Not in 1728 3144 4872

Total 4369 4896 9625

Male Female Total

Aust. 4066 4468 8534

Overseas 2061 2156 4217

Total 6127 6624 12 751

Freq

uenc

y

Number of goals

0123456

0 1 2 3 4 5

21–3011–20

31–40

Freq

uenc

y

41–5051–60

Number of goals

05

10152025

51–1001–50

101–150

Freq

uenc

y

151–200

201–250

Number of people

02468

10

4B➔

4E


528

A n s w e r s

answ

ers

d

Yes, both 0.275

e

Mathematics

f

P

(>90% Chem)

=

0.05

P

(>90% Maths)

=

0.1

9

157

10

31.8, or 32 visitors

11 a

7

b

18.3

12 a

Lines vary.

b

Factory 1 is cheaper at $43.21 (compared to Factory 2 at $56.61).

c

Factory 2 is cheaper at $168.16 (compared to Factory 1 at $216).

d

Factory 2 is marginally more linear.

Investigation — Modelling Olympic Games times

1

Scatterplot

2

Line of best fit

3

Prediction

The line of best fit predicts a time of 9.5 seconds in the year 2035. The Olympic Games closest to this year is 2036.

Investigation — The door game

Part 11

1

P(winning if stay) =

P(winning if change mind) =

If you change your mind you will double your change of winning from 1 in 3 to 2 in 3.

10 Quick Questions 2

1

23.3

2

21.5

3

16

4

29

5

5

6

7.93

7

No

8

Yes, 45 is an outlier.

9

Median, because the outlier inflates the mean.

10

The outlier makes the range very large.

Chapter review

1 a

Survey

b

Census

c

Census

d

Survey

2

D

3

Random sample — where the participants are chosen by luck.Stratified sample — where the participants are chosen in proportion to the entire population.

Systematic sample — where a system is used to select the participants.Accessibility sample — where those within easy access form the sample.Quota sample — where a quota is placed on the number in the sample.Judgemental sampling — where a judgement is made regarding those who should form the sample.Cluster sampling — where the sample is selected from clusters within the population.Capture–recapture sampling — used mainly to estimate populations in wildlife where an initial sample is tagged then another sample selected from the whole population.

4 a

Systematic

b

Random

c

Stratified

5

Check with your teacher.

6

Year 7 — 12, Year 8 — 12, Year 9 — 11, Year 10 — 10, Year 11 — 8, Year 12 — 7

7

2000

8

750

9 a

Barry — 2667 Viet — 1667 Mustafa — 1571

b

1968

10

B

11

1984

12

Check with your teacher.

15 a

140

b

30

c

90%

d

10%

16 a

130

b

33.8%

c

97.5%

17

A

18 a

200

b

96%

c

34

d

93 %

e

93%

19 a

9.7%

b

8.0%

c

No significant difference

20

B

21 a

b

Secondary students were much keener on having more holidays than were primary students.

22 a

Yes

b

Both are 17.5.

c

17 and 18

d

17 and 18

13---

23---

13 Test results


With virus

48 2 50

Without virus

149 1 150

Total

197 3 200

14 Test results


Telling truth

77 3 80

Telling lies

17 3 20

Total

94 6 100

Attitude Primary Secondary

Fewer

7.5% 4.3%

Same

43.3% 19.1%

More

49.2% 76.6%

Total

100% 100%

13---

A n s w e r s 529

answ

ers

23 a b No

c 0.1524 a A variety of answers

b 131

CHAPTER 5 NavigationSkills check1 Lines of latitude run parallel to the equator. Lines of

longitude run from one pole to the other and are east or west of Greenwich.

2 0° 3 0° 4 Latitude 5 C = 2πr

6 40 030 km 7 Tangent =

8 Speed =

9 The time at the prime meridian (0° longitude)10 A triangle which has 2 sides congruent, and base

angles congruent

Exercise 5A — Review of Earth geometry1 a (30°N, 60°W) b (40°S, 20°W)

c (30°S, 50°E) d (40°N, 60°W)e (20°N, 20°W) f (30°S, 20°E)

2 Any 2 meridians; for example, NDS, NGS; or any line of longitude; for example, 20°W

3 a 40° b 30°c 10° d 60°

4 a Johannesburg b Shanghaic Montreal d Perth

5 a (35°N, 118°W) b (35°S, 20°E)c (0°, 100°E) d (38°N, 115°E)

6 a 4448 km b 7784 kmc 6672 km d 7339 km

7 a 7784 km b 6450 km8 4226 km

Exercise 5B — Accurate position description1 a 27°9.6′S, 153°36′E b 27°S, 153°45.9′E

c 27°S, 153°36′E d 27°0.9′S, 153°37.6′Ee 27°1.1′S, 153°33.6′E f 27°8′S, 153°44.5′E

2 Sketch3 a Mt Sydney b Black Island

c Pinnacle Point4 a 20°2.2′S, 148°52.7′E b 20°4.3′S, 148°58.3′E

c 20°4.8′S, 148°52.2′E d 20°10′S, 148°53.6′Ee 20°10.5′S, 148°55′E

Exercise 5C — The nautical mile and the knot1 a 120′ b 150′ c 1422′ d 2871.7′2 a 9°43′ b 39°8.7′3 a 17°17′ b 57.3′

4 a J, D b A, H c H, Id i 50°N, 80°E ii 0°, 0° iii 60°S, 0°

iv 0°, 30°W v 50°N, 0°e i 2400 n mile ii 2400 n mile iii 5400 n mile

iv 9000 n mile v 9000 n milef i 6600 n mile ii 6600 n mileg i 3600 n mile ii 3600 n mile iii 3000 n mile

5 a 1650 n mile b 3750 n milec 7050 n mile d 1110 n mile

6 8 knots7 a 3.5 knots b 6.5 km/h8 a 3.85 knots b 12.6 knots

c 289 n mile d 52.1 n milee 30 hours f 10 minutes

9 a 7872 hoursb 6.4 km/h, 3.4 knots

10 a 3600 n mileb ii 3600 n mile

ii The Earth is a sphere and any arc joining 2 points on its surface subtending an angle of 60° must be separated by the same distance.

c 200 hours11 a 570 n mile b 4.63 knots12 3.08 am13 A separation of 1′ near the equator on a line of

latitude is greater than that further from the equator.

10 Quick Questions 11 Latitude 2 Latitude 3 60′4 1852 metres 5 150 n mile

6 Speed = 7 The knot

8 6 knots 9 5400 n mile 10 5 pm

Investigation — Distance to the horizon1 Angle PHC = 90° (PH is a tangent to the circle, so CH

is perpendicular to PH.)2 PC2 = CH2 + HP2 (by Pythagoras’ theorem)3 CH = AC (Both are radii of the Earth;

both = 6371 km.)4 a 25.2 km b 79.8 km c 112.9 km d 357.1 km5 As height increases, distance also increases. (On a flat

Earth, distance to horizon would be greater.)

Exercise 5D — Using the compass1 a 128°C b 292°C c 193°C d 40°C2 291°T 3 6°E4 a 120°C b 226°C c 4°W d 257°29′T

Exercise 5E — Compass bearings and reverse bearings1 a 50°T b 300°T c 230°T d 145°T2 a 230°T b 120°T c 50°T d 325°T5 a 6 n mile b 5 n mile c 11.2 n mile

d 11.4 n mile e 15.7 n mile f 10.9 n mile6 b 12 knots7 a 187°T b 176°C c 50 min d 356°C8 a Great Keppel Is. b North Keppel Is.9 a (23°5.6′S, 150°54′E) b (23°13′S, 150°58.2′E)

10 a 56°T b 46°Cc 7.2 n miles d 54 minutes

11 a 304°(C) , 1.8 n mile b 271°(C), 8.2 n milec 328°(C), 7 n mile d 296°(C), 11.5 n mile

Freq

uenc

y

Class centre

02468

1012

1 2 3 4 5 6

oppositeadjacent--------------------

distancetime

-------------------

distancetime

-------------------

5A➔

5E


530 A n s w e r san

swer

s10 Quick Questions 21 Compass 2 True north3 Magnetic variation 4 Compass5 True 6 60°T 7 158°C 8 180°9 Latitude 10 7 knots

Exercise 5F — Fixing position1 a Check with your teacher.

b 155°T c 7.5 n miled 15 knots e 3.5 n mile

2 a Check with your teacher.b 9.3 n mile c 18.6 knots

3 a 131°T, 18°T, 299°T b 198°T, 340°T, 265°Tc 11 knots d A 219°, B 293°, C 254°

4 a Tower 53°, Antenna 88°5 b 14 n mile c 14 n mile d 19.8 n mile

e i 243° ii 252°6 b 56 n mile c 250°

Exercise 5G — Come to the rescue!1 Man and Wife Rocks 108°T; Miall Island 220°T2 plot 3 9.4 n mile 4 28 minutes5 3.48 pm6 Vessel has moved 1.75 n mile out to sea to 23°6.3′S,

150°59′E7 2.5 n miles to the east8 The wind had greater impact on pushing the boat than

it did on the swimmer.

Exercise 5H — Transit fix1 b i 6 n mile ii 5.6 n mile iii 5.3 n mile

c Plot d 20 knots2 c B 250°, D 291°, E 316°3 b 7.5 n mile

Exercise 5I — Running fix1 b 6 n mile2 b 197°

Exercise 5J — Doubling the angle on the bow1 a i 50° ii 130° iii 80° iv 3 n mile

b i 100° ii 40° iii 7 n milec i 42° ii 96° iii 84° iv 11 n miled i 130° ii 25° iii 6.5 n mile iv 45°Te i 45° ii 90° iii 45° iv 10 n milef i 20° ii 140° iii 20° iv 8 n mile

v 8 n mile2 a 25° b 5 n mile3 b At 1300, 30°; at 1330, 60°

c 9 n mile d 9 n mile4 b 8 n mile c 6.2 n mile (from sketch)

d 7.03 am e 100°

10 Quick Questions 31 Two 2 Cocked hat 3 Transit line4 Isosceles 5 Front6 Angle on the bow7 9 knots 8 4 hours9 84 n miles 10 One

Exercise 5K — Dead reckoning1 a Check with your teacher.

b 20°08.3′S, 148°59.7′Ec Check with your teacher.d iii 20°07.3′S, 149°01.5′E

iii 20°06.3′S, 149°03.3′Eiii 20°05.3′S, 149°04.9′E

2 a 20°07.3′S, 149°14′Eb 20°00.2′S, 149°07′Ec 20°05.2′S, 149°09.2′Ed 20°01.4′S, 149°07.3′E

3 a 3.8 n mile b 11.4 knotsc ii 20°05.4′S, 149°04.2′E

ii 20°03.3′S, 149°01′E

Exercise 5L — The lighthouse and navigation1 a AB = 10.5 m b 1908 m

c 1°48′ d 1.345 n mile2 a 4 short flashes of light followed by a long period

of darkness every 20 secondsc 3105 m

3 a 2 flashes, then darkness every 12 secondsc 5156 m f 7.6 n mileg i 0.48° ii 0.36°

4 b 6875 m d 0.68°

Exercise 5M — Let’s go cruising1 a 11°18′ east b 11° east2 a 27°30.9′S, 153°20.7′E b 27°30.1′S, 153°22.4′E

c 27°32.7′S, 153°25.2′E d 27°30.6′S, 153°17.3′Ee 27°34.8′S, 153°21.6′E

3 a Coochiemudlo Island b The Bluffc Submerged rocks d Myora Light

4 a i 308° ii 338° iii 0° iv 266°b i 297° ii 327° iii 349° iv 255°c iii 5.1 n mile iii 5.1 n mile

iii 4.8 n mile iv 2.6 n mile5 a iii Yellow light flashes every 2.5 seconds

iii Red every 4 secondsiii Green every 6 seconds

b So that they can be readily identified as different from neighbouring lights.

6 a 5 n miles b 150°T, 139°Cc 33 minutes d 10.58 ame A southwest wind could push the vessel towards

the rocks near Goat Island.7 Approx. 15 n mile, so approx. 160 litres.8 a 351°T, 227°T b 27°32.8′S, 153°21.6′E

Exercise 5N — Air navigation1 a 26°15′S, 151°56′E b 26°40′S, 152°00′E

c 26°·17′S, 152°41′E d 26°33′S, 151°51′E2 a Tansey b The Bluff c Abbeywood3 a 1998 b 2457 c 20434 a 350°T, 339.5°C b 05°T, 355°C5 a 149°30′C

b Barambah Ck, Clonya, Murgon, Nanango.


A n s w e r s 531

answ

ers

Chapter review1 a A, 30°N, 60°W; B, 40°S, 20°W; C, 30°S, 50°E;

D, 40°N, 60°Wb NDS, NGS, NHS or any line of longitude (for

example, 40°W)c Fd PG, PN, PH, PC etc.

2 a 20°2.2′S, 148°52.7′Eb 20°10′S, 148°53.7′E

3 a 5400 n mile b 10 800 n milec 4200 n mile d 7920 n mile

4 a 1080 n mile b 5340 n mile5 a 360′ b 1110′ c 1695′ d 3457.4′6 540 n mile, 1000 km7 a 6 knots b 13.3 knots c 522 n mile

d 198.3 n mile e 50 hours f 15 minutes8 6 hours 24 minutes9 a 400 n mile b 180°T

c 5 hours d 6.45 pm10 a 114°C b 253°C

c 6°W d 206°T11 286°T12 b 9.8 n mile c 29.4 knots d 48°T13 a = 55°, b = 125°, c = 70°, PR = 7.4 n mile14 a 18° b 36°

c 13 n mile d 13 n mile15 b 50°, 100° c 12 n mile d 12 n mile16 a 5.57 m b 65.8 m

c 1.68° d 5810 m17 a 3 short flashes then long period of darkness every

16 secondsb 130 m c 16 n mile d 2480 m

18 a 8.1 n mile b 8°Tc Approx. 1 h 20 min trip, ETA 8.20 am

19 a 240°T b 16.5 n milec Plot d 11.45 am

CHAPTER 6 Land measurementSkills check1 Millimetre, centimetre, metre, kilometre2 Perimeter3 a 24 cm b 30 m c 15.6 cm

d 16.8 m e 12.6 m f 36.0 mg 38 m

4 a l2 b l × w c πr2

d b × h × e l × h5 a 1.5 cm b 0.18 m c 12 300 cm

d 680 m e 12 500 m6 a 40.7 m2 b 435.8 m2 c 51.7 m2

d 177 m2 e 25 m2

7 a 0.9397 b 0.9659 c 0.32498 c2 = b2 + a2

9 a 5 cm b 8 cm c 9.6 m

10 a b

c

11 Sine = Cosine =

Tangent =

12 a 6.8 cm b 7.7 m c 65.0 m

Exercise 6A — Perimeters and areas of triangles1 a 1.73 cm b 23.1 m c 11.4 m2 a 1.73 cm2 b 277 m2 c 55.3 m2

3 a 12.5 m2 b 4.5 m2

c 7443 m2 d 118.2 m2

4 a 26 m, 13 m b 90.9 m, 45.45 mc 42.4 km, 21.2 km

5 a 27.9 m2 b 250 m2 c 52.4 km2

Exercise 6B — Perimeters and areas of polygons1 a 5797 m2 b 1062 m2 c 27 952 m2

2 a 97.4 m2 b 3195 m2

3 Approx. 36 ha

Exercise 6C — Surveying on level ground without obstacles1 a 32 m b 28 m c 59 m

d 73 m e 47 m2 a 86.3 m b 107.5 m c 47.4 m

d 49.0 m3 Sketch4 a 120 m b 5

c i 48 m ii 39 m iii 37 m iv 32 m v 35 md i 65.8 m ii 44.7 m iii 34.4 m iv 90.2 me i 43.15° ii 1014 m2 iii 81.2°f Sketchg AB survey line established and measured. Staffs at

features Z and C, measurements taken. Staffs at V and D, measurements taken.

h 2340 m2

10 Quick Questions 11 5.47 m 2 × a × b × sin C

3 48.5 m2, 196.6 m2 45 24.2 m2 6 AB7 KF, JE, ID, HC8 36 m, 59 m, 73 m, 32 m, 84 m9 240 m2

10 a 51.2° b 59 m

12---

H

A

O

H

A O

H A

O

oppositehypotenuse--------------------------- adjacent

hypotenuse---------------------------

oppositeadjacent--------------------

12---

S S a–( ) S b–( ) S c–( )

5F➔

6C


532 A n s w e r san

swer

sExercise 6D — Surveying around obstacles1 a 42 m b 31 m c 52 m2 Sketch3 a Sketch

b The distance between the second and third staffs placed by Peter.

Exercise 6E — Plane table surveying: intersection or triangulation1 a 61 m

b i 43 m ii 28 m iii 106 m iv 124 mc i 065° ii 310° iii 180° iv 150°d 301 m e 0.38 ha

2 a 59 mb ii 28.5 m ii 31.5 m iii 32 m iv 73 m

v 49 mc 0.25 had i 15° ii 81° iii 151°

3 a 83 m b $1328 c 450 m2

4 a Sketchb i 100 m ii 66 m iii 50 m iv 90 m v 86 mc i 270° ii 310° iii 240°d 4300 m2

Exercise 6F — Plane table surveying: radiation and traversing1 a i 25 m ii 35 m iii 30.5 m iv 51.5 m

b i 0° ii 070° iii 180°c 1900 m2

2 a ii 23 m ii 72 m iii 51 m iv 12.5 mv 104 m vi 109 m vii 75 m

b 0.31 ha3 a Radiation b sketch

c A, 123°; B, 136°; C, 152°; D, 180°.d 3160 m2

4 a Sketch b traversingc i 212° ii 270°d i 107 m ii 77 m e 3800 m2

10 Quick Questions 21 Offset and triangulation2 41 m, 10 m3 Intersection (or triangulation), radiation, traversing4 Intersection5 Radiation6 Traversing7 287.5 mm8 4 triangles9 Area = = 310 m2

10 10 m2

Exercise 6G — Levelling: vertical measurements in relation to a datum1 a i 50.00 m ii 51.69 m

b 1.94 m c 53.63 md

2 a i 3.60 m ii 2.80 m iii 53.60 m iv 50.00 mb 50.80 m

3 a 61.25 m b 61.25 m c 61.25 md 61.25 m e 60.00 m f 59.50 mg 58.75 m h 58.25 m i 5.00 mj 10.00 m k 15.00 m

Exercise 6H — Topographic maps1 Easting 84, northing 462 a Maculata Park b oval

c building at quarry3 a GR 871464 b GR 854487

c GR 8134884 a 3350 m b 1250 m5 a 352° b 090°6 a Abattoirs, bridge over river on Warrego Highway,

then along river and over slag heapsb 155°. Yes. A scale diagram could be sketched and

trigonometry used to calculate angles.

Exercise 6I — Contour maps1 a 10 m b 80 m c 50 m

d Up a hill then down a steep descent, then up and down another smaller hill.

e Sketch f 52 a 93 m b 68 m3 a 20 m b 10.3 km c Sketch

d 20 e 293°f No, not if X and Y are at the surface.

4 a b 16.4°, steep

5 a 45° b 18.4° c 0.57°d 1.15° e 2.97°

6 a 1.27°

Exercise 6J — Cadastral maps and site plans1 a 630 m2 b 23.0 × 27.499 m

c 632.477 m2 d 1 : 1500e Rectangle of length 60 mm and width 42 mmf ii $57.88/m2 ii 850 m2

g ii Lot 109ii location, elevation, road frontage size, views

2 a 2100 m2, 83 perchesb 103.68 m2 c 56.3 md 0.049 or approx. e i rising ii 1800 mm iii 1.375°

Exercise 6K — Orienteering1 a 8° b 137° c 222° d 45°2 a 67 m b 136 m c 77 m d 130 m3 Any suitable set of 8 instructions.

Chapter review1 a 126 m2 b 165 m2 c 516 m2

d 2325 m2 e 8850 m2

2 0.2 ha3 a ii 150 m ii 52 m iii 63.2 m iv 13 m

v 75 m vi 141.9 mb ii 936 m2 ii 1533 m2 iii 4500 m2

iv 5912.5 m2

4 Sketch

Sta. BS IS FS HI RL Dist. Notes

A 3.63 53.63 50.00 0.00 TBM

B 1.94 53.63 51.69 20.00

S S a–( ) S b–( ) S c–( )

13.41----------

120------


A n s w e r s 533

answ

ers

5 a 84 mb i 050° ii 115° iii 295° iv 238° v 090°c 2000 m2 d 190 m

6 a i 43 m ii 48 m iii 46 m iv 56 m v 86 mb i 051° ii 090° iii 253°c 3200 m2

7 a i 3.90 m ii 2.70 m iii 53.60 m iv 50.00 mb 50.90 c Sketch

8 a i Industrial Estate ii Finlay Islandb 2.5 km c 153°

9 a 250 b 1 in 5c 11.3°, steep to moderate

10 a 45° b 26.6° c 1.1°d 2.9° e 7.2°

11 1012 a 90 m b 20 m c Sketch13 a 630 m2 b Sketch

CHAPTER 7 NetworksExercise 7A — Networks, nodes and arcs1 a ABDE b ABCE2 a b 487 km c 254 km

d i ii 357 min iii 191 min

iv

v

3 a 185 km b ii ii 321 min iii 143 min

c

d

4 a b $3.30 c $3.80

5 C

History of mathematics1 Men of Mathematics by E. T. Bell.2 The Nobel Prize and the Leroy P. Steele Prize.3 An algorithm is a procedure for solving a problem by

a number of steps.

Exercise 7B — Minimal spanning trees1 a b

c d

2 a Sturt b Rockdale c To Sturtd

Ulawatu Yallingup Black Rock Angourie Bargara

Ulawatu 0 120 100 209 254

Yallingup 120 0 220 118 160

Black Rock 100 220 0 109 154

Angourie 209 118 109 0 45

Bargara 254 160 154 45 0

Ulawatu Yallingup Black Rock Angourie Bargara

Ulawatu 0 85 75 157 191

Yallingup 85 0 160 80 114

Black Rock 75 160 0 82 116

Angourie 157 80 82 0 34

Bargara 191 114 116 34 0

U1

Ya

BR An

Ba100

120

109

118

45

160

U1

Ya

An

Ba

BR

75

85

82

80

34

120

Re

Mo

Pi

VG

Ma

Ga

Ce

75

4734

28

59

20

38

40

64

25

45

Re Pi Mo Ce VG Ma Ga

Re 0 62 58 104 108 147 179

Pi 62 0 41 65 46 85 123

Mo 58 41 0 46 76 126 121

Ce 104 65 46 0 30 90 75

VG 108 46 76 30 0 60 98

Ma 147 85 126 90 60 0 38

Ga 179 123 121 75 98 38 0

Re Pi Mo Ce VG Ma Ga

Re 0 47 44 84 81 119 143

Pi 47 0 25 45 34 75 100

Mo 44 25 0 40 59 97 99

Ce 84 45 40 0 20 58 59

VG 81 34 60 20 0 38 66

Ma 119 72 97 58 38 0 28

Ga 143 104 99 59 28 28 0

L

P B

T

K

2.401.50

1.50 1.80

1.80 2.00

2.40

A

B

D

C

4

5

8A

B

C

D

4

5

4

A D

B

C

E

17

15

12

18

A D

B

C

E

30

15

20

15

Yule

Zenith

Rockdale

Urchin

Walga

Xavier

View

PallasSturt

5052

67

52

55

5042

50

6D➔

7B


534 A n s w e r san

swer

s3 a 585 m b 245 m

c Check with your teacher.4 a b

c d

5 a

b

c

6 53 km7 54 km8 a 68 km b $1.7 million9 a $215 b $1740

10 B 11 D

Exercise 7C — Shortest paths1 a 20 b 38 c 74

d 45 e 28 f 1392 a 165 km b 202 km c 202 km3 a 37 b 90 c 32

d 72 e 30 f 444 a

b 80 min

10 Quick Questions 11 6 2 9

3 1 4

5 AC, AD, DF, CF 6 $16 6007 D–C–E–F 8

9 $16 200 10 B–A–C

Exercise 7D — Network flow1 a

b

c

d

234

5 a

b

E

D

C

A

B

F

20

2324

18

18

FDB

ECA

45 45

48 48

45

A B

C D E

F G40

50

23 2320 20

A B

C

E

D

F64

67

6

AD

F

B

E

C

G

17 15

15

12

10

13

E

F

G

DA

CB

12 1017

8

1510

C A

E

D G K

BA I

H

J

5 5

5

8 85 5

55

85

B

A

C

E

D

70

30

50

20

25

60

3025

a 23 b 16 c 16

a 6 b 3 c 3a i 250 ii No b i 150 ii Yesc i 24 ii Yes d i 15 ii No

From To Flow capacity

A B 4

A C 5

A D 3

B E 3

C B 2

C E 4

D C 2

D E 6


A B 4

A C 5

A D 3

B E 3

B C 2

C E 4

D C 2

D E 6

A B CDE F

A BC E F

100

50200

250 300

A B

D EC

50100

E

250 200 100S T UR

20

20

Q

15 12

125

10

O

M

R

N E

8

83

F2

8

86

5D

H

G J

E


A n s w e r s 535

answ

ers

c

d

6

7

8 B

10 Quick Questions 21 A–D–F 2 64 km3 B–C–E–F 4 45 km5 30 6 307 10 8 809 45 10 72

Chapter review1 a 7 b 102

3 Step 1 Choose any node at random and connect it to its closest neighbour.

Step 2 Choose any unconnected node which is closest to any connected node. Connect this node to the nearest connected node.

Step 3 Repeat Step 2 until all nodes are connected.4 A–B–E–C–D–F–G5

6 117 km7 ADFG, 408 ACFIK, 709 a 20 b 15 c 35

10 a 70 b 70 c 7011

12

CHAPTER 8 Critical path analysis and queuingSkills check1

2 a 117 km b 160 km

Exercise 8A — Critical path analysis1 a b

c


A B 4

A C 7

A D 3

A E 5

B E 3

C E 8

D B 2

D E 6


A B 4

A C 7

A D 12

A E 5

C F 7

D B 2

D E 6

D F 4

F E 8

a 10 b 10 c 18 d 22

a i ii 300

b i ii 150

c i ii 29

d i ii 25

50200

C300250

100

100

E

A B

D

50100

E

250 200 100

100

S T UR

20

20

Q

15

5

12

125

10

O

M

R

N E

8

108

3

F2

8

86

5D

H

G J

E

B

A

C

E

D

11

12

875

4

A B CD

E H I K

F G

13

15

10

14

3106

4

C

A B D F

E

13

15

10

13

10

16 3

14

3106

42

C

G

A B D F

E

H

B

A

C

D

4045

35

50

30

1520

30

A

B

C

ED G

F

EB

C D

A J I

HGF 7C➔

8A


536 A n s w e r san

swer

sd

2 a

b 49 minutes3 a B b D4 a 23 minutes b B, C, E, F, G5 A–C–F6 a B b D c A7 A–D–G8 a

b 61 minutes

9

10

11

12 a A–D–Gb Float (H) = 1, Float (J) = 3, Float (E) = 1,

Float (B) = 10, Float (C) = 1, Float (F) = 1c Activity B can be delayed 10 minutes, activity C

can be delayed 1 minute, activity E can be delayed 1 minute, activity F can be delayed 1 minute, activity H can be delayed 1 minute, activity J can be delayed 3 minutes.

13 a A–F–G–J–Kb M, L, H, C, E, B, D

Exercise 8B — Critical path analysis with backward scanning1 Critical path = B–D; Float (E) = 1 min,

Float (C) = 1 min, Float (A) = 1 min2 Critical path = B–F–G; Float (C) = 3 h,

Float (E) = 5 h, Float (A) = 3 h, Float (D) = 5 h3 D4 a 31 days b Critical path = A–C–E–G5 D6 C7 a

b c 85 minutes

d

8 C9 28%

10 a 29 b B–E–J–N11 a 35

b C–F–J–M–Q

c 3

Activity letterImmediate predecessor Time

A — 7

B — 9

C A 12

D B 8

E B 4

F C, D 9


A — 3

B — 4

C — 5

D A 6

E B, F 5

F C 8

G D 18

H E 8

J E 6


A — 10

B A 15

C A 12

D B 8

E C 10

F A 25

G D, E, F 6

H D, E, F 10

J G 11

K J 9

L G 8

M H, L 7

PON

RY

X Z

WVUQTS

A, 2

C, 4

F, 5G, 10 H, 12

E, 3D, 5

B, 20

A, 10 C, 12

F, 25

B, 15

E, 10 H, 10

G, 6 L, 8 K, 9

J, 11

M, 7

D, 825

0

41 52

6149352210

Activity letter Activity

Immediate predecessor Time

A Collect parts — 12

B Paint frame A 35

C Assemble brakes A 16

D Assemble gears B 20

E Install brakes C 12

F Install seat C 5

G Final assembly D, E 18

B, 35 D, 20

E, 12C, 16

F, 5

G, 18

A, 12

B, 35D, 20

E, 12C, 16

F, 5

G, 18

A, 12

85855528

6767

4747

121200A–B–D–G

2626

A, 15

E, 5

G, 9

P, 10

N, 6J, 7

H, 5

K, 5L, 3

M, 4Q, 5

D, 3

F, 9

X, 3B, 4

C, 100 0 10 10

4 11 13 16

24 25

35 35262619191615

24 27 30 30


A n s w e r s 537

answ

ers

12 a

b Float (D), Float (E) = 2, Float (G), Float (H) = 2

10 Quick Questions 11 C2 The earliest starting time of an activity is the earliest

time by which all prior activities can be completed.3 The latest start time of an activity is the latest time

the activity can start if the project is not to be delayed.

4 Float time = latest finish time − earliest start time −activity time.

5

6 67 148 99 3

10 B, F, D

Exercise 8C — Queues: one service point1 a

b 5 min c 9 min d 2 mine 2.28 min f 1.5 min

2 a

b 3 min c 7 min d 1 min

3 C 4 B5 a

b 2 c 4 min6 a 3 min b Yes

c

7 a 2 min b 3, 10 c 2 mind 4 min e 3 f 1g F h 3 min and 6 mini

(ix)

Time Customer

served ArrivalsLength of

queuePeople in

queue

8 D — 4 E F G H9 D I 5 E F G H I

10 E — 4 F G H I11 E J 5 F G H I J12 F K 5 G H I J K13 G — 4 H I J K14 G — 4 H I J K15 G L 5 H I J K L16 H — 4 I J K L

Time Customer


queuePeople in

queue

10 D — 4 E F G H11 E — 3 F G H 12 F — 2 G H 13 G — 1 H14 G — 1 H15 G — 1 H16 H — 0

D, 7J, 2E, 5 F, 3

A, 4

B, 7

C, 3

G, 3 H, 5

B, 4

F,2

C, 6

A, 5 D, 8

E, 50 0

6 9

14 14

6 6

44

Time Customer


queuePeople in

queue

0 A — 0 —

1 A — 0 —

2 A B 1 B

3 B — 0 —

4 B C 1 C

5 B — 1 C

6 C D 1 D

7 C — 1 D

8 C E 2 D E

9 D — 2 E

10 D F 2 E F

11 D — 2 E F

12 E G 2 F G

Time Customer


queuePeople in

queue

0 A — 0 —

3 A B 1 B

6 B C 1 C

9 B D 2 C D

12 C E 2 D E

15 C F 3 D E F

18 D G 3 E F G

Time Customer


queuePeople in

queue

0 A — 0 —

1 A B 1 B

2 B — 0 —

3 B C D 2 C D

4 B — 2 C D

5 C — 1 D

6 C E F 3 D E F

7 C — 3 D E F

8 D — 2 E F

9 D — 2 E F

10 E — 1 F

11 E — 1 F

12 E G 2 F G

13 E — 2 F G

14 F — 1 G

15 F — 1 G

16 F — 1 G

17 G — 0 — 8B➔

8C


538 A n s w e r san

swer

s8 9

10 17 min

10 Quick Questions 21 300 2 Greater than 1.5 minutes3 2 min 4 1 min5 3 min 6 5 min7 5 8 D, E, F9 F, G, H, I 10 3

Exercise 8D — Queues: multiple service points1 a 234 s (3 mins 54 s)

b 32 4 3 Yes4 a 4 had been completely served.

b 2 c 9 d 7 e 35 a

b 2 min c 5 min d 4 min6 a 2 min

b 5 minc

d 2 min

7 A 8 B9

Chapter review1 C 2 D 3 C 4 D5

6

The refit can be completed in 23 hours.

7

8

Time Teller 1 Teller 2 ArrivalsLength of

queue

9 G H J, K 3

10 G H L 4

11 G H 4

12 G H M 5

13 I J 3

14 I J N 4

15 K L 2

16 K L O 3

17 K L 3

Time Service 1 Service 2 ArrivalsPeople in

queueLength of

queue

0 A B E C, D, E 3

1 C B D, E 2

2 C D F E, F 2

3 C D E, F 2

4 E D G F, G 2

5 E F G

GHIJKL

F

BA

0

5

2

1 3

4 6

7

8

9

10

11

12

13

14 16

15

Minutes

Cus

tom

erEDC

••

••

•••

••

••

GH

F

BA

05

21 3

4 67

89

1011

1213

14 1615

Minutes

Cus

tom

er

EDC

•••

••

••

•

Time Nurse 1 Nurse 2 ArrivalsPeople in

queueLength of

queue

0.00 A B

0.45 A B C C 1

1.30 A B D C, D 2

2.00 C D 0

2.15 C D E E 1

3.00 C D F E, F 2

3.45 C D G E, F, G 3

4.00 E F G 1

4.30 E F H G, H 2

5.15 E F I G, H, I 3

6.00 G H J I, J 2

6.45 G H K I, J, K 3

7.30 G H L I, J, K, L 4

8.00 I J K, L 2

Activity letter Time

Earliest start time

Earliest finish time

Float time

Immediate predecessor

A 5 0 5 0 —

B 5 0 5 0 —

C 4 5 11 2 B

D 8 5 13 0 A

E 1 13 19 5 D, G

F 1 14 20 5 E

G 2 9 13 2 C

H 3 13 16 0 D

J 4 16 20 0 H

K 3 20 23 0 F, J

D, 8 E, 1

H, 3J, 4

F, 1 K, 3

G, 2

A, 3

B, 5C, 4

D, 8 E, 1

H, 3 J, 4

F, 1 K, 3

G, 2

A, 5

B, 5C, 4

2314 201350

5 9 16

D, 8 E, 1

H, 3 J, 4

F, 1 K, 3

G, 2

A, 5

B, 5C, 4

0 0 5 5 13 13 14 19 20 20 23 23

5 7 9 11 16 16


A n s w e r s 539

answ

ers

9 a

b 5 minc 3 mind (2 + 1 + 3 + 1 + 1 + 2) ÷ 6 = 1.67 mine (0 + 2 + 0 + 1 + 0 + 2 + 1 + 0 + 2 + 1) ÷ 10

= 0.9 min10

11

12 a Inter-arrival time = s = 3.6 s (less than service time)

b 313 a Inter-arrival time = 0.75 s

b Number of service points = 40

14 a

b 1 minc (0 + 0 + 0 + 1 + 0 + 1 + 1 + 0 + 1 + 1 + 1 + 0 +

1) ÷ 13 = 0.54 mind (0 + 1 + 2 + 3 + 3 + 4 + 3 + 4) ÷ 8 = 2.5 min

15 a

b

c The room would quickly become crowded.d Five people per minute

CHAPTER 9 Probability and the binomial distributionSkills check1 a 6 b 2 c 52 d 36

2 a b c

d e f

3 a 0.04 b 0.03 c 0.54d 0.72 e 0.67 f 0.125

4 a 0.125 b 0.188 c 0.75d 0.154 e 0.684 f 0.058

5 a Less than b Greater than or equal toc Greater than

Exercise 9A — Compound events — independent events

1

2

Time Customer ArrivalsPeople in

queueLength of

queue

0 A B B 1

1 A B 1

2 B C, D C, D 2

3 C E, F D, E, F 3

4 C D, E, F 3

5 C G D, E, F, G 4

6 D H, I E, F, G, H, I

5

7 E F. G. H. I 4

8 F J F, G, H, I, J

4

9 F K G, H, I, J, K

5

TimeCustomer

served ArrivalsPeople in

queueLength of

queue

0 A — — —

1 A B B 1

2 B C C 1

3 B D C, D 2

4 C E D, E 2

5 C F D, E, F 3

6 D G E, F, G 3

7 D H E, F, G, H 4

8 E I F, G, H, I 4

GHIJK

F

BA

0 521 3 4 6 7 8 9

Time hours

Tas

k

EDC

••

••

••

•••

••

36001000------------

TimePoint

1Point

2 ArrivalsPeople in

queueLength of

queue

0 A B C, D, E C, D, E 3

1 A C F, G D, E, F, G 4

2 A D H E, F, G, H 4

3 E D I, J F, G, H, I, J

5

4 E F K G, H, I, J 4

5 G H L I, J, L 3

6 G H M, N I, J, L, M, N

5

7 G I O J, L, M, N, O

5

a b

Arachnids(Spiders)

Arachnids(Scorpions)

Butterflies(Rainforest)

Butterflies(Temperate)

Moths

Tourinformation

Glow-worms(Caves)

Exit

Entry

A, 12 B, 13 C, 12 E, 4

D, 2 G, 6

H, 6I, 5 J, 12

F, 2

112------ 2

3--- 5

6---

136------ 1

54------ 5

16------

14---

12--- 1

2--- 8D➔

9A


540 A n s w e r san

swer

s3 S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

P(9) = =

4 7 is mostly likely, P(7) = =

5

6 {(H, H, H), (H, H, T), (H, T, H), (H, T, T), (T, H, H), (T, H, T), (T, T, H), (T, T, T)}

7 C

8 B

9 × × =

10

0.7 × 0.45 × 0.5 = 0.1575

11 0.3 × 0.55 × 0.5 = 0.0825

12

P(2 Tails) = 0.2025

13

a There are 8 paths.

14 D

15

a 0.12 = 12% b 0.32 = 32%

16 b, d, e

17

P(P − F) = 0.4 × 0.25 = 0.1

18 a P(no drug) = = 0.35

P(small dose) = = 0.40

P(large dose) = = 0.25

b, c, d

e Although few patients were tested, it appearsthat a greater percentage (80%) of those givena large dose of the drug recovered, whereasa much smaller percentage (29%) of those notgiven the drug recovered. 20% of people testedwere given a large dose of the drug andrecovered, 15% of people tested were given asmall dose and recovered, whereas only 10%of people were not given the drug and recovered.So it could be said that a patient is more likelyto recover if the drug is taken.

Exercise 9B — Compound events — mutually exclusive events1 a, d, e, g2

3

4 = 0.517

5 D

6 a = = b

7 a = b = c =

8 a 0.258 b 0.449 c 0.8659 a 0.037 b 0.296 c 0.667 d 0.333

10 a 0.32 b 0.46 c 0.3111 a 0.4999 b 0.9997 c 649 773

Exercise 9C — Compound events — Venn diagrams1

2 a = 0.275 b 0.4

a 0.2646 b 0.0204 c 0.1764

a b c d

436------ 1

9---

636------ 1

6---

18--- 3

8--- 7

8--- 3

8---

16--- 1

6--- 1

6--- 1

216---------

New York up

New York down

Tokyo up

Tokyo up

Tokyo down

Tokyo down

Australia up

Australia up

Australia up

Australia up

Australia down

Australia down

Australia down

Australia down

0.7 × 0.45 × 0.5 = 0.16

0.7 × 0.45 × 0.5 = 0.16

0.7 × 0.55 × 0.5 = 0.19

0.7 × 0.55 × 0.5 = 0.19

0.3 × 0.45 × 0.5 = 0.07

0.3 × 0.45 × 0.5 = 0.07

0.3 × 0.55 × 0.5 = 0.08

0.3 × 0.55 × 0.5 = 0.08

0.55 × 0.55 = 0.3025

0.55 × 0.45 = 0.2475

0.45 × 0.55 = 0.2475

0.45 × 0.45 = 0.2025

H

H

T

H

T

HH

HT

TH

TT

0.550.450.55

0.45

0.55

0.45T

H

H

T

H

T

H

TH

T

TH

T

T

H

P(H, H, H) =

P(H, H, T) =

P(H, T, H) =

P(H, T, T) =

P(T, H, H) =

P(T, H, T) =

P(T, T, H) =

P(T, T, T) =

1—81—81—81—81—81—81—81—8

0.6 x 0.2 = 0.12

0.6 x 0.8 = 0.48

0.4 x 0.2 = 0.08

0.4 x 0.8 = 0.32

W

B

B'

B

B'

WB

WB'

MB

MB'

0.60.80.2

0.8

0.2

0.4M

a = b c

a b = c d e 1

a 4 b 5 c 8

P

T

S

0.4 x 0.3 = 0.120.4 x 0.25 = 0.100.4 x 0.45 = 0.18

PBPFPL

B

LF

0.25 x 0.3 = 0.0750.25 x 0.25 = 0.0630.25 x 0.45 = 0.112

SBSFSL

B

LF

0.35 x 0.3 = 0.1050.35 x 0.25 = 0.0880.35 x 0.45 = 0.157

TBTFTL

B

LF

1440------

1640------

1040------

0.35 x 0.286 = 0.10

0.35 x 0.714 = 0.250.40 x 0.375 = 0.15

0.40 x 0.625 = 0.250.25 x 0.80 = 0.20

0.25 x 0.20 = 0.05

N

NR

NR'SR

SR'LR

LR'

R

R'

L

SR

R'

R

R'

452------ 1

13------ 6

13------ 15

52------

1645------ 35

45------ 7

9--- 29

45------ 19

45------

16 76+178

------------------

59 13+148

------------------ 72148--------- 18

37------ 19

37------

1236------ 1

3--- 24

36------ 2

3--- 15

36------ 5

12------

317 o'clock 11 o'clock

5580 34

S55

200---------


A n s w e r s

541

answ

ers

3

The events are mutually exclusive and the Venn diagram could have been drawn as two circles which did not overlap.

4

0.18

5

0.9009

6

C

78

0.27, much higher probabilities of winning with roulette.

9

36

10

9


1 2 3

0.36 or 36%

4

0.16 or 16%

5

0.48 or 48%

6

7 8

4

9

14

10

Exercise 9D — The binomial distribution using Pascal’s triangle

1 a

1 7 21 35 35 21 7 1

b

1 9 36 84 126 84 36 9 1

2 a

1 5 10 10 5 1

b

1 7 21 35 35 21 7 1

3 a

1

b

4

c

70

d

1

e

110

4 a

0.0256

b

0.1176

c

0.125

d

0.0132

e

0.0720

f

0.0156

5 a

0.0179

b

0.0284

c

0.3456

6 a

0.2344

b

0.3125

7 a

0.2322

b

0.8936

8 a

0.4019

b

0.8038

9 a

0.5

b

0.2734

10 a

0.5

b

(0.5)

10

11 a

0.25

b

(0.75)

6

c

0.0330

12 a

0.2090

b

0.0413

13

0.1342

Career profile — Gail Twemlow

1

Selling betting tickets, calculating dividends, Cashbook and basic accounting

2

A boxed trifecta is more expensive because you have more chances of winning.

3

Reading cashbooks and using computer screens to follow what money has gone through the system.

Investigation — Pascal’s triangle

1

2

The triangle is symmetrical about a vertical line through the centre.

3

Number of entries in row

=

row number

+

1

4

Odd-numbered rows have an even number of entries.Even-numbered rows have an odd number of entries.

5

An odd number of trials has two middle numbers of the same value.An even number of trials has one middle number.

6

Sum of numbers in row

=

2

row number

(2 to the power of the row number)

7

Yes. 11

3

=

1331Yes. 11

4

=

14 641

8

They are square numbers.

9

Fibonacci’s sequence

10

Use the diagonal1

36

1015

11

The triangular numbers are ocated in the diagonal shown in question 10.

12

Sum of 10 lies in the position below the 4 to the right. This pattern continues. The same pattern continues for the numbers in the second diagonal.

a

37%

b

63%

IPT 16TS

031 29

S

S 0.51A

0.180.08 0.23

S

A 0.9009B

0.00090.0291

0.0691

S

A14

46

7

302

C

B

S 103

8

9

5

84

N

C

16--- 1

6---

126------

713------

120------

1 Row 0

1

1

11

1

11

1

1

1 10 45 120 210 252 210 120 45

9 36 84 126 126 84 36

8 28 56 70 56 28

7 21 35 35 216 15 20 15 6

5 10 10

4 63 3

4

5

7

8

9

10

1

121

1

1

11

1

11

Row 1Row 2

Row 3

1

1

1

11

1

11

1

11 10 45 120 210 252 210 120 45

9 36 84 126 126 84 36

8 28 56 70 56 28

7 21 35 35 216 15 20 15 6

5 10 10

4 63 3

4

5

7

8

910

1

121

1

1

11 2

3 5

11

1

11

2113

89

34

8

55

1 + 3 = 4 = 22

3 + 6 = 9 = 32

6 + 10 = 16 = 42

9B

➔

9D

542 A n s w e r san

swer

s13

s = 2r + 1 − 114 a Multiples of 2

b Multiples of 3

History of mathematics — Blaise Pascal1 16 years old2 A calculating machine3 The Puy de Dôme mountain4 For probability, permutations and combinations5 A brain tumour and stomach ulcer

Exercise 9E — Binomial probabilities through tables1 a 0.0008 b 0.9527 c 0.5793 d 0.99072 a 0.8281 b 0.9389 c 0.9887 d 0.95323 a 0.5000 b 0.6964 c 0.4142 d 0.00004 0.86845 a 0.6550 b Between 0.6167 and 0.96666 a 0.6230 b 0.5881 c 0.5000 d 0.0781

e 0.0139 f 0.00347 a 0.0404 b 0.9536 c 0.99408 a 0.2

b i 0.2013 ii 0.3222 iii 0.1074c The probability of this is very slight; there may

be some error in the sampling technique.

9 0.1275

10 a b c Approximately 0.2


1 2 1 4 6 4 1

3 p = , q = , n = 4 4 4p3q

5 0.0154 6 0.3602 or 36% 7 0.94088 0.0592 9 0.0139 10 0.0026

Chapter review1 D 2 C 3 C4 a

b c d

5 a b

6 0.3975 7 0.4928 8 0.99999

10 D 11 C12 a 0.5625 b 0.375 c 0.843 7513 D

14 a i ii

b i ii

15 B16 617 P(both A and B) = 0.25

18 D19 a

b

20 D21 a Two outcomes, success or failure, same event

repeatedb Tossing a coin 6 timesc Rolling a die and noting the upper-most face

22 a 0.402 b 0.03223 0.01624 a 0.39 b 0.39 c 0.7825 a 0.8725 b 0.804226 a 0.2 b 0.370427 a 0.12 b 0.2517

Row (r) 0 1 2 3 4 5 6

Sum (s) 1 3 7 15 31 63 127

Sum 2 3 4 5 6 7 8 9 10 11 12

Prob.

a 0.5 b c

13--- 1

3---

10

14---

16--- 5

6---

136------ 2

36------ 3

36------ 4

36------ 5

36------ 6

36------ 5

36------ 4

36------ 3

36------ 2

36------ 1

36------

12--- 2

9--- 1

9---

18--- 3

8---

1536------ 32

36------

2549------ 20

49------

2042------ 20

42------

0.05 0.25 0.25

A

S

B

18 12 13

Yes No

S

3043------


A n s w e r s 543

answ

ers

CHAPTER 10 The normal distribution and games of chanceSkills check1 Answers will vary. Check with your teacher.2 a 6.7 b 2.4 c 0.4 below the mean3 Distribution b4 45 to 555 a the value of x is larger than 40

b the value of x is less than or equal to 40c The value of x is larger than 20 and less than 30.

Exercise 10A — z-scores1 32 −23 a 0 b 1 c −2 d 3 e −14 a 10.5 b 13.7 c 16.9 d 7.3 e 0.95 −0.276 1.57 a −0.48 b 1.44 c 0.08 d −2.24 e 2.88 a 10.3 s b 10.58 s c 10.37 s

d 9.88 s e 10.251 s f 10.524 s9 a = 19.55, sn = 1.76 b 1.68

b = 56, σn = 20.1c i 0.30 ii 2.2 iii −2.0

11 B12 B13 C14 a = 64.7, σn = 11.4

b Highest score z = 2.66, Lowest score z = −1.7315 English 1, Mathematics 1.31, Biology 1.5,

Computing studies −2, Visual arts 0.67, Music −0.8

Exercise 10B — Comparison of scores1 a English 1.25, Maths 1.33

b Maths mark is better as it has a higher z-score.2 2nd test, Barbara’s z-score was −0.33 compared to

−0.5 in the first test.3 C4 D5 Course A, z-score of −0.8 compared to −0.75 on

course B6 a Atlanta 0.44, Sydney 1

b In Atlanta because of the lower z-score7 C8 B

9 a Mathematics = 59.5, sn = 17.9

Chemistry = 59.6, sn = 16.8b Mathematics 0.25, Chemistry 0.20. So

Mathematics is the better result.10 Kory is the better candidate with a z-score of 1.5

compared with 0.875 for Ricardo.

10 Quick Questions 11 22 −23 −1.034 2.955 One standard deviation above the mean6 Two standard deviations below the mean7 508 89 English 1.25, Maths 1.4

10 Maths

Exercise 10C — Distribution of scores1 a 68% b 95% c 99.7%2 a 68% b 95% c 99.7%3 95%4 16%5 a 68% b 16% c 0.15%6 21.1 and 33.97 a 68% of the values have a z-score between −1

and 1.b 95% of the values have a z-score between −2

and 2. c 99.7% of the values have a z-score between −3

and 3.8 B9 A

10 0.15%11 a 16% b 16%12 a 95% b 16% c 34%

d 15.85% e 83.85%13 a 95 g to 105 g b 92.5 g to 107.5 g14 163 cm − 181 cm15 Faulty, as the one chosen has a z-score greater than 316 2.6 kg − 5 kg

Exercise 10D — Standard normal tables1 a 0.8413 b 0.9192 c 0.9641

d 0.1587 e 0.0446 f 0.2417g 0.6826 h 0.9544 i 0.9974j 0.1359 k 0.0215 l 0.8664

2 a 0 b −0.75 c −1 d 1.53 a 0.5 b −0.5 c −1.3 d 1.24 a 0.8413 b 0.9452 c 0.5

d 0.0013 e 0.3413 f 0.81855 a 0.6406 b 0.8577 c 0.5 d 0.3594

e 0.2812 f 0.6188 g 0.0509 h 0.05546 a 0.0500 b 0.1335 c 0.0294 d 0.86657 a 0.7486 b 0.9082 c 0.2514 d 0.65688 a 75% b 37 c 469 a i 0.9332 ii 0.8413 iii 0.1210

b 22.6%10 a 0.7881 b 0.3446 c 0.1151

d 0.7881 e 0.5403

10 a Amount ($) Class centre Frequency

0–<20 $10 2

20–<40 $30 8

40–<60 $50 19

60–<80 $70 15

80–<100 $90 6

x

x

x

x

x

9E➔

10D


544 A n s w e r san

swer

s11 4.75%12 0.38%, assuming weights to be normally

distributed.

Exercise 10E — Odds1 a $105 b $105 c $429 d $300

e $15 f $33.33 g $66.662 a $140 b $175 c $507 d $420

e $60 f $83.33 g $216.66

3 a b c

d e f

4 a 2:1 b 4:1 c 3:2 on d 5:2e 7:5 f 2:1 on g 6:4 h 11:9

5 a Evens b 5:1 c 3:1d 12:1 e 3:1

6 a 2:1 b 3:1 c Evensd 7:5 e 2:1 on f 5:2 on

7 a $36 000 b i $13 500 ii $22 5008 a $160 b $71.11 c $80

Exercise 10F — Two-up1 2 3

4 5 6

7 a TH, TH, TH, TH b

8

9 a b c d e 0.48

10 Not quite, probability of winning = 0.48

Exercise 10G — Roulette1 a b

2 a 19:18 b 31:6c No, slightly lower

3 a $60 b 0 c $200 d $754 Nothing happens except when the ball lands on 0,

then he loses both bets.

Exercise 10H — Common fallacies in probability1 a 1/16 b 1/32 c 1/22 a i 0.32 ii 0.24

b Her chances of winning any match remain 0.75.4 a Events are not independent.

b How many people in this group own surfboards?

Exercise 10I — Mathematical expectation1 3.6 2 3.13 15.6 4 −40 cents5 −$2.50 6 $0.627 −13.5 cents 8 $1.609 7

10 Quick Questions 21 Very probable 2 2.5%3 970 mL 4 0.95055 0.0495 6 0.04957 0.9010 89 $500 10 $1 loss

Chapter review1 −22 a 0 b 1 c −2 d 3 e −1 3 1.874 a 0.17 b 1.83 c −3 d −1.75 e −2 5 a = 1130, σn = 334.2

b i −0.39 ii 2.05 iii −2.62 iv −1.13 v 3.076 a = 20.1, σn = 2.1

b Highest = 1.86, Lowest = −1.95 7 a 1.5 b 1 c Physics, higher z-score8 a Geography: −0.8, Business studies: −0.53

b Business studies: higher z-score9 Numeracy: lower z-score

10 a 68% b 95% c 99.7%11 a 68% b 95% c 99.7%12 a 34% b 47.5% c 2.5%

d 0.15% e 97.35%13 Faulty, as it is more than three standard deviations

from the mean.14 a 0.9032 b 0.9918 c 0.1587

d 0.0668 e 0.207515 a 0 b 1 c −0.83 d 5.3316 a 0.5 b 0.8413 c 0.1587

d 0.0485 e 0.1293 f 0.377917 a 0.5 b 0.8413 c 0.7357 d 0.234218 a Horses 5 and 6

b Horses 1, 3 and 519 Test this by simulation on a computer.20 a Same b Same21 a i 0.49 ii 0.25 iii 0.25

b No22 a $1.67

b About $2.65 (if the operator’s percentage is based on this figure)

45--- 1

4--- 3

5---

27--- 3

10------ 2

3---

14--- 1

4--- 1

2---

12--- 1

8--- 1

16------

116------

132------

18--- 1

16------ 1

32------ 1

64------

1837------ 6

37------

25---

x

x
