Year 12 Maths A Textbook - Answers

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Year 12 Maths A Textbook - Answers

  1. 1. Answers 515 Answers CHAPTER 1 Simple and compound interest Exercise 1C — Graphing simple interest functions Skills check 1 a 4 a 450 d 1.157 625 $8.75 $4 0.0725 0.125 1 -- years 6 5 4 ----- years 12 b 525 e 1.083 c f c f c f $1.50 $0.25 0.2 0.001 2 -- years 3 -2 1 years 2 b 1000 500 0 2 a 16 000 14 000 12 000 10 000 8000 6000 4000 2000 0 3 4 Years 5 1 2 3 4 5 $1600 $3200 $4300 $6400 $8000 0 1 2 3 4 5 6 7 8 9 10 Years c 1600 d $16 000 3 a 6000 4000 2000 0 0 1 2 3 4 Years 5 2 3 4 Years 5 1000 Interest ($) b 800 600 400 200 0 c 0 1 4000 3000 2000 1000 0 0 1 2 3 4 Years 5 1A ➔ d Yes ($1281.60) 2 No. of years Interest c $5425 6 $352 9 C 13 A 1 0 Interest ($) b Exercise 1B — Finding P, R and T c 80% e 3.36% c 3 months 3 4 5 $1200 $1600 $2000 1500 Exercise 1A — Simple interest 1 a $3070 b $4400 d $236.36 e $2500 2 a 10% b 6.25% -d 2.125% or 2 1 % 8 b 18 months 3 a 1 year d 7 years e 1 month 4 $1515.79 5 $2133.33 7 24 months 8 3 years 10 B 11 B 12 D 14 a Yes ($1112.50) b No c Yes ($1600 in 23 months) 2 $800 2000 c 21 000 f 1.877 1 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 2 a $103.50 b $2700 c $325 d $131.25 3 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 16 $584.50 17 a The Big-4 Bank offers the best rates. -b The Big-4 Bank charges 11 1 % p.a. for a loan 3 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 1 years 2 1 $400 Interest ($) b e b e b e Interest ($) $1 $0.25 7.25 0.002 1 -- years 4 1 2 ----- years 12 Interest ($) 1 a d 2 a d 3 a d No. of years Interest answers Maths A Yr 12 - Answers Page 515 Wednesday, September 11, 2002 3:41 PM 1C
  2. 2. 516 Answers 10 Quick Questions 1 Interest ($) 20 000 1 $800 5 $7000 9 54c 15 000 10 000 5000 1 0 2 3 4 Years 5 Interest ($) 400 300 200 100 0 b Investment ($) 0 3 4 Years 5 3500 0 1 2 3 Years 4 5 1 a 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 b 5000 2 a 2 3 4 Years 1 2 3 4 5 $8400 $8820 $9261 $9724 $10 210 3 4 5 5 1 2 3 4 5 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 0 1 2 3 4 Years 5 8 a $4500, $5000, $5200 c $25 900 25 000 20 000 15 000 10 000 25 000 20 000 15 000 b $875 0 9 5000 4000 Future value ($) b 6.2% 5.8% 5.2% 4500 2 0 1 2 3 4 5 6 7 8 9 10 Years 3 a 500 0 1 5 0 Future value ($) 1000 2 3 4 Years 30 000 3.75% 3.5% 3% 1500 Interest ($) b 1 No. of years Future value ($) No. of years 0 Future value $12 960 $13 997 $15 117 $16 326 $17 632 b 7 a 10 000 9500 9000 8500 8000 0 0 1 No. of years Future value 8% 6% 5% 10 000 0 0 1 2 3 4 Years 5 4000 3000 2000 0 0 1 2 3 Years 4 5 4 $1.90 8 $1998.75 Exercise 1E — Graphing compound interest functions 3000 Interest ($) b 2 4000 0 6 a 1 3 $2700 7 $215.44 1 $6655 2 $17 173.50 3 $2938.66 4 a $4630.50 b $9274.19 c $24 488.80 d $13 503.78 e $12 588.72 5 $70 555.26 6 $502 7 $14 059.91 8 $31 850.33 9 a $1003.69 b $8985.76 c $181 402.12 d $20 039.67 e $1 264 568.95 10 B 11 B 12 B 13 C 14 a $15 746.40 b $15 793.09 c $15 817.40 d $15 833.99 15 a 0.0219% b $108 320.71 c $8320.71 d $320.71 16 a $4720 b $4726.24 c Compounding interest 4 288 5 a 2 $1260 6 $138 10 $173.58 Exercise 1D — Calculation of compound interest 0 Future value ($) d Interest ($) answers Maths A Yr 12 - Answers Page 516 Wednesday, September 11, 2002 3:41 PM 0 0 1 2 3 4 Years 5
  3. 3. 517 Answers Chapter review 4 a 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 $2688 2700 2600 2500 2400 2300 2200 2100 2000 0 Interest 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 b Years Future value ($) 1800 1600 1400 1200 600 500 400 300 200 100 0 0 1 2 3 4 5 6 7 8 9 10 Years 6 a No. of years Interest ($) b Compound interest earned ($) 5 a 0 0 1 2 3 4 5 6 7 8 9 10 Years 1 2 3 4 5 Interest (4%) $320 $653 $999 $1359 $1733 Interest (6%) $480 $989 $1528 $2100 $2706 Interest (8%) $640 $1331 $2078 $2884 $3755 b Interest ($) 12 000 Interest (8%) Interest (6%) Interest (4%) 10 000 1 2 3 4 Years 7 a No. of years Future value ($) 1 Annually $4240 Six-monthly $4244 3 $4764 $4776 4 $5048 $5068 5 $5352 $5376 Annually Six-monthly 5500 5000 4500 4000 0 0 1 2 3 Years 4 5 10 Quick Questions 2 1 4 7 10 $2051.28 $429.69 $6792.58 $14 700.68 2 $17 253 5 $2315.25 8 $6805.66 3 4 5 $1350 $1800 $2250 1 2 3 4 5 Future value $20 800 $21 632 $22 497 $23 397 $24 333 5 2 $4496 $4504 2 $900 $455 C B 0 1 2 3 4 5 6 7 8 9 10 Years No. of years b 0 4500 4000 3500 3000 2500 2000 1500 1000 500 0 1 $450 $42 c 450 d $4500 13 $2778.30 14 $5700.47 15 a $3932.39 b $12 596.90 c $14 457.96 d $23 031.76 e $5334.67 16 $756.94 17 a $12 024.02 b $12 052 c Compounding interest 18 a 8000 0 b b $2820 c e $7617.58 b $39 780 c 5 B 6 8 15 months 9 11 C 12 a No. of years 3 $3437.50 6 $315.25 9 $6812.41 Exercise 1F — Nominal and effective interest rates 28 000 27 000 26 000 25 000 24 000 23 000 22 000 21 000 20 000 0 0 1 2 3 4 5 6 7 Years 8 9 10 c $29 600 19 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. effective c 3.96% p.a. effective Choose 3.895% p.a. compounding monthly. CHAPTER 2 Appreciation and depreciation Skills check 1 a d 2 a d 3 a d 4 a d 5 a d 1.1 0.9625 $4 41c $44 $18 40% 4.5% 1.06 0.97 b e b e b e b e b e 1.01 0.9775 $5 $1.10 $3 $135.94 87.5% 51.4% 0.94 1.03 c f c f c f c f c f 1.0175 0.9975 $3.95 $4.37 $7.64 $3940 64.5% 1.56% 1.04 0.97 1D ➔ 1 4.06% p.a. 2 4.08% p.a. 3 3.79% p.a. 4 a 4.27% p.a. effective rate b 4.26% p.a. effective rate So 4.2% p.a. compounding quarterly is better. 5 5.095% effective rate, 5.12% effective rate. Choose 5% p.a. compounding monthly. Future value ($) Future value ($) b 1 $1000 2 a $1296 d $4.05 3 a $7280 4 D 7 6.5% 10 B answers Maths A Yr 12 - Answers Page 517 Wednesday, September 11, 2002 3:41 PM 1F
  4. 4. 518 Answers 1 $20 800 2 a $618 d $579.60 3 a $878.05 4 $117.90 7 $1.20 10 $500 b e b 5 8 11 $48.15 $932.40 $901.76 $619 $122.80 $2350 6 $2.52 9 D 12 $2460 8 B 9 a Value ($) 3 a Value ($) 22 000 18 000 4 14 000 5 V = 50 000 − 8000A 26 000 3 0 2 4 6 8 10 Age (years) 30 000 2 10 000 b See part d. c Age (years) Value ($) New (0) 50 000 40 000 30 000 20 000 10 000 0 7000 6000 5000 4000 3000 2000 1000 0 12 228 5 c 4 15 360 4 d 19 200 3 c 9 years 24 000 2 0 2 4 6 8 10 Age (years) 30 000 1 0 2 4 6 8 10 Age (years) b $20 000 4 a V = 6400 − 2000A b Value ($) 0 2 4 6 8 10 Age (years) 1 Value ($) Value ($) 50 000 40 000 30 000 20 000 10 000 0 iii $102 400 New (0) b V = 100 000 − 10 000A 2 200 000 160 000 120 000 80 000 40 000 0 Age (years) Value ($) 100 000 80 000 60 000 40 000 20 000 0 ii $128 000 b c $1.91 Exercise 2B — Modelling depreciation 1 a 7 a i $160 000 iv $81 920 Value ($) Exercise 2A — Inflation and appreciation 9 830 30 000 24 000 18 000 12 000 6 000 0 Straight line value Declining balance value 0 2 4 6 8 10 Age (years) e 6 years 0 1 2 3 4 Age (years) b $2000 1 $20 000 2 a $1000 d $145 3 a $7 125 000 4 $10 600 5 8 years 6 a 6 years c 8 years 7 $2500/year 8 a $4000/year 9 $900/year 10 $25 000 11 a $110 000 12 $78 000 Value ($) 20 000 16 000 12 000 8 000 4 000 0 Exercise 2C — Straight line depreciation b $17 000 c 7 5 a 6 a Value ($) answers Maths A Yr 12 - Answers Page 518 Wednesday, September 11, 2002 3:41 PM 0 2 4 6 8 10 Age (years) 620 000 480 000 360 000 240 000 120 000 0 0 2 4 6 8 10 Age (years) b $10 300 e $32 000 b $3 750 000 c $270 000 b 5 years d 7 years b $12 500/year c $14 500/year b $26 500 c $1450
  5. 5. Answers Exercise 2D — Declining balance or diminishing value method of depreciation 1 2 3 4 5 6 7 8 9 10 11 12 $20 480 a $2220 7 years $383 000 a $5900 d $62 100 $6174 $676 000 a $14 600 A C a $5360 c $3591 5 years 3 a 100 000 90 000 85 000 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 c $1200 b $20 400 b $2640 d $1769 b 0 19 700 Straight line 100 000 value 80 000 Diminishing value 60 000 40 000 20 000 0 0 2 4 6 8 10 Age (years) Value ($) 2 $2350/year 4 $21 000 R 5 S = Vo  1 – --------  100 100 000 2 1 $650 3 7 years Diminishing value ($) New (0) ii $390 b $68 100 e $3900 Straight line value ($) 1 b i $750 10 Quick Questions 1 7 $11 000 9 $37 500 Age of equpment (years) 519 T 6 $7250 8 $389 000 10 16 years 4 Age of computer (years) Salvage value at 35% ($) 3520.00 2860.00 2 2816.00 2288.00 3 2252.80 1830.40 4 1802.24 1464.32 5 1 a $1638.50 b $9537.50 c $34 870 2 a Age of car Straight line Diminishing (years) value ($) value ($) Salvage value at 20% ($) 1 Exercise 2E — Depreciation tables 1441.79 1171.46 New (0) 40 000 40 000 1 35 000 32 000 2 30 000 25 600 6 1153.43 937.16 3 25 000 20 500 7 922.75 749.73 4 20 000 16 400 8 738.20 599.79 5 15 000 13 100 9 590.56 479.83 6 10 000 10 500 10 472.45 383.86 7 5 000 8 400 5 Salvage 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 Years 8 Value ($) b answers Maths A Yr 12 - Answers Page 519 Wednesday, September 11, 2002 3:41 PM 40 000 32 000 24 000 16 000 8 000 0 0 Straight line value Diminishing value c After 6 years 2A ➔ 0 2 4 6 8 10 Age (years) 6 700 2E
  6. 6. 520 Answers 6 Salvage value — declining balance ($) Tax deduction ($) 1 200 000 50 000 2 160 000 40 000 4 746 3 128 000 32 000 3 560 4 102 400 25 600 2 670 5 81 920 20 480 65 536 16 384 Salvage value ($) Tax deduction ($) 1 33 750 11 250 2 25 313 8 438 3 18 984 6 328 4 14 238 5 10 679 6 8 009 Years Age of truck (years) 7 6 007 2 002 6 8 4 505 1 502 7 52 429 13 107 8 41 943 10 486 9 33 554 8 389 10 26 844 6 710 7 a Age of truck (years) New (0) Straight line value ($) Diminishing value ($) 250 000 225 000 200 000 2 200 000 160 000 3 175 000 128 000 4 150 000 102 400 5 125 000 81 920 6 100 000 65 536 7 75 000 52 429 8 50 000 41 943 9 25 000 33 554 10 b 250 000 1 Value ($) answers Maths A Yr 12 - Answers Page 520 Wednesday, September 11, 2002 3:41 PM 0 26 844 250 000 Straight line 200 000 value 150 000 100 000 50 000 Diminishing 0 value 0 2 4 6 8 10 Age (years) c 8 a $10 000 b 9 a $3000 b $75 1 -4 Exercise 2F — Future and present value of an annuity 1 $7049.37 2 a $6691.13 b $16 859.14 c $6158.56 d $3974.56 e $17 713.21 f $3530.21 3 $4472.93 4 $90 237.49 5 a $20 326.23 b $24 297.37 c $45 881.32 d $69 770.03 6 C 7 $4067.23 8 a $4524.37 b $7068.59 c $1930.08 9 $4787.76 10 a $1324.00 b $23 932.35 c $7503.81 d $62 953.50 11 a 4% b 10 c $6003.05 12 a $4103.92 b $5535.38 c $7546.74 13 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 $6918.50 16 a $1845.09 b $12 289.20 c $4455.79 d $16 604.40 Salvage value — straight line ($) 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 1 A = 20 000  1 + --------  100 6 100 000 25 000 2 7 75 000 25 000 8 50 000 25 000 9 25 000 25 000 10 0 25 000 Age of truck (years) c $2500 c $1600 d $750 10 Quick Questions 2 1 3 5 7 9 $15 937.42 $13 537.79 $5084.04 $22 094.93 $10.63 2 4 6 8 10 $15 937.42 $1435.91 $19 277.16 $8513.56 $13 295.75 Investigation — A growing investment T
  7. 7. 521 Answers The investment will grow to $30 000 in 7 years. 8 10 000 8000 6000 4000 2000 0 Value ($) 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 0 2 4 6 8 10 Age (years) 9 10 11 14 15 16 $6500 a $1300 b $15 000 c $235 000 12 years 12 $250/year 13 After 6 years $20 880 $474 000 a $23 620 b $1000 c $24 290 d $27 210 e $49 380 17 a $167 100 b $432 900 18 a Salvage value Salvage Age — straight line value — (years) ($) 15% p.a. ($) New (0) Value ($) 200 000 160 000 120 000 80 000 40 000 0 61 400 60 000 52 200 50 000 44 400 40 000 37 700 7 30 000 32 100 8 20 000 27 200 9 10 000 23 100 10 5 a 70 000 6 0 2 4 6 8 10 Age (years) 72 300 5 3 $7900 b V = 200 000 − 20 000A 80 000 4 2 $149.90 85 000 3 Chapter review 100 000 90 000 2 Investing $15 000 at 9.1% p.a. will reach $30 000 in 8 years. 100 000 1 1 $83.44 4 a 0 19 700 0 2 4 6 8 10 Age (years) b V = 3500 − 250A 6 a c $1250 Salvage value — 15% p.a. 0 2 4 6 8 10 Age (years) 19 60 000 50 000 40 000 30 000 20 000 10 000 0 0 2 4 6 8 10 Age (years) 1485 2 2020 995 3 1353 667 907 446 0 907 20 $41 039.20 21 a $4399.95 c $1842.84 22 $2242.95 23 a $1516.32 c $4055.45 b $34 641.25 d $51 014.25 b $14 047.20 d $11 177.64 2F ➔ c 9 years 3015 5 c 17 years Tax deduction ($) 4 0 2 4 6 8 10 Age (years) Salvage value ($) 1 b $10 500 7 a Value ($) Salvage value — straight line Year Value ($) 16 000 12 000 8 000 4 000 0 b $6500 100 000 80 000 60 000 40 000 20 000 0 Value ($) Value ($) b 3 000 2 000 1 000 0 answers Maths A Yr 12 - Answers Page 521 Wednesday, September 11, 2002 3:41 PM 2F
  8. 8. answers Maths A Yr 12 - Answers Page 522 Wednesday, September 11, 2002 3:41 PM 522 Answers CHAPTER 3 Consumer credit and investments 4 a Month 1 2 3 4 5 6 7 8 9 10 11 12 1 a d 2 a d 3 a d 4 a b c 5 a -b 11 2 0.1 0.0375 18 2 i 30 i 45 i 25 Feb. $440 c e 1 ----12 1 2 -2 1 ----52 f b e b e ii ii ii b 0.01 0.005 33 69 30 33 24 Aug. $20 c f c f iii iii iii c 1 -------365 1 ----26 0.025 0.0125 8 15 31 56 13 Apr. 37.5% Exercise 3A — Flat rate interest 1 a $700 b $1200 c $7500 d $2850 e $390 2 $1584 3 $5000 4 a $4060 b $21 330 c $1803.75 d $308.25 e $275 000 5 a $1650 b $3850 c $693 d $6193 6 a $1600 b $6600 c $137.50 7 a $800 b $2800 c $53.85 8 a $2000 b $6000 c $2160 d $8160 e $226.67 9 $43.33 10 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 loans 1 a $800 b $79 950 2 a $312.50, $49 848.99 b $151.01 c $311.56, $49 697.04 d $151.95 3 Principal ($) Month 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 Interest ($) 637.50 636.55 635.58 634.61 633.64 632.66 631.66 630.67 629.66 628.65 627.63 626.60 Balance owing ($) 84 872.73 84 744.51 84 615.32 84 485.16 84 354.03 84 221.92 84 088.81 83 954.71 83 819.60 83 683.48 83 546.34 83 408.17 Month 1 2 3 4 5 6 7 8 9 10 11 12 Skills check Principal ($) 85 000.00 84 872.73 84 744.51 84 615.32 84 485.16 84 354.03 84 221.92 84 088.81 83 954.71 83 819.60 83 683.48 83 546.34 Principal ($) 85 000.00 84 837.50 84 673.78 84 508.83 84 342.65 84 175.22 84 006.53 83 836.58 83 665.35 83 492.84 83 319.04 83 143.93 Interest ($) 637.50 636.28 635.05 633.82 632.57 631.31 630.05 628.77 627.49 626.20 624.89 623.58 Balance owing ($) 84 837.50 84 673.78 84 508.83 84 342.65 84 175.22 84 006.53 83 836.58 83 665.35 83 492.84 83 319.04 83 143.93 82 967.51 b c $440.66 $243 123 a $302 308.80 b $241 500 c $60 808.80 A B a $112 034 b $86 072 c $61 789.40 d $39 329.60 10 a Smith – $6000, Jones – $9000 5 6 7 8 9 b i Smith family Principal Interest Balance owing ($) ($) ($) Month 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
  9. 9. Answers Jones family b ii Principal Interest Balance owing ($) ($) ($) 523 Exercise 3E — Bonds, debentures and term deposits 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 $3200 2 $315 3 $472.50 4 $1500 $1800 6 $612.50 7 B 8 A C 10 D 11 C 12 B a $22.15 b $84.99 c $297 a i $1406.25 ii $1350 iii $1321.88 b No difference 4 48 929.06 387.36 48 566.42 Exercise 3F — Bank savings accounts 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 b $9.47 1 a $2066.10 b 12 cents 2 a 8 cents 3 9% 4 1998 Transaction Debit 8 47 461.16 375.73 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 1 May 3 May 7 May 17 May 26 May 31 May 2 June 8 June 19 June 21 June 23 June 30 June 1 July 4 July 18 July 26 July 31 July 47 086.89 9 1 5 9 13 14 45 559.95 c $3134.16 10 Quick Questions 1 1 3 5 7 9 answers Maths A Yr 12 - Answers Page 523 Wednesday, September 11, 2002 3:41 PM $420 $7025 $5525 $8342.75 $1640 2 4 6 8 10 $1135.20 $975 $2817.75 $231.74 $265 577 Exercise 3C — The cost of a loan 1 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 1 7 a $231 546 b $200 745.60 c $145 593.60 8 Loan 2 – they will save $6041 9 C 10 a $341 376 b $337 578 11 D 12 a $562 279.20 b 6.25% c 5.8% 5 a $34.03 6 1998 4 Aug 5 Aug 10 Aug 15 Aug 18 Aug 20 Aug 30 Aug 31 Aug 1 Sept 2 Sept 5 Sept 5 Sept 15 Sept 20 Sept 29 Sept 30 Sept Exercise 3D — Loan repayments Credit 1740.60 11.02 Balance 2132.20 1671.93 1902.09 1010.89 2751.49 2757.18 3172.28 955.28 526.78 509.98 2250.58 2253.45 2276.25 2239.53 1958.57 3699.17 3710.19 Credit Balance 1410.20 1410.20 1082.60 1150.80 1000.54 2410.74 1789.74 1609.74 1615.99 3026.19 2845.55 3306.82 2979.22 4389.42 3768.62 5178.82 5196.60 460.27 230.16 891.20 1740.60 5.69 415.10 2217.00 428.50 16.80 1740.60 2.87 22.80 36.72 280.96 b $14.45 Transaction Salary Health fund Health fund Electricity a/c Salary Rent Visa Interest Salary Telephone a/c Tax refund Health fund Salary Rent Salary Interest 7 a i $6.25 b i $4.79 c i $10.94 Debit 327.60 68.20 150.26 1410.20 620.80 180.00 6.25 1410.20 180.64 461.27 327.60 1410.20 620.80 1410.20 17.78 ii $13.35 ii $4.76 ii $16.86 iii $7.10 iii –$0.03 iii $5.92 10 Quick Questions 2 1 3 5 7 9 $2835 $299 191.20 6.0% $264.40 $5825 2 4 6 8 10 160% p.a. 10.3% $45.23 $375 $2.23 3A ➔ 1 $674.25 2 a $90.46 b $341.25 c $819.84 d $1101.00 e $1515.54 3 a $400 b $3600 c $123.05 4 They will not need to increase their repayments. 5 a $1510.20 b $1620.14 6 Yes. The repayment is $744 and the most he can afford is $750. 7 a $7000 b $1750 c $178 000 8 a $733.40 b $174.80 Balance B/F Cheq 4217 Deposit Cheq 4218 Wages Interest Deposit Cheq 4220 Cheq 4219 Cheq 4222 Wages Interest Deposit Cheq 4221 Cheq 4223 Wages Interest 3F
  10. 10. 524 Answers b $3656.25 b $365.63 b $124 228.12 c $12 000 c $1200 c $448 800 b $3175.50 b $272 415.50 b $278 375 c $12 832.50 c $554 952.50 c $8682.50 $19 131.25 $5071 25 c/share 6 c/share a $1.224 million c 43.52 c/share 10 a $5.22 million c $1.66/share 11 $3.276 million 12 4.57% 13 Dividend 2 4 6 8 b $2511.25 $1540 $1.50/share 29.27 c/share $2.176 million b $9.28 million Share price $8.40 $7.40 $23.40 $15.76 $0.76 $0.56 $0.78 $1.20 $1.09 $0.04 2.91% 15 a 6.6% a $1.14 8.5% a 0.59% c 6.44 c/share a $77.50 b a $60 b a 5000 b 23.3 D Dividend yield 6.7% 10.5% 5.1% 6.9% 5.3% 16 $364 b $1.06/share b $5.928 million b $10.64 d 0.61% 1.2% c 82.6 1.2% c 83.3 0.75% c 133.3 Month b $1.20 4 a 16.00 14.00 12.00 10.00 n b r r y n l g p t v c n b r r y n Ja Fe a p a Ju Ju u Se Oc o e Ja Fe a p a Ju 1– 1– 1–M 1–A 1–M 1– 1– 1–A 1– 1– 1–N 1–D 1– 1– 1–M 1–A 1–M 1– Month b $16.00 5 a 14.50 Share price ($) 1 3 5 7 9 21 22 23 24 2.04 2.02 2.00 1.18 1.16 1.14 1.12 1.10 1.08 1.06 1.04 1.02 1.00 14.00 13.50 13.00 12.50 12.00 1– Ja 1– n Fe 1– b M 1– ar A 1– pr M 1– ay Ju 1– n J 1– ul A 1– ug Se 1– pt O 1– ct N 1– ov De c Exercise 3H — Investing in the stock market 14 17 18 19 20 3 a 1– J 1– an F 1– eb M 1– ar A 1– pr M 1– ay J 1– un 1– Jul A 1– ug Se 1– pt O 1– ct N 1– ov D 1– ec J 1– an F 1– eb M 1– ar A 1– pr M 1– ay J 1– un 1– Jul A 1– ug Se 1– pt Oc t a $2448.75 a $244.88 a $77 256.37 $80 750 a $1873 a $189 123 a $169 692.50 $8745 $127 500 $146 450 $289 500 Share price ($) 1 2 3 4 5 6 7 8 9 10 11 Share price ($) Exercise 3G — Investing in real estate Month b $14.50 History of mathematics — The Wall Street Crash 1 Soaring share prices were suddenly reversed. 2 Share prices declined rapidly. 3 People stopped investing, banks and businesses collapsed, unemployment rose and Hitler came to power. History of mathematics — The Dow Jones Industrial Average 1 a 1 Wall Street Journal journalists Charles Dow and Eddie Jones. 2 30. 3 Sum of 30 stock prices divided by 0.2252. 4 Technology, telecommunications. 7.30 7.10 6.90 6.70 6.50 6.30 6.10 5.90 5.70 1– M ay 1– Ju n 1– Ju l 1– Au g 1– Se pt 1– Oc t Share price ($) Exercise 3I — Graphing share performance Month Share price ($) b $7.00 2 a 4.50 4.00 3.50 3.00 2.50 1– Ju 1– n J 1– ul A 1– ug Se 1– pt O 1– ct N 1– ov D e 1– c Ja 1– n Fe 1– b M 1– ar A 1– pr M ay answers Maths A Yr 12 - Answers Page 524 Wednesday, September 11, 2002 3:41 PM b $1.75 Chapter review 1 a $1120 d $39.60 2 $6760 3 $191.02 4 6.15% 5 a $1250 b $7187.50 c $1281.60 e $12 285.00 b $124 873.64
  11. 11. Answers 6 a Balance owing ($) 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 6 128 881.67 859.21 128 653.51 857.69 128 423.83 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 Month Principal ($) Interest ($) 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 Month b 7 8 9 10 11 12 c $5137.21 a $596 844 a 7.25% c 25.65% a $18 223.20 Loan 2 $21.15 a $316.75 c $2369.11 a $2453.49 B $2700 $5000 a $1.08 a $3.33 $3075 $270 662.50 $1.93/share b $18 884 b 13.70% d 14.11% b $4723.20 c 7% 15 18 21 25 28 31 b $599.40 d $5100 b $2618.06 A 16 D $694.17 19 $2520 D b $1.15 b $3.64 $401.63 26 $236 425.45 $15 832.50 29 $46 687.50 $14.74/share 32 5.22% 16.50 16.00 15.50 Month b $18.00 CHAPTER 4 Populations, samples, statistics and probability Skills check 1 2 3 4 5 a 0.375 b 0.083 c 0.813 d 0.590 a 75% b 12.5% c 42.5% d 4% Answers will vary. a 4 b 4 c 3 d 7 e 1 a a=8 b b=9 c c = 22.5 d d = 17.5 e e = 10.5 6 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 6 e 6 f 8 g 6 h 2 Investigation — Australia’s population and housing census 1 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 samples 1 Census, sample 2 Census — every member of the population participates. 3 Survey 4 a Survey b Survey c Census d Census e Survey 5 a Survey b Census c Census d Survey 6 Survey 3G ➔ 13 14 17 20 22 23 24 27 30 34 81.8 c/share b 2.5% c 40.4 128 653.51 7 33 1.6% 35 a $260 36 a 17.00 Share price ($) Interest ($) 1– Ja 1– n Fe 1– b M 1– ar A 1– pr M 1– ay Ju 1– n J 1– ul Au 1– g Se 1– pt O 1– ct No 1– v De c Principal ($) 525 answers Maths A Yr 12 - Answers Page 525 Wednesday, September 11, 2002 3:41 PM 4A
  12. 12. answers Maths A Yr 12 - Answers Page 526 Wednesday, September 11, 2002 3:41 PM 526 Answers Exercise 4B — Samples and sampling 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 80, 84, 70, 85, 79, 54, 56, 51, 81, 67 Range of answers Range of answers Should be three different sets of numbers. a Random sampling b Judgemental sampling c Accessibility sampling d Systematic sampling e Stratified sampling a Systematic b Stratified c Systematic d Random e Stratified A C Year 7— nine, Year 8 — eleven, Year 9 — nine, Year 10 — eight, Year 11 — seven, Year 12 — six 36 men and 24 women Age 20–29 30–39 40–49 50–59 Male 10 7 12 1 Female 7 8 3 2 4000 13 400 a 10 000 b 25 000 c 3663 No — estimated population 20 000 a 625 b 500 c 625 a 833 b 1000 c 882 d 905 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 data 1 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 324 330 = ---------------------- × 100% 7 636 319 = 4.2% 4 Analyse data from the 2001 census. Exercise 4D — Contingency tables 1 Test results Accurate 1 3 5 7 9 Census Census Systematic sample Cluster sample Quota sample 2 4 6 8 10 Survey Random sample Stratified sample Judgemental sample Random sample Not accurate Total 98 2 100 Without virus 388 12 400 Total 10 Quick Questions 1 486 14 500 With virus Test results 2 Exercise 4C — Bias 1 Check with your teacher. 2 Check with your teacher. 3 a Sample does not represent characteristics of population. b No control over responses c Unrepresentative sample d Abnormal conditions e 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. Accurate Not accurate Total Telling truth 777 23 800 Telling lies 156 44 200 Total 933 67 1000 3 4 a 1000 a 200 e 94% 5 B 8 a b 75 c 96.7% d 60% b 44 c 90.9% d 5.1% f Check with your teacher. 6 D 7 A Test results Accurate Not accurate Total 48 2 50 Bags with no prohibited items 145 5 150 Total 193 7 200 Bags with prohibited items b i 96% ii 3.3% iii 4% iv 96.5%
  13. 13. 527 Answers 9 a ii 10 a b ii c ii 12 a Education b Education Female Total In 2641 1752 4393 Not in 1728 3144 4872 4369 4896 9625 Female 3 685 4 071 239 389 240 029 Male Other Male Total Male Female Total 3 685 4 071 7 756 Other 239 389 240 029 479 418 Total 243 074 244 100 answers Maths A Yr 12 - Answers Page 527 Wednesday, September 11, 2002 3:41 PM b i 35.8% ii 39.9% c No — more than 39% of the labour force are female. 13 a Male Female Total 487 174 Aust. Male Other Total d 1.5% 98.5% 98.3% 100% Female Total Education 47.5% 52.5% 100% Other 49.9% 50.1% 100% 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 Male Female Other b 184 287 355 776 4 087 764 3 008 492 Male Education 2156 4217 6127 6624 12 751 b No — 33% of the males in the community were born overseas, while 49% of the people born overseas were male. 100% Male Education 2061 Total 1.7% 8534 Female 355 776 1 a Yes b 3 c Yes, both equal 3 d 3 2 a No b 5–9 and 20–24 c No d 5–9 and 20–24 e 25–29 3 a b Yes 540 063 Other 4 272 051 3 364 268 7 636 319 0 1 2 3 4 5 Number of goals d Yes, both equal 2.5 b 2 25 20 15 10 5 0 c 5 b 0.73 0 0 0 0 0 –2 –3 –4 –5 –6 11 21 31 41 51 4 087 764 3 008 492 7 096 256 Total 6 5 4 3 2 1 0 c 0 and 5 e 0 4 a 2 5 a Total 184 287 Exercise 4E — Applications of statistics and probability Frequency Education 4468 Overseas Female 4066 Frequency c Number of goals Male Education Other Total d Female 4.3% 10.6% 95.7% Frequency c 6 A 7 a 89.4% 100% b No 10 8 6 4 2 0 50 00 50 00 50 1– 1–1 1–1 1–2 1–2 5 10 15 20 100% Number of people Male Female Total Education 34.1% 65.9% 100% Other 57.6% 42.4% 100% 151–200 and 201–250 0.67 0.17 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. 4B ➔ e 10.6% f 65.9% g Same as g in question 10. Comments should note increases in percentage and the reasons for this. c d e 8 a 4E
  14. 14. answers 528 Answers 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 4 5 6 7 8 9 2 Line of best fit 10 12 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. a Systematic b Random c Stratified Check with your teacher. Year 7 — 12, Year 8 — 12, Year 9 — 11, Year 10 — 10, Year 11 — 8, Year 12 — 7 2000 750 a Barry — 2667 Viet — 1667 Mustafa — 1571 b 1968 B 11 1984 Check with your teacher. 13 Test results Accurate Not accurate Total With virus 48 2 50 Without virus 149 1 150 Total 197 3 200 Accurate Not accurate Total Telling truth 77 3 80 Telling lies 3 Prediction 17 3 20 14 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) = 1 3 P(winning if change mind) = 2 -3 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 3 5 7 9 10 23.3 2 21.5 16 4 29 5 6 7.93 No 8 Yes, 45 is an outlier. Median, because the outlier inflates the mean. 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. Test results Total 15 16 17 18 19 a a A a a c 20 B 21 a 140 130 94 b 30 b 33.8% 6 c 90% c 97.5% 200 b 96% c 34 9.7% b 8.0% No significant difference Attitude Primary 100 d 10% -d 93 1 % e 93% 3 Secondary Fewer 7.5% 4.3% Same 43.3% 19.1% More 49.2% 76.6% Total 100% 100% 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
  15. 15. 529 Answers Frequency 23 a 12 10 8 6 4 2 0 b No 1 2 3 4 5 6 Class centre c 0.15 24 a A variety of answers b 131 CHAPTER 5 Navigation Skills check 1 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 = 2π r opposite 6 40 030 km 7 Tangent = ------------------adjacent distance 8 Speed = -----------------time 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 geometry 1 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 Shanghai c 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 km c 6672 km d 7339 km 7 a 7784 km b 6450 km 8 4226 km Exercise 5B — Accurate position description 1 a 27°9.6′S, 153°36′E c 27°S, 153°36′E e 27°1.1′S, 153°33.6′E 2 Sketch 3 a Mt Sydney c Pinnacle Point 4 a 20°2.2′S, 148°52.7′E c 20°4.8′S, 148°52.2′E e 20°10.5′S, 148°55′E b 27°S, 153°45.9′E d 27°0.9′S, 153°37.6′E f 27°8′S, 153°44.5′E b Black Island b 20°4.3′S, 148°58.3′E d 20°10′S, 148°53.6′E Exercise 5C — The nautical mile and the knot b 150′ c 1422′ b 39°8.7′ b 57.3′ d 2871.7′ 10 Quick Questions 1 1 Latitude 2 Latitude 4 1852 metres 5 150 n mile distance 6 Speed = -----------------time 8 6 knots 9 5400 n mile 3 60′ 7 The knot 10 5 pm Investigation — Distance to the horizon 1 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 km 5 As height increases, distance also increases. (On a flat Earth, distance to horizon would be greater.) Exercise 5D — Using the compass 1 a 128°C 2 291°T 4 a 120°C b 292°C b 226°C c 193°C 3 6°E c 4°W d 40°C d 257°29′T Exercise 5E — Compass bearings and reverse bearings 1 a 50°T b 300°T c 230°T d 145°T 2 a 230°T b 120°T c 50°T d 325°T 5 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 mile 6 b 12 knots 7 a 187°T b 176°C c 50 min d 356°C 8 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°C c 7.2 n miles d 54 minutes 11 a 304°(C) , 1.8 n mile b 271°(C), 8.2 n mile c 328°(C), 7 n mile d 296°(C), 11.5 n mile 5A ➔ 1 a 120′ 2 a 9°43′ 3 a 17°17′ 4 a J, D b A, H c H, I d 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 mile f i 6600 n mile ii 6600 n mile g i 3600 n mile ii 3600 n mile iii 3000 n mile 5 a 1650 n mile b 3750 n mile c 7050 n mile d 1110 n mile 6 8 knots 7 a 3.5 knots b 6.5 km/h 8 a 3.85 knots b 12.6 knots c 289 n mile d 52.1 n mile e 30 hours f 10 minutes 9 a 7872 hours b 6.4 km/h, 3.4 knots 10 a 3600 n mile b 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 hours 11 a 570 n mile b 4.63 knots 12 3.08 am 13 A separation of 1′ near the equator on a line of latitude is greater than that further from the equator. answers Maths A Yr 12 - Answers Page 529 Wednesday, September 11, 2002 3:41 PM 5E
  16. 16. answers Maths A Yr 12 - Answers Page 530 Wednesday, September 11, 2002 3:41 PM 530 Answers 10 Quick Questions 2 1 3 5 9 Exercise 5K — Dead reckoning Compass 2 True north Magnetic variation 4 Compass True 6 60°T 7 158°C 8 180° Latitude 10 7 knots Exercise 5F — Fixing position 1 a b d 2 a b 3 a c 4 a 5 b e 6 b Check with your teacher. 155°T c 7.5 n mile 15 knots e 3.5 n mile Check with your teacher. 9.3 n mile c 18.6 knots 131°T, 18°T, 299°T b 198°T, 340°T, 265°T 11 knots d A 219°, B 293°, C 254° Tower 53°, Antenna 88° 14 n mile c 14 n mile d 19.8 n mile i 243° ii 252° 56 n mile c 250° Exercise 5G — Come to the rescue! 1 2 5 6 Man and Wife Rocks 108°T; Miall Island 220°T plot 3 9.4 n mile 4 28 minutes 3.48 pm Vessel has moved 1.75 n mile out to sea to 23°6.3′S, 150°59′E 7 2.5 n miles to the east 8 The wind had greater impact on pushing the boat than it did on the swimmer. i 6 n mile ii 5.6 n mile iii 5.3 n mile Plot d 20 knots B 250°, D 291°, E 316° 7.5 n mile Exercise 5I — Running fix 1 b 6 n mile 2 b 197° Exercise 5J — Doubling the angle on the bow 1 a b c d e f 2 a 3 b c 4 b d i 50° ii 130° iii 80° iv 3 n mile i 100° ii 40° iii 7 n mile i 42° ii 96° iii 84° iv 11 n mile i 130° ii 25° iii 6.5 n mile iv 45°T i 45° ii 90° iii 45° iv 10 n mile i 20° ii 140° iii 20° iv 8 n mile v 8 n mile 25° b 5 n mile At 1300, 30°; at 1330, 60° 9 n mile d 9 n mile 8 n mile c 6.2 n mile (from sketch) 7.03 am e 100° 10 Quick Questions 3 1 4 6 7 9 Two 2 Isosceles 5 Angle on the bow 9 knots 8 84 n miles 10 Cocked hat Front 4 hours One 2 a b c d 3 a c Check with your teacher. 20°08.3′S, 148°59.7′E Check with your teacher. iii 20°07.3′S, 149°01.5′E iii 20°06.3′S, 149°03.3′E iii 20°05.3′S, 149°04.9′E 20°07.3′S, 149°14′E 20°00.2′S, 149°07′E 20°05.2′S, 149°09.2′E 20°01.4′S, 149°07.3′E 3.8 n mile b 11.4 knots ii 20°05.4′S, 149°04.2′E ii 20°03.3′S, 149°01′E Exercise 5L — The lighthouse and navigation 1 a AB = 10.5 m b 1908 m c 1°48′ d 1.345 n mile 2 a 4 short flashes of light followed by a long period of darkness every 20 seconds c 3105 m 3 a 2 flashes, then darkness every 12 seconds c 5156 m f 7.6 n mile g i 0.48° ii 0.36° 4 b 6875 m d 0.68° Exercise 5M — Let’s go cruising Exercise 5H — Transit fix 1 b c 2 c 3 b 1 a b c d 3 Transit line 1 a 11°18′ east b 11° east 2 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′E e 27°34.8′S, 153°21.6′E 3 a Coochiemudlo Island b The Bluff c 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 mile 5 a iii Yellow light flashes every 2.5 seconds iii Red every 4 seconds iii 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°C c 33 minutes d 10.58 am e 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 navigation 1 a c 2 a 3 a 4 a 5 a b 26°15′S, 151°56′E b 26°40′S, 152°00′E 26°·17′S, 152°41′E d 26°33′S, 151°51′E Tansey b The Bluff c Abbeywood 1998 b 2457 c 2043 350°T, 339.5°C b 05°T, 355°C 149°30′C Barambah Ck, Clonya, Murgon, Nanango.
  17. 17. 531 Answers Chapter review 1 a A, 30°N, 60°W; B, 40°S, 20°W; C, 30°S, 50°E; D, 40°N, 60°W b NDS, NGS, NHS or any line of longitude (for example, 40°W) c F d PG, PN, PH, PC etc. 2 a 20°2.2′S, 148°52.7′E b 20°10′S, 148°53.7′E 3 a 5400 n mile b 10 800 n mile c 4200 n mile d 7920 n mile 4 a 1080 n mile b 5340 n mile 5 a 360′ b 1110′ c 1695′ d 3457.4′ 6 540 n mile, 1000 km 7 a 6 knots b 13.3 knots c 522 n mile d 198.3 n mile e 50 hours f 15 minutes 8 6 hours 24 minutes 9 a 400 n mile b 180°T c 5 hours d 6.45 pm 10 a 114°C b 253°C c 6°W d 206°T 11 286°T 12 b 9.8 n mile c 29.4 knots d 48°T 13 a = 55°, b = 125°, c = 70°, PR = 7.4 n mile 14 a 18° b 36° c 13 n mile d 13 n mile 15 b 50°, 100° c 12 n mile d 12 n mile 16 a 5.57 m b 65.8 m c 1.68° d 5810 m 17 a 3 short flashes then long period of darkness every 16 seconds b 130 m c 16 n mile d 2480 m 18 a 8.1 n mile b 8°T c Approx. 1 h 20 min trip, ETA 8.20 am 19 a 240°T b 16.5 n mile c Plot d 11.45 am CHAPTER 6 Land measurement Skills check b H A O O A H c H A O opposite adjacent 11 Sine = -------------------------Cosine = -------------------------hypotenuse hypotenuse opposite Tangent = ------------------adjacent 12 a 6.8 cm b 7.7 m c 65.0 m Exercise 6A — Perimeters and areas of triangles 1 a 1.73 cm b 23.1 m c 11.4 m 2 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 m c 42.4 km, 21.2 km 5 a 27.9 m2 b 250 m2 c 52.4 km2 Exercise 6B — Perimeters and areas of polygons 1 a 5797 m2 2 a 97.4 m2 3 Approx. 36 ha b 1062 m2 b 3195 m2 c 27 952 m2 Exercise 6C — Surveying on level ground without obstacles 1 a 32 m b 28 m c 59 m d 73 m e 47 m 2 a 86.3 m b 107.5 m c 47.4 m d 49.0 m 3 Sketch 4 a 120 m b 5 c i 48 m ii 39 m iii 37 m iv 32 m v 35 m d i 65.8 m ii 44.7 m iii 34.4 m iv 90.2 m e i 43.15° ii 1014 m2 iii 81.2° f Sketch g 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 1 1 5.47 m 3 5 7 8 9 10 2 1 -2 × a × b × sin C 4 S(S – a)(S – b)(S – c) 48.5 m , 196.6 m 24.2 m2 6 AB KF, JE, ID, HC 36 m, 59 m, 73 m, 32 m, 84 m 240 m2 a 51.2° b 59 m 2 2 5F ➔ 1 Millimetre, centimetre, metre, kilometre 2 Perimeter 3 a 24 cm b 30 m c 15.6 cm d 16.8 m e 12.6 m f 36.0 m g 38 m 4 a l2 b l×w c πr 2 -d b×h× 1 e l×h 2 5 a 1.5 cm b 0.18 m c 12 300 cm d 680 m e 12 500 m 6 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 0.3249 8 c2 = b2 + a2 9 a 5 cm b 8 cm c 9.6 m 10 a answers Maths A Yr 12 - Answers Page 531 Wednesday, September 11, 2002 3:41 PM 6C
  18. 18. answers Maths A Yr 12 - Answers Page 532 Wednesday, September 11, 2002 3:41 PM 532 Answers Exercise 6D — Surveying around obstacles 1 a 42 m b 31 m c 52 m 2 Sketch 3 a Sketch b The distance between the second and third staffs placed by Peter. Exercise 6E — Plane table surveying: intersection or triangulation 1 a b c d 2 a b c d 3 a 4 a b c d 61 m i 43 m ii 28 m iii 106 m iv 124 m i 065° ii 310° iii 180° iv 150° 301 m e 0.38 ha 59 m ii 28.5 m ii 31.5 m iii 32 m iv 73 m v 49 m 0.25 ha i 15° ii 81° iii 151° 83 m b $1328 c 450 m2 Sketch i 100 m ii 66 m iii 50 m iv 90 m v 86 m i 270° ii 310° iii 240° 4300 m2 Exercise 6F — Plane table surveying: radiation and traversing 1 a b c 2 a b 3 a c d 4 a c d i 25 m ii 35 m iii 30.5 m iv 51.5 m i 0° ii 070° iii 180° 1900 m2 ii 23 m ii 72 m iii 51 m iv 12.5 m v 104 m vi 109 m vii 75 m 0.31 ha Radiation b sketch A, 123°; B, 136°; C, 152°; D, 180°. 3160 m2 Sketch b traversing i 212° ii 270° i 107 m ii 77 m e 3800 m2 Offset and triangulation 41 m, 10 m Intersection (or triangulation), radiation, traversing Intersection Radiation Traversing 287.5 mm 4 triangles Area = S ( S – a ) ( S – b ) ( S – c ) = 310 m2 10 m2 Exercise 6G — Levelling: vertical measurements in relation to a datum 1 a i 50.00 m ii 51.69 m b 1.94 m c 53.63 m d Sta. BS A 3.63 B IS FS HI RL Dist. Notes 53.63 50.00 0.00 TBM 1.94 53.63 51.69 20.00 i 3.60 m ii 2.80 m iii 53.60 m iv 50.00 m 50.80 m 61.25 m b 61.25 m c 61.25 m 61.25 m e 60.00 m f 59.50 m 58.75 m h 58.25 m i 5.00 m 10.00 m k 15.00 m Exercise 6H — Topographic maps 1 Easting 84, northing 46 2 a Maculata Park b oval c building at quarry 3 a GR 871464 b GR 854487 c GR 813488 4 a 3350 m b 1250 m 5 a 352° b 090° 6 a Abattoirs, bridge over river on Warrego Highway, then along river and over slag heaps b 155°. Yes. A scale diagram could be sketched and trigonometry used to calculate angles. Exercise 6I — Contour maps 1 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 5 2 a 93 m b 68 m 3 a 20 m b 10.3 km c Sketch d 20 e 293° f No, not if X and Y are at the surface. 1 4 a --------b 16.4°, steep 3.41 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 plans 10 Quick Questions 2 1 2 3 4 5 6 7 8 9 10 2 a b 3 a d g j 1 a c e f g 2 a b d e 630 m2 b 23.0 × 27.499 m 632.477 m2 d 1 : 1500 Rectangle of length 60 mm and width 42 mm ii $57.88/m2 ii 850 m2 ii Lot 109 ii location, elevation, road frontage size, views 2100 m2, 83 perches 103.68 m2 c 56.3 m 1 0.049 or approx. ----20 i rising ii 1800 mm iii 1.375° Exercise 6K — Orienteering 1 a 8° b 137° c 222° 2 a 67 m b 136 m c 77 m 3 Any suitable set of 8 instructions. Chapter review d 45° d 130 m 1 a 126 m2 b 165 m2 c 516 m2 2 2 d 2325 m e 8850 m 2 0.2 ha 3 a ii 150 m ii 52 m iii 63.2 m iv 13 m v 75 m vi 141.9 m b ii 936 m2 ii 1533 m2 iii 4500 m2 iv 5912.5 m2 4 Sketch
  19. 19. 533 Answers 5 a 84 m b 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 m b 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 m b 50.90 c Sketch 8 a i Industrial Estate ii Finlay Island b 2.5 km c 153° 9 a 250 b 1 in 5 c 11.3°, steep to moderate 10 a 45° b 26.6° c 1.1° d 2.9° e 7.2° 11 10 12 a 90 m b 20 m c Sketch 13 a 630 m2 b Sketch c Re Re Pi Mo Ce VG Ma Ga 62 58 104 108 147 179 0 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 47 44 84 81 119 143 d 1 a ABDE 2 a 120 Ya U1 b 487 km c 254 km 160 118 Pi 47 0 25 45 34 75 100 44 25 0 40 59 97 99 84 45 40 0 20 58 59 VG b ABCE 0 Ce Exercise 7A — Networks, nodes and arcs Re Mo CHAPTER 7 Networks 81 34 60 20 0 38 66 Ma 119 72 97 58 38 0 28 Ga 143 104 99 59 28 28 0 L 2.40 4 a 2.40 Ya 85 U1 5 C History of mathematics Ba 75 34 An 82 iv Ulawatu Ulawatu 1.80 T ii 357 min iii 191 min 120 80 BR Yallingup Black Rock Angourie Bargara 0 120 100 209 254 Yallingup 120 0 220 118 160 Black Rock 100 220 0 109 154 Angourie 209 118 109 0 Bargara 254 160 154 45 1 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. 45 0 Exercise 7B — Minimal spanning trees 1 a b B 4 A A Ulawatu Yallingup Black Rock Angourie Bargara 0 85 75 157 191 85 0 160 80 114 157 80 82 0 34 191 114 116 34 0 c 12 A 17 B 3 a 185 km b ii 75 Pi ii 321 min iii 143 min Ma 34 47 38 45 Re 75 160 0 82 116 Bargara VG 28 25 Ga 20 C 15 2 a Sturt d 4 C D Ce 30 C B E b Rockdale 15 D 15 E c To Sturt Pallas 55 Sturt Urchin 6D ➔ 40 20 A 18 52 Mo d Xavier Yule 50 52 42 50 Zenith Walga 67 View Rockdale 50 64 59 D 5 D 5 C Yallingup Black Rock Angourie B 4 8 v Ulawatu c $3.80 B 1.50 109 d i b $3.30 1.80 K 2.00 45 An 1.50 P Ba 100 BR answers Maths A Yr 12 - Answers Page 533 Wednesday, September 11, 2002 3:41 PM 7B
  20. 20. answers Maths A Yr 12 - Answers Page 534 Wednesday, September 11, 2002 3:41 PM 534 Answers 10 Quick Questions 1 3 a 585 m b 245 m c Check with your teacher. 4 a b A E B E 45 B 45 D 48 48 F 2 9 4 5 AC, AD, DF, CF 7 D–C–E–F 6 $16 600 8 C E F D A B C E F A B A 18 9 $16 200 23 24 C c 45 18 D 20 C 1 6 3 1 A Exercise 7D — Network flow F d B C E D 17 6 4 G 40 5 a 6 A 1 a B 6 C F b G E 20 E A B 8 F 8 G I 5 5 J 8 5 5 a J 6 8 E 8 F 2 a 23 3 a 6 4 a i 250 c i 24 H E 5 G E R 5 8 5 5 A 5 50 12 3 D 5 8 C 8 17 C 15 U 12 O 10 Q d 10 B 10 100 T 5 M 15 D 15 N 20 B 12 200 S E 300 E c 12 c 250 R D 250 100 C A 50 200 D F 10 b B 15 A 13 100 A 7 23 23 20 20 C D E 50 F 10 B–A–C 2 H b 16 b 3 ii No ii Yes c 16 c 3 b i 150 ii Yes d i 15 ii No 6 7 8 9 10 1 a d 2 a 3 a d 4 a 20 45 165 km 37 72 b e b b e 30 B 50 38 28 202 km 90 30 D 25 30 70 A 25 74 139 202 km 32 44 3 B 2 E 4 D C 2 D b 3 E E 6 From To Flow capacity A B 4 A C 5 D 3 B E 3 B C 2 C 20 60 C b 80 min E c f c c f 5 D B Exercise 7C — Shortest paths 4 C A b $1.7 million b $1740 11 D B A A 53 km 54 km a 68 km a $215 B Flow capacity C K To A G From C D E 4 D C 2 D E 6
  21. 21. 535 Answers c From To Flow capacity A B 4 A C 7 A D 3 A E 5 B E 3 C E B E To B C 7 A D 12 A E 5 C F 7 D B 2 D E 6 D 6 a 10 7 a i F E 100 250 d 22 S D 250 200 N T 15 300 100 5 20 Q G 1 J F 15 E 14 45 40 D 35 30 A C 2 a 117 km 8 E 8 2 D 3 20 15 B b 160 km Exercise 8A — Critical path analysis 1 a b B D E H C c 64 km 45 km 30 80 72 C A D B E F J G H I 7C ➔ 2 4 6 8 10 G F A B A–D–F B–C–E–F 30 10 45 B 10 Skills check ii 25 10 Quick Questions 2 1 3 5 7 9 ii 29 30 6 F 13 10 50 5 8 4 CHAPTER 8 Critical path analysis and queuing R 3 H 3 ii 150 50 E 15 E 14 10 C U F 3 2 13 c 35 c 70 10 D E 12 10 10 D 8 12 O 4 G 16 5 M 8 B 6 10 6 E d i 13 12 A 100 20 A b 15 b 70 C 100 100 c i 117 km ADFG, 40 ACFIK, 70 a 20 a 70 ii 300 B C R c 18 50 200 b i 6 7 8 9 10 11 8 b 10 A E A B C E H I K D 4 F 8 4 7 C 4 A D 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–G 5 F G Flow capacity A 11 12 6 From b 10 B A 2 D 1 a 7 2 5 8 D d Chapter review answers Maths A Yr 12 - Answers Page 535 Wednesday, September 11, 2002 3:41 PM 8A
  22. 22. answers Maths A Yr 12 - Answers Page 536 Wednesday, September 11, 2002 3:41 PM 536 d Answers N O S R X B, 20 Q W U V Y 2 a 12 a A–D–G b Float (H) = 1, Float (J) = 3, Float (E) = 1, Float (B) = 10, Float (C) = 1, Float (F) = 1 c 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–K b M, L, H, C, E, B, D P T Z F, 5 G, 10 A, 2 H, 12 C, 4 D, 5 3 4 5 6 7 8 E, 3 b 49 minutes a B a 23 minutes A–C–F a B b D A–D–G a B, 15 0 A, 10 10 25 b D b B, C, E, F, G c A D, 8 22 C, 12 Exercise 8B — Critical path analysis with backward scanning E, 10 41 G, 6 35 H, 10 J, 11 52 L, 8 49 M, 7 K, 9 61 F, 25 1 Critical path = B–D; Float (E) = 1 min, Float (C) = 1 min, Float (A) = 1 min 2 Critical path = B–F–G; Float (C) = 3 h, Float (E) = 5 h, Float (A) = 3 h, Float (D) = 5 h 3 D 4 a 31 days b Critical path = A–C–E–G 5 D 6 C 7 a Activity Immediate b 61 minutes 9 10 11 letter Activity letter A B C D E F Activity letter A B C D E F G H J Immediate predecessor — — — A B, F C D E E Time 3 4 5 6 5 8 18 8 6 Activity letter A B C D E F G H J K L M Immediate predecessor — A A B C A D, E, F D, E, F G J G H, L Time 10 15 12 8 10 25 6 10 11 9 8 7 Time 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 Time 7 9 12 8 4 9 predecessor A Immediate predecessor — — A B B C, D Activity Final assembly D, E 18 b c 85 minutes D, 20 B, 35 A, 12 G, 18 E, 12 C, 16 F, 5 d B, 35 12 12 67 67 C, 16 G, 18 E, 12 28 55 8 9 10 11 A–B–D–G D, 20 A, 12 0 0 47 47 85 85 F, 5 C 28% a 29 b B–E–J–N a 35 b C–F–J–M–Q 4 11 E, 5 13 16 G, 9 B, 4 0 0 C, 10 X, 3 10 10 24 25 F, 9 A, 15 15 16 D, 3 H, 5 19 19 P, 10 J, 7 26 26 26 M, 4 26 K, 5 c 3 24 27 N, 6 L, 3 30 30 35 35 Q, 5
  23. 23. Answers 12 a 3 C 5 a B, 7 A, 4 C, 3 E, 5 F, 3 10 Quick Questions 1 1 C 2 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 6 A, 5 D, 8 F,2 B, 4 4 4 0 0 14 14 E, 5 6 Customer served Arrivals Length of queue People in queue 0 1 2 3 4 5 6 7 8 9 10 11 12 H, 5 b Float (D), Float (E) = 2, Float (G), Float (H) = 2 C, 6 A A A B B B C C C D D D E — — B — C — D — E — F — G 0 0 1 0 1 1 1 1 2 2 2 2 2 — — B — C C D D DE E EF EF FG b 2 a 3 min c c 4 min b Yes Time Customer served Arrivals Length of queue People in queue 0 3 6 9 12 15 18 A A B B C C D — B C D E F G 0 1 1 2 2 3 3 — B C CD DE DEF EFG 6 9 6 7 8 9 10 4 B Time J, 2 D, 7 G, 3 537 6 14 9 3 B, F, D Exercise 8C — Queues: one service point 1 a Time Customer served Arrivals Length of queue People in queue 8 9 10 11 12 13 14 15 16 D D E E F G G G H — I — J K — — L — 4 5 4 5 5 4 4 5 4 EFGH EFGHI FGHI FGHIJ GHIJK HIJK HIJK HIJKL IJKL b 5 min e 2.28 min 2 a c 9 min f 1.5 min d 2 min Time Customer served Arrivals Length of queue People in queue 10 11 12 13 14 15 16 D E F G G G H — — — — — — — 4 3 2 1 1 1 0 EFGH FGH GH H H H c 7 min d 1 min 7 a 2 min d 4 min g F i Time 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 b 3, 10 c 2 min e 3 f 1 h 3 min and 6 min Customer (ix) served Arrivals A A B B B C C C D D E E E E F F F G — B — CD — — EF — — — — — G — — — — — Length of queue People in queue 0 1 0 2 2 1 3 3 2 2 1 1 2 2 1 1 1 0 — B — CD CD D DEF DEF EF EF F F FG FG G G G — 8B ➔ b 3 min answers Maths A Yr 12 - Answers Page 537 Wednesday, September 11, 2002 3:41 PM 8C
  24. 24. 538 Answers 9 A B• C• D E F G H I J K L • 0 2 • • 4 1 3 • • 6 5 • 8 7 Customer 8 Customer answers Maths A Yr 12 - Answers Page 538 Wednesday, September 11, 2002 3:41 PM A• B • C • D • • E • F • G • H 0 2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15 7 A 9 • 0.00 300 2 min 3 min 5 F, G, H, I 2 4 6 8 10 B C C 1 A B D C, D 2 C D C D E E 1 C D F E, F 2 3.45 C D G E, F, G 3 4.00 E F G 1 4.30 Exercise 8D — Queues: multiple service points A 3.00 Greater than 1.5 minutes 1 min 5 min D, E, F 3 B 2.15 10 Quick Questions 2 A 2.00 Minutes Nurse 2 0.45 • 10 12 14 16 9 11 13 15 Nurse 1 1.30 • 10 17 min 1 a b 2 4 4 a b 5 a Arrivals E F H G, H 2 0 5.15 4 had been completely served. 2 c 9 d 7 Time Teller 1 Teller 2 Arrivals 9 G H J, K G, H, I 3 J I, J 2 G H K I, J, K 3 G H L I, J, K, L 4 8.00 Length of queue I H 7.30 e 3 F G 6.45 3 Yes E 6.00 234 s (3 mins 54 s) 3 I J K, L 2 Chapter review 3 10 G H 11 G H 12 G H 13 I J 14 I L J 1 5 C 2 D 15 K 16 K 6 A, 5 17 K 5 L O J, 4 13 E, 1 5 H, 3 9 C, 4 14 F, 1 20 K, 3 23 J, 4 16 The refit can be completed in 23 hours. 4 7 2 0 0 A, 5 5 5 D, 8 13 13 E, 1 G, 2 3 B, 5 5 7 3 c 5 min D, 8 G, 2 B, 5 3 L K, 3 C, 4 5 L 4 D F, 1 H, 3 G, 2 4 N E, 1 B, 5 4 M 3 C D, 8 A, 3 0 b 2 min 6 a 2 min b 5 min c People in Length of queue queue Time Minutes 1 3 5 7 9 8 B d 4 min 8 Activity letter Time C, 4 9 11 14 19 F, 1 20 20 K, 3 23 23 H, 3 J, 4 16 16 Earliest Earliest start finish Float Immediate time time time predecessor Service 1 Service 2 Arrivals 0 A B 1 C B 2 C D 3 C D 4 E D 5 E F d 2 min E People in Length of queue queue 5 0 5 0 — B 5 0 5 0 — C Time A 4 5 11 2 B D 8 5 13 0 A 2 E 1 13 19 5 D, G E, F 2 F 1 14 20 5 E E, F G 3 D, E F C, D, E 2 G 2 9 13 2 C 2 H 3 13 16 0 D J 4 16 20 0 H K 3 20 23 0 F, J F, G G
  25. 25. 539 Answers 9 14 a Time Customer Arrivals 0 A B a People in Length of queue queue Point Point 1 2 Time People in Length of queue queue Arrivals 1 A 2 B 3 C 4 C 0 B C, D, E C, D, E 3 A C F, G D, E, F, G 4 2 A D H E, F, G, H 4 3 1 A 1 B E D I, J F, G, H, I, J 5 K G, H, I, J 4 B 1 C, D C, D 2 E, F D, E, F 3 4 E F D, E, F 3 5 G H L I, J, L 3 G H M, N I, J, L, M, N 5 G I O J, L, M, N, O 5 5 D, E, F, G 4 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 Task G 7 10 C 6 6 b c d e F K G, H, I, J, K 5 b 1 min c (0 + 0 + 0 + 1 + 0 + 1 + 1 + 0 + 1 + 1 + 1 + 0 + 1) ÷ 13 = 0.54 min d (0 + 1 + 2 + 3 + 3 + 4 + 3 + 4) ÷ 8 = 2.5 min a 15 Moths Entry 5 min 3 min (2 + 1 + 3 + 1 + 1 + 2) ÷ 6 = 1.67 min (0 + 2 + 0 + 1 + 0 + 2 + 1 + 0 + 2 + 1) ÷ 10 = 0.9 min K J I H G F E D C B A • • • • • • • • • • Tour information Butterflies (Rainforest) Arachnids (Scorpions) • Butterflies (Temperate) Glowworms (Caves) Arachnids (Spiders) Exit b D, 2 A, 12 B, 13 C, 12 G, 6 E, 4 H, 6 I, 5 J, 12 F, 2 0 1 2 3 4 5 6 7 8 9 Time hours 11 Time Customer served 0 A — — — 1 A B B 1 2 B C C 1 c The room would quickly become crowded. d Five people per minute People in Length of Arrivals queue queue CHAPTER 9 Probability and the binomial distribution Skills check 1 a 6 b 2 c 52 d 36 3 B D C, D 2 2 a 1 ----12 4 C E D, E 2 d 1 ----36 e 1 ----54 5 C F D, E, F 3 6 D G E, F, G 3 7 D H E, F, G, H 4 I F, G, H, I 4 0.04 0.72 0.125 0.154 Less than Greater than 0.03 c 0.54 0.67 f 0.125 0.188 c 0.75 0.684 f 0.058 b Greater than or equal to E 3 a d 4 a d 5 a c b e b e 8 b 2 -3 c 5 -6 f 5 ----16 Exercise 9A — Compound events — independent events 1 1 -4 2 a 1 -2 b 1 -2 8D ➔ ----------a Inter-arrival time = 3600 s = 3.6 s (less than 1000 service time) b 3 13 a Inter-arrival time = 0.75 s b Number of service points = 40 12 answers Maths A Yr 12 - Answers Page 539 Wednesday, September 11, 2002 3:41 PM 9A
  26. 26. answers Maths A Yr 12 - Answers Page 540 Wednesday, September 11, 2002 3:41 PM 540 Answers 3 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)} 4 -P(9) = ----- = 1 36 9 6 -4 7 is mostly likely, P(7) = ----- = 1 6 36 5 a 0.2646 b 0.0204 c 0.1764 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)} a 1 -8 3 -8 b c 7 -8 d 17 P S T P(P − F) = 0.4 × 0.25 = 0.1 18 a P(no drug) = 9 = 0.35 14 ----40 P(small dose) = 16 ----40 = 0.40 P(large dose) = 3 -8 10 ----40 = 0.25 b, c, d 7 C 8 B 1 -6 B PB 0.4 x 0.3 = 0.12 F PF 0.4 x 0.25 = 0.10 L PL 0.4 x 0.45 = 0.18 B SB 0.25 x 0.3 = 0.075 F L SF 0.25 x 0.25 = 0.063 SL 0.25 x 0.45 = 0.112 B F TB 0.35 x 0.3 = 0.105 L TF 0.35 x 0.25 = 0.088 TL 0.35 x 0.45 = 0.157 R' N × × 1 -6 1 -6 = lia ustra k Yor New New Yor k A Australia down p yo u Tok Tokyo dow n up dow lia up Austra Australia down 0.7 × 0.45 × 0.5 = 0.16 R' LR' 0.25 x 0.20 = 0.05 e Although few patients were tested, it appears that a greater percentage (80%) of those given a large dose of the drug recovered, whereas a much smaller percentage (29%) of those not given the drug recovered. 20% of people tested were given a large dose of the drug and recovered, 15% of people tested were given a small 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 likely to recover if the drug is taken. 0.7 × 0.55 × 0.5 = 0.19 0.3 × 0.45 × 0.5 = 0.07 0.3 × 0.55 × 0.5 = 0.08 0.3 × 0.45 × 0.5 = 0.07 0.3 × 0.55 × 0.5 = 0.08 0.7 × 0.45 × 0.5 = 0.1575 11 0.3 × 0.55 × 0.5 = 0.0825 12 0.55 HH 0.55 × 0.55 = 0.3025 H H T 0.55 T 0.45 HT 0.55 × 0.45 = 0.2475 H 0.55 TH 0.45 × 0.55 = 0.2475 0.45 T 0.45 TT 0.45 × 0.45 = 0.2025 P(2 Tails) = 0.2025 1 H P(H, H, H) = — 8 1 — T P(H, H, T) = 8 1 H P(H, T, H) = — 8 H H T 1 — T P(H, T, T) = 8 1 H P(T, H, H) = — 8 1 8 T P(T, H, T) = — 1 H P(T, T, H) = — 8 H T T T a There are 8 paths. 14 D 15 0.2 B W M 0.6 B' 0.8 B 0.2 1 a, d, e, g 1 4 2 a ----- = ----13 52 16 ----45 16 + 76 ----------------178 8 WB 0.6 x 0.2 = 0.12 WB' 0.6 x 0.8 = 0.48 MB 0.4 x 0.2 = 0.08 4 5 D 6 a 7 a 8 9 10 11 a a a a b = 7 -9 c 0.8 = 1 -3 0.258 0.037 0.32 0.4999 = 72 -------148 = 18 ----37 24 ----36 b d 19 ----45 e 1 b = 19 ----37 2 -3 c 15 ----36 = 5 ----12 b 0.449 c 0.865 b 0.296 c 0.667 d 0.333 b 0.46 c 0.31 b 0.9997 c 649 773 Exercise 9C — Compound events — Venn diagrams 1 a 4 2 a S b 5 MB' 0.4 x 0.8 = 0.32 b 0.32 = 32% 29 ----45 15 ----52 = 0.517 59 + 13 ----------------148 12 ----36 35 ----45 c 6 ----13 b 7 o'clock 11 o'clock 31 B' 16 b, d, e 1 P(T, T, T) = — 0.4 a 0.12 = 12% Exercise 9B — Compound events — mutually exclusive events 3 a 13 SR' 0.40 x 0.625 = 0.25 LR 0.25 x 0.80 = 0.20 R L 0.7 × 0.55 × 0.5 = 0.19 Australia up Aus trali a do wn n R' 0.7 × 0.45 × 0.5 = 0.16 up Australia Australia down Tokyo up Tok yo d ow n up NR' 0.35 x 0.714 = 0.25 SR 0.40 x 0.375 = 0.15 R S 1 -------216 10 NR 0.35 x 0.286 = 0.10 R 80 55 34 55 -------200 c 8 = 0.275 b 0.4
  27. 27. 3 IPT TS 31 0 b (0.75)6 b 0.0413 11 a 0.25 12 a 0.2090 13 0.1342 S 16 541 c 0.0330 29 Career profile — Gail Twemlow The events are mutually exclusive and the Venn diagram could have been drawn as two circles which did not overlap. 4 S 0.18 S A 0.51 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. answers Answers 0.08 0.18 0.23 Investigation — Pascal’s triangle 5 1 0.9009 S A 1 1 4 7 14 6 3 2 0 C 9 C 10 3 9 8 5 4 8 N 10 Quick Questions 1 1 1 -6 2 4 0.16 or 16% 7 10 1 -6 3 0.36 or 36% 1 ----26 5 0.48 or 48% 6 8 4 7 ----13 1 ----20 9 14 Exercise 9D — The binomial distribution using Pascal’s triangle 1 7 21 1 9 36 1 5 10 1 7 21 1 b 0.0256 0.0132 0.0179 0.2344 0.2322 0.4019 0.5 0.5 35 84 10 35 4 4 6 1 4 35 21 7 1 126 84 36 9 1 5 1 35 21 7 1 c 70 d 1 e 110 b 0.1176 c 0.125 e 0.0720 f 0.0156 b 0.0284 c 0.3456 b 0.3125 b 0.8936 b 0.8038 b 0.2734 b (0.5)10 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 = 2row number (2 to the power of the row number) 7 Yes. 113 = 1331 Yes. 114 = 14 641 8 They are square numbers. 9 Fibonacci’s sequence 1 12 35 1 1 8 13 1 2 1 21 34 1 3 3 1 55 1 4 6 4 1 89 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 10 Use the diagonal 1 1 + 3 = 4 = 22 3 3 + 6 = 9 = 32 6 6 + 10 = 16 = 42 10 15 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. 9B ➔ 1 a b 2 a b 3 a 4 a d 5 a 6 a 7 a 8 a 9 a 10 a 1 3 1 B S 2 3 Row 0 Row 1 Row 2 Row 3 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 6 C b 63% 7 a 37% 8 0.27, much higher probabilities of winning with roulette. 9 36 10 1 1 0.0291 0.0009 0.0691 A 1 B 0.9009 9D
  28. 28. answers Maths A Yr 12 - Answers Page 542 Wednesday, September 11, 2002 3:41 PM 542 13 Answers Row (r) 0 1 2 3 4 5 6 Sum (s) 1 3 7 15 31 63 127 s = 2r + 1 − 1 14 a Multiples of 2 9 0.1275 1 10 b  --  3 1 10 a -3 c Approximately 0.2 10 Quick Questions 2 1 1 -4 2 1 4 6 4 1 4 4p3q 5 -6 1 -6 3 p= ,q= ,n=4 5 0.0154 8 0.0592 6 0.3602 or 36% 7 0.9408 9 0.0139 10 0.0026 Chapter review 1 D 4 a Sum 2 3 4 5 6 7 8 9 10 11 12 Prob. 1 ----36 2 ----36 3 ----36 4 ----36 5 ----36 6 ----36 5 ----36 4 ----36 b b Multiples of 3 5 a 6 9 10 12 13 14 2 C 1 -2 c 0.3975 a 0.5 D a 0.5625 D ----a i 25 49 b i 2 -9 b 1 -8 ii 16 years old A calculating machine The Puy de Dôme mountain For probability, permutations and combinations A brain tumour and stomach ulcer 3 ----36 2 ----36 1 ----36 8 1 -9 0.9999 c 32 ----36 c 0.843 75 20 ----49 ii 20 ----42 P(both A and B) = 0.25 A 1 2 3 4 5 d 7 0.4928 ----b 15 36 11 C b 0.375 20 ----42 C 3 -8 15 B 16 6 17 S History of mathematics — Blaise Pascal 3 0.05 B 0.25 0.25 18 D 19 a S Yes 18 No 12 13 Exercise 9E — Binomial probabilities through tables 1 2 3 4 5 6 a 0.0008 b 0.9527 c 0.5793 d 0.9907 a 0.8281 b 0.9389 c 0.9887 d 0.9532 a 0.5000 b 0.6964 c 0.4142 d 0.0000 0.8684 a 0.6550 b Between 0.6167 and 0.9666 a 0.6230 b 0.5881 c 0.5000 d 0.0781 e 0.0139 f 0.0034 7 a 0.0404 b 0.9536 c 0.9940 8 a 0.2 b i 0.2013 ii 0.3222 iii 0.1074 c The probability of this is very slight; there may be some error in the sampling technique. b 30 ----43 20 D 21 a Two outcomes, success or failure, same event repeated b Tossing a coin 6 times c Rolling a die and noting the upper-most face 22 a 0.402 b 0.032 23 0.016 24 a 0.39 b 0.39 c 0.78 25 a 0.8725 b 0.8042 26 a 0.2 b 0.3704 27 a 0.12 b 0.2517
  29. 29. Answers CHAPTER 10 The normal distribution and games of chance Skills check 1 2 3 4 5 Answers will vary. Check with your teacher. a 6.7 b 2.4 c 0.4 below the mean Distribution b 45 to 55 a the value of x is larger than 40 b the value of x is less than or equal to 40 c The value of x is larger than 20 and less than 30. Exercise 10A — z-scores 1 2 3 4 5 6 7 8 3 −2 a 0 b 1 c −2 a 10.5 b 13.7 c 16.9 −0.27 1.5 a −0.48 b 1.44 c 0.08 a 10.3 s b 10.58 s d 9.88 s e 10.251 s 9 a x = 19.55, sn = 1.76 e −1 e 0.9 d −2.24 e 2.8 c 10.37 s f 10.524 s b 1.68 $10 $30 19 60–<80 $70 15 80–<100 $90 6 10 Quick Questions 1 1 2 3 4 5 6 7 8 9 10 b x = 56, σn = 20.1 c i 0.30 ii 2.2 iii −2.0 B B C a x = 64.7, σn = 11.4 b Highest score z = 2.66, Lowest score z = −1.73 English 1, Mathematics 1.31, Biology 1.5, Computing studies −2, Visual arts 0.67, Music −0.8 Exercise 10B — Comparison of scores 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a 68% b 95% c 99.7% a 68% b 95% c 99.7% 95% 16% a 68% b 16% c 0.15% 21.1 and 33.9 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. B A 0.15% a 16% b 16% a 95% b 16% c 34% d 15.85% e 83.85% a 95 g to 105 g b 92.5 g to 107.5 g 163 cm − 181 cm Faulty, as the one chosen has a z-score greater than 3 2.6 kg − 5 kg Exercise 10D — Standard normal tables 1 2 3 4 5 6 7 8 9 10 a d g j a a a d a e a a a a b a d 0.8413 0.1587 0.6826 0.1359 0 b 0.5 b 0.8413 0.0013 0.6406 b 0.2812 f 0.0500 b 0.7486 b 75% i 0.9332 22.6% 0.7881 0.7881 b 0.9192 e 0.0446 h 0.9544 k 0.0215 −0.75 c −0.5 c b 0.9452 e 0.3413 0.8577 c 0.6188 g 0.1335 c 0.9082 c b 37 ii 0.8413 b 0.3446 e 0.5403 c f i l 0.9641 0.2417 0.9974 0.8664 −1 d 1.5 −1.3 d 1.2 c 0.5 f 0.8185 0.5 d 0.3594 0.0509 h 0.0554 0.0294 d 0.8665 0.2514 d 0.6568 c 46 iii 0.1210 c 0.1151 9E ➔ 1 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 C 4 D 5 Course A, z-score of −0.8 compared to −0.75 on course B 6 a Atlanta 0.44, Sydney 1 b In Atlanta because of the lower z-score 7 C 8 B 2 −2 −1.03 2.95 One standard deviation above the mean Two standard deviations below the mean 50 8 English 1.25, Maths 1.4 Maths Exercise 10C — Distribution of scores 8 $50 9 a Mathematics x = 59.5, sn = 17.9 Chemistry x = 59.6, sn = 16.8 b Mathematics 0.25, Chemistry 0.20. So Mathematics is the better result. 10 Kory is the better candidate with a z-score of 1.5 compared with 0.875 for Ricardo. 2 40–<60 15 Frequency 20–<40 11 12 13 14 Class centre 0–<20 10 a Amount ($) d 3 d 7.3 543 answers Maths A Yr 12 - Answers Page 543 Wednesday, September 11, 2002 3:41 PM 10D
  30. 30. answers Maths A Yr 12 - Answers Page 544 Wednesday, September 11, 2002 3:41 PM 544 11 12 Answers 4.75% 0.38%, assuming weights to be normally distributed. Exercise 10E — Odds 1 a e 2 a e 3 a $105 $15 $140 $60 b f b f 4 -5 2 -7 d 4 a e 5 a d 6 a d 7 a 8 a $105 $33.33 $175 $83.33 -b 1 4 c g c g $429 d $300 $66.66 $507 d $420 $216.66 -c 3 5 3 ----10 e 2 -3 f 2:1 b 4:1 c 3:2 on 7:5 f 2:1 on g 6:4 Evens b 5:1 c 12:1 e 3:1 2:1 b 3:1 c 7:5 e 2:1 on f $36 000 b i $13 500 ii $160 b $71.11 c d 5:2 h 11:9 3:1 Evens 5:2 on $22 500 $80 Exercise 10F — Two-up 1 -4 2 1 -4 3 1 -2 4 1 -2 5 1 -8 6 1 ----16 7 a TH, TH, TH, TH 1 8 ----32 1 -8 b 1 ----16 c b 1 ----32 d 1 ----64 1 ----16 e 0.48 Exercise 10G — Roulette 18 ----37 b 6 ----37 2 a 19:18 b 31:6 c No, slightly lower 3 a $60 b 0 c $200 d $75 4 Nothing happens except when the ball lands on 0, then he loses both bets. Exercise 10H — Common fallacies in probability 1 a 1/16 b 1/32 c 1/2 2 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? 3.6 15.6 −$2.50 −13.5 cents 7 2 4 6 8 3.1 −40 cents $0.62 $1.60 10 Quick Questions 2 1 3 5 7 9 Very probable 970 mL 0.0495 0.9010 $500 Chapter review 1 2 3 4 5 7 8 10 Not quite, probability of winning = 0.48 1 a 1 3 5 7 9 6 1 9 a Exercise 10I — Mathematical expectation 9 10 11 12 13 14 15 16 17 18 19 20 21 22 2 4 6 8 10 2.5% 0.9505 0.0495 2 -5 $1 loss −2 a 0 b 1 c −2 d 3 e −1 1.87 a 0.17 b 1.83 c −3 d −1.75 e −2 a x = 1130, σn = 334.2 b i −0.39 ii 2.05 iii −2.62 iv −1.13 v 3.07 a x = 20.1, σn = 2.1 b Highest = 1.86, Lowest = −1.95 a 1.5 b 1 c Physics, higher z-score a Geography: −0.8, Business studies: −0.53 b Business studies: higher z-score Numeracy: lower z-score a 68% b 95% c 99.7% a 68% b 95% c 99.7% a 34% b 47.5% c 2.5% d 0.15% e 97.35% Faulty, as it is more than three standard deviations from the mean. a 0.9032 b 0.9918 c 0.1587 d 0.0668 e 0.2075 a 0 b 1 c −0.83 d 5.33 a 0.5 b 0.8413 c 0.1587 d 0.0485 e 0.1293 f 0.3779 a 0.5 b 0.8413 c 0.7357 d 0.2342 a Horses 5 and 6 b Horses 1, 3 and 5 Test this by simulation on a computer. a Same b Same a i 0.49 ii 0.25 iii 0.25 b No a $1.67 b About $2.65 (if the operator’s percentage is based on this figure)

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