Two pages of formulas will be provided with your HSC exam, listing the following: Area of an annulus: A = π(r 2 r 2 )

Transcription

1 !NNC Maths Year 12 Ch 11 Page 413 Wednesday, October 4, :33 AM 11 Revision Congratulations. If you are reading this page, then you have probably finished the General Mathematics course and are now preparing for your final exams. This chapter summarises and revises all of the topics covered in Years 11 and 12. It will concentrate mainly on work from the HSC Course, but content from the Preliminary Course will also be included because up to 30% of the HSC exam may test Preliminary work. Remember the four stages in studying Maths: 1. Practise your Maths: do the exercises to strengthen your understanding. 2. Rewrite your Maths: revise your topic summaries. 3. Attack your Maths: work on your weak areas. 4. Test your Maths: practise on past exams and mixed review assignments. 1 A Practice HSC Exam (of 2 -- hours duration) is provided at the end of this book in addition 2 to the four Practice Papers (mini-exams of 1 hour s duration) after chapters 3, 6, 9 and 11. This chapter revises the following topics: Measurement Area and volume The sine and cosine rules Geometry of the Earth Financial mathematics Credit and loans Long term investing FORMULAS LIST Algebraic modelling Equations and functions Functions and graphs Data analysis Statistical distributions The normal distribution and correlation Probability Probability Two pages of formulas will be provided with your HSC exam, listing the following: Area of an annulus: A = π(r 2 r 2 ) Area of an ellipse: A = πab θ Area of a sector: A = πr 2 θ Arc length of a circle: l = πr Surface area of a sphere: A = 4πr 2 h Simpson s rule: A -- (d f + 4d m + d l ) 3 1 Volume of a cone: V = -- πr 2 h Volume of a cylinder: V = πr 2 h Volume of a pyramid: V = -- Ah Volume of a sphere: V = -- πr a b c 1 Sine rule: = = Area of a triangle: A = -- ab sin C sin A sin B sin C 2 Cosine rule: c 2 = a 2 + b 2 a 2ab cos C or cos C = 2 + b 2 c ab REVISION 413

2 !NNC Maths Year 12 Ch 11 Page 414 Wednesday, October 4, :33 AM vertical change in position Gradient of a straight line: m = horizontal change in position Gradient-intercept form of a straight line (equation of a straight line): y = mx + b Simple interest: I = Prn Compound interest: A = P(1 + r) n (A = final balance) ( 1 + r) Future value of an annuity: A = M n r ( 1 + r) Present value of an annuity: N = M n 1 A or N = r( 1 + r) n ( 1 + r) n Straight line depreciation: S = V 0 Dn Declining balance depreciation: S = V 0 (1 r) n Σx Mean of a distribution: x = or x = n x x z-score formula: z = s Probability of an event: P(event) = Σfx Σf number of favourable outcomes total number of outcomes AREA AND VOLUME MEASUREMENT Preliminary Course Metric units of measurement Error in measurement Significant figures, scientific notation Repeated percentage changes Areas of plane figures Sketching solid shapes and their nets Volumes of solids: V = Ah; V = πr 2 h; 1 1 V = Ah; V = πr h; V = -- πr Surface area Capacity and volume: 1 cm 3 = 1 ml; 1 m 3 = 1000 L Ratios: unitary method and dividing in a given ratio Similar figures and scale factors Scale drawings and building plans HSC Course Composite areas Field diagrams θ Arc length of a circle: l = πr 360 Area of a sector and annulus: θ A = πr 2 ; A = π(r 2 r 2 ) 360 Area of an ellipse: A = πab h Simpson s rule: A -- (d f + 4d m + d l ) 3 Surface area of a cylinder (open and closed) Surface area of a sphere: A = 4πr 2 Volumes of composite solids Errors in area and volume calculations Exercise 11-01: Measurement 1. Convert: (a) 3640 L to kl (b) 742 kg to g (c) 3.5 hours to minutes (d) m to mm 2. Find (i) the absolute error and (ii) the percentage error of each measurement. (a) 8.5 m (b) 15 g (c) 2.35 ml (d) 2000 t (to the nearest 10 t) 414 NEW CENTURY MATHS GENERAL: HSC

3 !NNC Maths Year 12 Ch 11 Page 415 Wednesday, October 4, :33 AM 3. Find the areas, correct to 2 significant figures, of the following shapes. (a) (b) 5 m 15 cm 3 m 22 cm 8 m 4. Calculate the volume of each solid correct to 2 significant figures. (a) 13 cm (b) 30 cm 30 cm 30 cm 9 cm 2 m 1 m (c) 2.7 m 0.8 m 1.5 m 5 m 5. An offset survey of a park was conducted and the notebook entry is shown here. All measurements are in metres. (a) Draw a neat sketch of the park. (b) Find the area of the park. (c) Find the perimeter of the park to 2 significant figures. K 52 P R 47 A 6. A pizza with diameter 20 cm is cut into 8 equal slices. (a) Find the length (PQ) of the crust of 1 slice correct to 1 decimal place. (b) Calculate the area of the slice correct to 2 significant figures. R P Q 7. What is the surface area of a can of peaches with the top removed? Answer correct to 2 significant figures. 14 cm PEACHES 12 cm 8. Use two applications of Simpson s rule to find the area of the land bounded by William Creek and the Oodnadatta Track. Answer to the nearest square metre. William Creek 9.2 km 7.5 km 8.6 km 7.7 km 6.8 km 8 km 8 km 8 km 8 km Oodnadatta Track REVISION 415

4 !NNC Maths Year 12 Ch 11 Page 416 Wednesday, October 4, :33 AM 9. Find the surface areas of these closed solids. (a) (b) 300 mm 500 mm 1400 mm (c) 1.2 m 50 mm 200 mm 350 mm 50 mm 8.4 m 3000 mm 10. The plan shows a three-bedroom house. Measurements are all in millimetres N Kitchen Living Rumpus Laundry Proposed deck Bed 3 Bed 1 Bed 2 (a) What is the area of bedroom 1 in square metres (correct to 2 decimal places)? (b) How many internal hinged doors are there in the house? (c) What is the area of the proposed deck (to the nearest square metre)? (d) The bathroom, toilet and laundry are to be tiled. How many square metres of tiles are needed? Answer to the nearest square metre. 11. A cube has a measured side of 4.2 m. (a) What is the absolute error of this measured value? (b) Between what two values could this measurement lie? (c) What is the largest possible volume of this cube? (d) What is the smallest possible surface area of the cube? 12. The radius of the Earth is approximately 6400 km. What is its surface area? Answer in scientific notation correct to 4 significant figures. 13. A lotto win of $5 million was divided between 4 people in the ratio 1 :3:2:4. What is the value of the largest share? WC Linen Bath NEW CENTURY MATHS GENERAL: HSC

5 !NNC Maths Year 12 Ch 11 Page 417 Wednesday, October 4, :33 AM 14. At 4 pm, a tower 25 m high casts a 35 m shadow while a tree casts a shadow 12 m long. (a) What is the height of the tree (correct to 2 significant figures)? (b) If the diagrams were drawn to a scale of 1 : 50, what length would represent the tower s shadow? (c) At 9:30 am, the shadow of the tower is only 20 m. What is the length of the tree s shadow at this time (to 2 significant figures)? 25 m 35 m 12 m THE SINE AND COSINE RULES MEASUREMENT Preliminary Course Pythagoras theorem Right-angled triangle trigonometry: the sine, cosine and tangent ratios Degrees and minutes Finding missing sides and angles in right-angled triangles Angles of elevation and depression Exercise 11-02: The sine and cosine rules HSC Course Problems involving right-angled triangle trigonometry Problems involving maps and bearings Trigonometry with obtuse angles a b c The sine rule: = = sin A sin B sin C The cosine rule: c 2 = a 2 + b 2 2ab cos C or a cos C = 2 + b 2 c ab Finding missing sides and angles in nonright-angled triangles Problems involving the sine and cosine rules 1 Area of a triangle: A = -- ab sin C 2 Surveying: offset survey (field diagrams), radial (plane table and compass) surveys 1. Test whether this triangle is right-angled cm 2 cm 9.9 cm 2. A TV screen has a height of 30.4 cm and a diagonal of length 51 cm. Calculate its width correct to 1 decimal place. 51 cm 30.4 cm REVISION 417

6 !NNC Maths Year 12 Ch 11 Page 418 Wednesday, October 4, :33 AM 3. A stairwell is inclined at 34 to the ground floor and has a horizontal length of 4.5 m. Graham wants to place a bookcase of length 1.45 m underneath the stairwell. What is the height h of the tallest bookcase that can fit underneath the stairwell? Write your answer correct to 2 decimal places. 34 h 4.5 m 1.45 m 4. Petersville (P) is 80 km due east of Queensfield (Q). Rossmore (R) is on a bearing of 055 from Queensfield and 300 from Petersville. Which diagram correctly illustrates this? A. R B. R C. R Q km P Q km P 5. A hockey player is standing directly in front of the centre of the goal. He is told that he can score as long as he keeps within a 48 angle. What is his distance, d, from the goal line (correct to 2 decimal places)? Q km 48 P d 3.6 m 6. A vertical pole has a height of 3.5 m. Find the length of its shadow correct to 1 decimal place, when the angle of elevation of the sun is Use the sine rule to find r in the diagram, to the nearest 0.1 m r 8. Jesse walked 3.3 km due south of camp, then turned and walked on a bearing of 129 until he was 5.5 km from camp. Calculate the bearing of Jesse from camp, to the nearest degree m N Camp 3.3 km 5.5 km 129 Jesse 9. In the following questions, find θ (to the nearest degree) if it is obtuse. 2 (a) sin θ = (b) cos θ = -- (c) tan θ = NEW CENTURY MATHS GENERAL: HSC

7 !NNC Maths Year 12 Ch 11 Page 419 Wednesday, October 4, :33 AM 10. A pendulum of length 38 cm swings a horizontal distance of 24 cm. (a) Use the cosine rule to calculate the angle θ swept by the pendulum, to the nearest minute. 38 cm (b) Hence use the arc length formula θ l = πr 360 to calculate the length of the arc traced by the pendulum correct to 1 decimal place. 11. Emad surveyed a block of land and drew the notebook diagram shown. Measurements are in metres. (a) Draw a neat sketch of the block. P 25 (b) Calculate the perimeter of the block, to the nearest metre. (c) Calculate the area of the block. 12. Find the acute angle A that makes each statement true. (a) sin 163 = sin A (b) tan 142 = tan A 13. A rhombus has sides of length 15 cm. Two adjacent sides make an angle 50. Use the 1 area formula A = -- ab sin C to calculate the area of the rhombus to 2 significant figures. 2 B 14. This is a radial survey of the field ABCD. (a) What is the size of DOC? 35 m (b) Hence calculate the length of DC (to the nearest metre) using the cosine rule. 38 m C A 72 O θ l 24 cm 34 m A K 47 m 50 R 15. Two ships sail from the same port. One sails 20 M on a bearing of 140 while the other sails 24 M on a bearing of 200. Calculate the distance between the ships, to the nearest 0.1 M. 16. At point A, Bethany observes the top of a control tower at an angle of elevation of 35. She walks to B, 100 m closer to the tower, where the angle of elevation is 64. (a) Explain why ATB = sin 35 (b) Hence show that TB = sin sin 35 sin 64 (c) Hence show that the height h of the tower is h = and evaluate h sin 29 to the nearest metre. T D h A m B C REVISION 419

8 !NNC Maths Year 12 Ch 11 Page 420 Wednesday, October 4, :33 AM GEOMETRY OF THE EARTH MEASUREMENT HSC Course Latitude and longitude θ Arc length of a circle: l = πr 360 Great circle distances Nautical miles and knots: 1 M = km; 1 knot = km/h Longitude and time differences International time zones Exercise 11-03: Geometry of the Earth 1. Match the following coordinates to the points illustrated on the diagram of the Earth. (a) (50 N, 0 ) (b) (15 N, 0 ) (c) (35 S, 50 E) (d) (15 N, 50 E) (e) (35 S, 0 ) (f) (50 N, 50 E) Greenwich meridian R P Q 50 S 50 N 15 N T U 35 S 2. Dallas, USA has coordinates (32 N, 96 W) and Havana, Cuba has coordinates (23 N, 82 W). (a) State which city is closer to: (i) the Equator (ii) the North Pole (iii) the Greenwich meridian (b) Calculate the difference in longitude between Dallas and Havana and state which city is further east. (c) San Diego, USA is 21 due west of Dallas. What are the coordinates of San Diego? 3. State whether the circles drawn through the following points are great or small circles. (a) A and C (b) B and D 40 W 10 W (c) A and B (d) C and D Equator A B C D 32 S θ 4. For the diagram in question 3, use the arc length formula l = πr to calculate the 360 following circle distances, given that the radius of the Earth is 6400 km and the radius of the 32 S parallel of latitude is 5400 km. Express all answers to the nearest kilometre. (a) AB (b) CD (c) AC (d) BD 5. Port Moresby, Papua New Guinea (9 S, 147 E) and Wagga Wagga, NSW (35 S, 147 E) lie on the same meridian of longitude. Calculate, to the nearest kilometre, the shortest distance between the two cities. 420 NEW CENTURY MATHS GENERAL: HSC

9 !NNC Maths Year 12 Ch 11 Page 421 Wednesday, October 4, :33 AM 6. In India, the city of Calcutta is 6 south and 11 east of New Delhi, which has coordinates (28 N, 77 E). What are the coordinates of Calcutta? 7. (a) Do Baghdad, Iran (33 N, 44 E) and Islamabad, Pakistan (33 N, 73 E) lie on the same line of latitude or longitude? (b) Is this line a great circle or small circle? (c) Calculate the distance between the two cities along this circle if the radius of the circle is 5340 km. (d) Calculate the time difference between these two cities in hours and minutes. (e) If it is 3:50 pm in Baghdad, what is the local time in Islamabad? 8. Calculate in nautical miles the great circle distance between Quito, Ecuador (0, 78 W) and Kampala, Uganda (0, 32 E). 9. A plane flies along the 18 N parallel of latitude. (a) How many nautical miles is it from the Equator? (b) Convert this distance to kilometres given that 1 M = km. 10. What distance (in kilometres) do you travel for each 1 along a great circle? 11. What is the local time in Darwin (12 S, 130 E) when it is 6 pm Tuesday in Greenwich? 12. A ship is sailing along the 120 E meridian. (a) How many nautical miles will it travel after covering 6 of latitude? 1 (b) What is its speed in knots if it takes 8 -- hours to travel this distance (correct to 3 1 decimal place)? 13. A ship s local time is 8:20 am when GMT is 1:00 pm. What is its longitude position? 14. (a) Which states and territories observe Australian Eastern Standard Time (AEST)? (b) When it is 9:30 pm AEST, what is the time in: (i) South Australia? (ii) Western Australia? (iii) Northern Territory? 15. A plane leaves Japan on Friday at 9 am and takes 6 hours to fly east to Hawaii. (a) When the plane crosses the International Date Line, what day does it become? (b) If Japan is GMT + 9 and Hawaii is GMT 10, what is the local time in Hawaii when the plane arrives there? EQUATIONS AND FUNCTIONS ALGEBRAIC MODELLING Preliminary Course Expanding algebraic expressions Solving linear equations, including those involving algebraic fractions The linear function y = mx + b The gradient and vertical intercept Independent and dependent variables Line of best fit HSC Course Algebraic terms Scientific notation Equations involving powers and roots Guess, check and refine method of solving equations, e.g. (1.05) x = 2 Substituting into formulas Equations and formulas Changing the subject of a formula Linear functions and linear models Intersection of lines, break-even points REVISION 421

10 !NNC Maths Year 12 Ch 11 Page 422 Wednesday, October 4, :33 AM Exercise 11-04: Equations and functions 1. Simplify these expressions. (a) 4xy 3y + 2y + xy (b) 3kt tk 8 (c) 2b 2 8b b 2 (d) 3p 2 6p (e) 18a 2 (f) 4p 2 7 p ab 2 2. Expand and simplify these expressions. (a) 8(c + 7) 3(c + 1) (b) 2(4b 2) + 3(b 4) (c) 5(w + 5) 2(2w 9) (d) 4h(h + 4) + 2(h + 10) 3. Express these numbers in scientific notation. (a) (b) (c) Express these numbers in normal form. (a) (b) (c) Evaluate these expressions, expressing the answers correct to 2 significant figures. (a) ( ) ( ) (b) (c) The velocity V m/s required for a rocket to escape the Earth s gravity is V = 2gr where g = 9.8 m/s 2 (gravitational acceleration) and r = m (radius of the Earth). Calculate the escape velocity of the rocket in m/s correct to 3 significant figures. 7. Solve these equations. 2x x 1 (a) 8(2r 5) = 12r + 10 (b) = (c) 5x = 324 (d) 3 6r = 3 8. Heron s formula for calculating the area of a triangle with side lengths a, b, c is 1 A = ss ( a) ( s b) ( s c) where s = -- (a + b + c) 2 (a) Use the formula to calculate the area of a triangle with sides 5 cm, 12 cm and 13 cm. 1 (b) Evaluate A if A = -- bh and b = 5, h = (c) What type of triangle is described in (a)? Justify your answer. 9. Make y the subject of the following equations. 12 p (a) 2x = 150 2y (b) = (c) 3x = 10y + 4 y The number of days, D, that fresh milk will keep if stored at a temperature of T C is 6 D = T + 1 (a) Calculate how many days fresh milk will keep if stored at 1 C. (b) If a carton of milk lasted 5 days, at what temperature was it stored? 11. Solve these equations (to 1 decimal place) using a guess, check and refine method. (a) (1.5) x = 25.6 (b) (0.73) n = 0.08 (c) 3 w = Washing removes 28% of a deep stain at each wash. After further washes, the percentage of the original stain that remains is given by the formula P = 100(0.72) w where w is the number of washes. (a) How much (to the nearest percentage) of the original stain remains after 3 washes? (b) After how many washes will there be less than 20% of the original stain? 422 NEW CENTURY MATHS GENERAL: HSC

11 !NNC Maths Year 12 Ch 11 Page 423 Wednesday, October 4, :33 AM 13. Find the equation of the linear function represented by this table of values. n d The maximum distance d m that a ball travels if thrown with velocity v m/s is v d = g where g = 9.8 m/s 2. At what velocity was a ball thrown if it travelled a distance of 45 m? 15. Thanh-Lee is comparing the plans of two mobile phone companies. OzExpress has a monthly access fee of $5 and charges $0.75 per minute, while No-Tel has a monthly access fee of $15 and charges $0.35 per minute. The monthly costs C of these plans can be represented by the following linear functions: OzExpress: C = 0.75t + 5 No-Tel: C = 0.35t + 15 (a) Graph both functions on the same axes for values of t from 0 to 60. (b) If Thanh-Lee usually makes 45 minutes of calls each month, which plan is better for her? (c) Use your graph to find the cost of making 45 minutes of calls under each plan. (d) Use your graph to find the duration of calls where both plans charge the same. (e) To what type of caller would you recommend: (i) the OzExpress plan? (ii) the No-Tel plan? FUNCTIONS AND GRAPHS ALGEBRAIC MODELLING Preliminary Course Rates, concentrations and converting rates Direct linear variation y = kx and its graph Conversion graphs, stepwise and piecewise linear graphs HSC Course The linear function y = mx + b The quadratic function y = ax 2 + bx + c Maximum and minimum values of a quadratic function The cubic function y = ax 3 The exponential function y = b(a x ) Exponential growth and decay a The hyperbolic function y = -- x Applications of functions (algebraic modelling) Direct variation Inverse variation Exercise 11-05: Functions and graphs 1. Classify each of these functions as being linear (L), quadratic (Q), cubic (C), exponential (E) or hyperbolic (H). (a) y = 4x 3 (b) R = 7(2.1) t (c) R = 2s + 1 (d) y = (x 4) 2 x (e) H = (f) T = n REVISION 423

12 !NNC Maths Year 12 Ch 11 Page 424 Wednesday, October 4, :33 AM 2. (a) What type of function is y = 2x 2 + 2? (b) Graph y = 2x 2 + 2, showing the x- and y-intercepts and the vertex. (c) What is the maximum value of y = 2x 2 + 2? 3. The function M = 1.5(1.2) x describes the mass M kg of a baby orang-utan at age x months, for values of x from 0 to 6. (a) Calculate the mass of the orang-utan at: (i) 3 months (ii) birth (b) By a guess, check and refine method, find when the orang-utan will have a weight of 3.5 kg. Answer correct to 1 decimal place. (c) Why do you think this function is not a good model for values of x greater than 6? 1 4. (a) What type of function is y = x ? (b) Graph y = -- x (c) What is the value of y when x = 3? 5. At the 2000 Olympic Games, Ian Thorpe swam 400 m in 3 min 41 s. (a) Calculate his average speed in metres per second, correct to 2 decimal places. (b) How far would he travel (to the nearest metre) if he maintained this speed for 5 min? y 6. What could be the equation of this graph? A. y = x B. y = (x 2) 2 C. y = x 2 2 D. y = x 2 2x x 7. Alan wants to drive from Sydney to Canberra, a distance of 288 km. His car s petrol consumption rate is 11.4 L/100 km and petrol costs 86 cents/l. Calculate, to the nearest cent, the cost of petrol for the journey. 8. A rectangular playground is to be made using 24 m of fence wire. (a) If the width of the playground is W m, find a Perimeter is 24 m. W formula for the length L in terms of W. (b) Hence show that the area, A, of the playground is given by A = 12W W 2 L. (c) Graph A = 12W W 2. Use it to find the maximum possible area of the playground. (d) What are the length, width and shape of the playground that give maximum area? (e) Explain why the formula A = 12W W 2 is only valid for values of W from 0 to (a) What type of function is y = 4 x? (b) Graph y = 4 x. (c) What is the value of y when x = 1.5? (a) What type of function is y = --? x 5 (b) Graph y = --, indicating any asymptotes. x (c) What is the value of y when x = 2.5? 11. Convert the speed of light, km/s: (a) to m/s (b) to km/h 12. (a) What is the y-intercept of all exponential curves of the form y = a x? (b) What is the x-intercept? 424 NEW CENTURY MATHS GENERAL: HSC

13 !NNC Maths Year 12 Ch 11 Page 425 Wednesday, October 4, :33 AM 13. This step graph shows the cost of sending parcels of different masses. Maree wants to send two parcels of weight 1.5 kg and 0.8 kg to Julian. How much would she save if she sent them together as one parcel rather than separately? Cost ($) Weight (kg) 14. The distance, D km, to the visible horizon is directly proportional to the square root of the observer s height, h m, above sea level. (a) Write a formula for D in terms of h and a constant k. (b) Find k to 4 decimal places if, from a height of 2 m, the distance to the visible horizon is 5 km. (c) Hence find the height at which the distance to the visible horizon is 40 km. 15. The intensity, I, of light under water drops to 40% of its previous value after each metre. This can be modelled by the function I = 100(0.6) x where x is the number of metres under cloudy water and I is the percentage of the original intensity of the light. (a) Complete a table of values for this function for values of x from 0 to 10, and draw a graph to represent this function. (b) Find what percentage of the original intensity exists under 3.5 m of cloudy water: (i) using the graph (ii) using the formula (correct to 2 decimal places) (c) Estimate the distance under water when the light intensity drops to half: (i) using the graph (ii) using the formula and a guess, check and refine method (to 1 decimal place) 0 CREDIT AND LOANS FINANCIAL MATHEMATICS Preliminary Course Wages, salaries and overtime Commission, piecework and royalties Bonuses and allowances Household bills and budgeting Income tax and GST HSC Course Flat rate loans Buying on terms Reducing balance loan tables Loan repayment tables Comparing loans Credit card payments REVISION 425

14 !NNC Maths Year 12 Ch 11 Page 426 Wednesday, October 4, :33 AM Exercise 11-06: Credit and loans 1. Use the tables below to answer the following questions. Taxable income Tax on this income $1 $6000 Nil $6001 $ cents for each $1 over $6000 $ $ $ cents for each $1 over $ $ $ $ cents for each $1 over $ $ and over $ cents for each $1 over $ Taxable income Medicare levy $1 $ Nil $ $ cents for each $1 over $ $ or over 1.5% of whole taxable income (a) Gayle had a taxable income of $ in the last financial year. (i) Calculate her income tax payable. (ii) What is her Medicare levy? (b) Jayden had a gross income of $ She made the following tax deductions: Work clothing $328 Car expenses $115 Self-education $84 Books $45 (i) What is her taxable income? (ii) How much tax must she pay? 2. Softa ice-cream company employs these people. Overtime is paid at time and a half. Weekly wages Name Pay rate, $P per hour Normal hours worked per week, N Overtime hours, A Gross weekly pay, $W Kelly $ $355 Sue $ A Drew $ B Kim $ C $ (a) What was Sue s gross weekly pay (A)? (b) What was Drew s gross weekly pay (B)? (c) How many overtime hours (C) did Kim work? 3. The graph shows the amount of GST payable in Canada for prices up to $100. (a) Use the graph to find the amount of GST (to the nearest 10 cents) payable on goods worth: (i) $48 (ii) $95 (iii) $32 (b) Use the graph to find the value of goods (to the nearest $) if GST is: (i) $3 (ii) $6.50 (iii) $4.20 GST of 7% ($) % GST in Canada Cost of item, p ($) 426 NEW CENTURY MATHS GENERAL: HSC

15 !NNC Maths Year 12 Ch 11 Page 427 Wednesday, October 4, :33 AM 4. The Ozcard monthly credit card statement is shown here for Mr T. Rex, who has an interest-free period of 45 days. OZCARD Mr T. Rex Statement for period 16/04/00 to 15/5/00 Opening balance Date Transactions DON DREW MENSWEAR PAYMENT RECEIVED THANK-YOU ZORBA S TAVERN HARDWARE HOUSE (a) How much was paid by Mr Rex and when did he pay this? (b) What was the total of his purchases during this statement period? (c) What is the due date for payment? (d) If Mr Rex does not make his payment until 15 June 2000, how much interest will he be charged on his purchases made during this statement period? 5. National Park workers are paid $19.80 per hour and get extra allowances for unfavourable conditions as shown in the table. Type of work Additional payment per hour Firefighting $19.80 Working at a height over 20 m $1.50 Dirty work $ (a) Paulo worked a 35-hour week with 6 -- hours of these spent firefighting and 2 hours 2 spent working at a height of 25 m. What was his weekly pay? (b) Tuncil worked a 40-hour week with 5 hours of these working at a height of 22 m and 2 hours doing dirty work. How much did he earn that week? 6. The table shows the monthly repayment necessary to pay off a loan of $1000 Monthly repayment to pay off a loan of $1000 Closing balance Account number Credit limit Available credit Annual % rate Monthly % rate % 1.875% Payment due date Past due Minimum payment due Date paid Amount paid 30/05/ Annual interest Term of loan rate (%) 1 year 2 years 3 years 4 years What is the monthly repayment if you obtain a loan of: (a) $1000 for 3 years at 13% p.a.? (b) $8600 for 4 years at 10% p.a.? (c) $3275 for 2 years at 12% p.a.? REVISION 427

16 !NNC Maths Year 12 Ch 11 Page 428 Wednesday, October 4, :33 AM 7. Employees received a bonus for exceeding a company s production targets. If 40% of $ profit was shared between the 720 employees, how much did each employee receive? Answer to the nearest dollar. 8. The graph shows the progress of a loan for fortnightly and monthly repayments. (a) What is the amount borrowed? (b) How long will it take to repay the loan making fortnightly repayments? (c) Describe in 1 2 sentences the benefits of fortnightly repayments versus monthly repayments. 9. The table shows the monthly repayment for various amounts borrowed over 25 years. Monthly repayment of loans over 25 years Amount of Interest rate p.a. loan ($) 8% 8.5% 9% 9.5% 10% Find (i) the total amount repaid and (ii) the total interest paid for a loan over 25 years of: (a) $ at 9% p.a. (b) $ at 8% p.a. (c) $ at 8.5% p.a. 10. Tammy buys a stereo for $1800 on interest-free terms over 2 years. If she pays 20% deposit, what is her monthly repayment? 11. Lola borrowed $ to purchase her first apartment. The bank offered to lend her the money at 8% p.a. with monthly repayments of $1400. (a) Copy and complete the table showing the progress of her loan for the first 6 months. No. of months, n Principal, $P Interest, $I $(P + I) $(P + I R) (b) How much had she paid off the principal after 6 repayments? (c) How much interest did she pay: (i) in the first 3 months? (ii) in the next 3 months? Loan amount ($) Fortnightly repayment Monthly repayment Years 428 NEW CENTURY MATHS GENERAL: HSC

17 !NNC Maths Year 12 Ch 11 Page 429 Wednesday, October 4, :33 AM 12. Simone borrowed $8200 to buy a jet ski. She took out a flat rate loan at 15% p.a. interest and repaid it in weekly instalments over 3 years. (a) What was her weekly instalment? (b) How much interest did she pay altogether? 13. Ozibank s fees and charges for home loans are shown in the table. Ozibank home loan fees and charges Application fee (variable rate) $800 Application fee (standard/combination) $500 Stamp duty 0.4% of amount borrowed and 3% of purchase price Loan maintenance fee (highest balance): up to $ $ $ $ $ $ $ over $ Charge per month: $8 $10 $13 $15 $18 Mr and Mrs Kettle obtained a variable rate home loan of $ from Ozibank. (a) What was their application fee for the loan? (b) How much stamp duty did they pay on the amount borrowed? (c) What is their loan maintenance fee for the first 3 months? 14. The table shows the progress of a loan with an annual repayment and interest compounded annually. Year Loan amount at start of year ($) Annual interest ($) Loan amount plus interest ($) Annual repayment ($) Balance at end of year ($) (a) What is the interest rate per annum? (b) How much will be owing at the beginning of the 4th year? (c) What will be the annual interest in the 4th year? (d) How much will have been paid off the principal at the end of 4 years? (e) How much will be owing at the beginning of the 5th year? 15. The Dragon Credit Union published the following table for flat rate loans. Carole borrowed $ over 5 years. (a) How much does she repay per month? (b) What is the total amount to repay the loan? (c) What is the interest charged? (d) Calculate the flat interest rate per annum. Years to repay loan Monthly repayment (per $1000) 1 $ $ $ $ $24.58 REVISION 429

18 !NNC Maths Year 12 Ch 11 Page 430 Wednesday, October 4, :33 AM LONG TERM INVESTING FINANCIAL MATHEMATICS Preliminary Course Simple interest: I = Prn Compound interest: A = P(1 + r) n ; I = A P Interest tables and graphs Account fees and charges Investing in shares Inflation and appreciation HSC Course Future value of an annuity: ( 1 + r) A = M n r Present value of an annuity: ( 1 + r) N = M n 1 A or N = r( 1 + r) n ( 1 + r) n Annuity tables Loan repayments Straight line method of depreciation: S = V 0 Dn Declining balance method of depreciation: S = V 0 (1 r) n Depreciation graphs and tables Calculating tax deductions Exercise 11-07: Long term investing 1. Find the simple interest earned on the following investments. (a) $ at 6% p.a. for 8 years (b) $8240 at 5.6% p.a. for 3 years (c) $6320 at 1% per month for 2 years 2. Calculate (i) the final amount and (ii) the compound interest earned for the following investments. (a) $ at 7% p.a. for 5 years (b) $ at 8% p.a. compounded quarterly for 10 years (c) $8760 at 9.5% p.a. compounded fortnightly for 3 years week High Low Company Last sale Move Buy Sell Div. cents Div. yield % AGL Angus and Coote Just Jeans Qantas Seven Network Sydney Gas TAB Westpac The table gives information about share prices for eight companies. (a) What is the market price (as shown by the last sale) of TAB shares? (b) If the Qantas dividend was paid today, how much would the dividend be for a shareholder with 2000 shares? (c) What is the lowest price in the last year for Angus and Coote shares? 430 NEW CENTURY MATHS GENERAL: HSC

19 !NNC Maths Year 12 Ch 11 Page 431 Wednesday, October 4, :33 AM 4. A grandmother puts $100 into her granddaughter s account on her 12th birthday then each year up to and including her 18th birthday. If the account earns 8% p.a. interest, complete the table to find how much is in the account on her 18th birthday. Age Balance at start of year ($) Interest ($) Deposit ($) Balance at end of year ($) The Overflow Oil Company set up a fund to pay $1000 at the end of each year to the employee who had suggested the best improvement to work conditions. How much (to nearest dollar) should the company invest now at 6% p.a. to achieve this for 10 years? 6. Layla contributes $500 per month for 8 years to an account earning 6.5% p.a. compounded monthly. (a) What amount (to nearest dollar) will she have in the account at the end of 8 years? (b) How much interest will she earn in the 8 years? (c) What single sum of money could she invest now at 6.5% p.a. compounded monthly to achieve the same result? 7. Fred invested $ in an account earning 7.2% p.a. for 3 years. At the end of 3 years he deposited another $5000 in the account. If the interest rate then increased to 7.5% p.a., how much did he have in the account at the end of 6 years? 8. Soumaya and Katerina both started working at age 20. Soumaya invested $1000 each year at 10% p.a. interest for the next 20 years. Katerina made no investment for the first 10 years but then invested $2000 per year at 10% p.a. interest for the next 10 years. (a) How much did each girl deposit into her own account over 20 years? (b) What was the value of Soumaya s 20-year investment? (c) What was the value of Katerina s 10-year investment? (d) What advice on investment could you give a person starting work at age 20, based on these answers? 9. Find the difference in interest when $1000 is invested for 12 years at 5.5% p.a.: (i) simple interest rate (ii) interest compounded annually 10. Johann and Yoko were given two options: take $ in cash now or $5000 per year for the next 12 years. If the current interest rate is 8%p.a., which is the better option? Why? 11. Craig bought a new truck in 1996 for $ Age (years) Value ($) The table shows the value of the truck at the end of each year for the next 5 years (a) What is the depreciation rate of his truck? (b) Use the declining balance method of depreciation and this rate to predict the value of Craig s truck when it is 12 years old REVISION 431

20 !NNC Maths Year 12 Ch 11 Page 432 Wednesday, October 4, :33 AM 12. A vehicle has a current value of $ Work out which calculation gives the better trade-in value after 5 years: (i) straight line method of depreciation at $2000 p.a. or (ii) declining balance method of depreciation at 20% p.a. 13. A B C D E 1 Annuity table 2 Annual payment $1 3 Interest rate 8% 4 5 Year Payment ($) Amount at start of year ($) Interest ($) Amount at end of year ($) The table shows the progress of an annuity with an annual contribution of $1. (a) Find the missing values in cells D8, C9, D9. (b) What would be the amount at the end of 5 years if the annual payment were: (i) $1000? (ii) $3800? (c) What lump sum invested for 5 years at 8% p.a. will grow to $6.34 in 5 years? Give your answer to the nearest cent. 14. (a) The graph shows the decreasing value of a bulldozer as it depreciates over 10 years at 15% p.a. From the graph: (i) what was the salvage value of the bulldozer after 5 years? (ii) by how much did it depreciate in the 4th year? (iii) how long did it take to depreciate to half its initial value? (b) Use the declining balance formula for depreciation to predict the value of the bulldozer when it is 15 years old. Value ($) Depreciation of bulldozer Time (years) 15. The table shows the value of a pottery kiln, bought new for $15 000, at the end of each year for the first 6 years of its life. (a) How much is the kiln worth at the end of 3 years? Time (years) Value ($) (b) What is the amount of depreciation in the 4th year? (c) What is the declining balance rate of depreciation of the kiln? (d) Plot the values on graph paper and draw the curve of depreciation (e) What is the vertical intercept and what does it represent? (f) After which year will the value fall below $1000? (Hint: Use a guess, check and refine method.) NEW CENTURY MATHS GENERAL: HSC

21 !NNC Maths Year 12 Ch 11 Page 433 Wednesday, October 4, :33 AM STATISTICAL DISTRIBUTIONS DATA ANALYSIS Preliminary Course Interpreting and constructing graphs Types of data: categorical versus quantitative (discrete, continuous) Types of samples: random, systematic, stratified Frequency histograms and polygons Dot plots, stem-and-leaf plots, radar charts, box-and-whisker plots Quartiles, deciles and percentiles The population and sample standard deviations: σ n and σ n 1 respectively Five-number summary: lower extreme, lower quartile, median, upper quartile, upper extreme Cumulative frequency histogram and polygon: median and interquartile range HSC Course Σx Σfx The mean: x = or x = n Σf The median and mode The range, interquartile range and standard deviation Features of a statistical display: shape, skewness Investigating the effect of outliers Comparing data sets: double stem-andleaf plots, box-and-whisker plots Radar charts and area charts Two-way tables Multiple displays Exercise 11-08: Statistical distributions 1. Classify the data as categorical, quantitative and discrete, or quantitative and continuous. (a) weights of gym members (b) weights of gym members measured to the nearest kilogram (c) colour of a person s hair (d) religious denomination (e) names of five favourite movies (f) number of movies seen in the last year 2. The swim times (in minutes) for 6 triathletes were: Carolyn Jane Margaret Nancy Lynne Rosemary (a) What is the mean swim time? (b) What is the standard deviation of swim times? (c) If Lynne increased her time by 2 minutes to 34 minutes, what effect would this have on the mean and standard deviation? (d) If all triathletes increased their times by 2 minutes, what effect would this have on the mean and standard deviation? 3. The population of a country town has twice as many women as men. (a) When surveying this population, what type of sample would be the most appropriate: systematic, stratified or random? Why? (b) If there are 200 women and 100 men in this town and a sample of 45 were selected, how many women should the sample include? 4. A questionnaire was designed to survey people s drinking habits. Design three questions to find out whether a person drinks alcohol, how frequently and how much. REVISION 433

22 !NNC Maths Year 12 Ch 11 Page 434 Wednesday, October 4, :33 AM 5. The cumulative frequency polygon shows the speeds of motorists when passing through a school zone. 25 Cumulative frequency (no. of motorists) Speed (km/h) (a) How many motorists were observed? (b) What was the median speed of the motorists? (c) How many motorists were travelling at 50 km/h or less? (d) How many motorists were observed travelling at over 60 km/h? (e) The point (40, 10) is a point on this graph. What do the coordinates represent? 6. The resting pulse rates (beats/minute) of gym members were taken and recorded in the table. Men Women (a) Find the mean pulse rate and standard deviation of pulse rates for each group. (b) Find a five-number summary for each group and construct box plots on the same diagram. (c) The fitter a person, the lower their resting pulse. Use the information you have found to comment on the suggestion that Women are generally fitter than men as they have a lower resting pulse. 7. The area chart shows the number and predicted number of car air conditioners (CACs) using ozone-depleting hydrocarbons in eastern and southern Asia. No. of CACs (millions) CACs using ozone-depleting hydrocarbons India China Others Year (a) Which country is expected to be the largest user of this type of CAC in 2010? (b) In 2030, how many of this type of CAC are predicted to be used in: (i) China? (ii) India? (c) How many CACs are expected to be used in eastern and southern Asia in 2025? (d) What percentage of these are expected to be used in China? 434 NEW CENTURY MATHS GENERAL: HSC

23 !NNC Maths Year 12 Ch 11 Page 435 Wednesday, October 4, :33 AM 8. The pie chart shows the breakdown of NSW tax dollars in (a) How much did the NSW government collect in taxes in ? (b) What percentage of this was from gambling and betting? Stamp duties $3939m (c) What percentage of the total taxation was collected from stamp duties? (d) Give one example of a situation where stamp duty is payable. (e) What is the sector angle (nearest degree) Payroll tax $3440m that would represent the land tax sector? NSW state taxation Petroleum, tobacco and liquor licences $1977m Gambling and betting $1419m Land tax $912m Other taxes and levies Total: $13 829m $2142m Source: NSW Auditor-General s report to Parliament for The weekly sales (in dollars) at the Choo Choo café over a 15-week period are listed: (a) Determine the: (i) range (ii) median sales (iii) interquartile range (iv) sample standard deviation (v) mean (b) The café is situated near a railway station and there was a train strike one week. What were the likely sales for this week? Justify your answer. 10. The dot plots represent steel rods made by two machines in an engineering factory. Rods produced by 2 machines A B cm If rods have a diameter greater than 8.01 cm or smaller than 7.99 cm they are discarded. (a) How many rods from each machine would be discarded? (b) If one machine was to be replaced, which one should it be? Why? 11. The radar chart shows the money earned (in dollars) by Doug and Doug and Pete Jan Pete as part-time pizza delivery boys. (a) How much did Pete earn in Dec Feb August? (b) How much did Doug earn in September? (c) In which month(s) did Pete earn Nov Oct Mar Apr more than Doug? Sep May (d) By comparing the area enclosed Doug s by each line graph, decide who Aug Jun earnings in $ Pete s earned the most for the year. Jul earnings in $ REVISION 435

24 !NNC Maths Year 12 Ch 11 Page 436 Wednesday, October 4, :33 AM 12. The weekly repair bills over 1 year for a photocopier in a large TAFE college were analysed and a grouped frequency distribution table was prepared. (a) Copy the table and add a column for the class centres. (b) Use your calculator to calculate the mean weekly repair bill. (c) What is the standard deviation of repair bills? (d) Draw a grouped frequency histogram and polygon and clearly label each one. Weekly bill ($) Frequency The diagrams show the age distribution of road user casualties. No. of casualties Drivers Ages No. of casualties Passengers Ages No. of casualties 200 Pedestrians Ages (a) Comment on the shape of the distributions of driver and passenger casualties, mentioning skewness, clustering and any outliers. (b) At what age are you most likely to be injured in a road accident if you are: (i) a driver? (ii) a passenger? (c) Comment on the shape of the distribution of pedestrian casualties. (d) If you were running a road safety campaign to help lower the number of casualties on our roads, whom would you target and why? 14. The waiting times (in minutes) for 20 patrons Movie Magic Fine Flicks to purchase tickets at two cinema complexes were recorded. The double stem-and-leaf display shows the results (a) What is the range of waiting times at Fine Flicks? (b) What is the median waiting time at Movie Magic? (c) One time is missing from Fine Flicks. Give a possible value for the missing time. 436 NEW CENTURY MATHS GENERAL: HSC

25 !NNC Maths Year 12 Ch 11 Page 437 Wednesday, October 4, :33 AM (d) A patron is selected at random from Movie Magic. What is the probability that this person waited more than 10 minutes for tickets? (e) At which cinema complex would you have a shorter wait? Justify your answer. 15. (a) Describe in 1 2 lines what the graph is Depth of Coopers Creek showing. (b) What is the depth of the creek at 9 am? 10 (c) At what time(s) is the depth 6 m? (d) A bridge is 12 m above the creekbed. What is the distance between the bridge and the water level at 4 pm? 5 (e) A boat has a mast that reaches 5 m above the water level. What is the earliest time that it can pass under the bridge? Depth (m) 0 6 am 9 am noon 3 pm 6 pm Time THE NORMAL DISTRIBUTION AND CORRELATION DATA ANALYSIS HSC Course The normal distribution Properties of the normal distribution Percentages of scores in a normal distribution within one (68%), two (95%) and three (99.7%) standard deviations of the mean x x z-scores: z = s Comparing z-scores of different normal distributions Scatterplots Correlation Lines of best fit: the median regression line Making predictions Exercise 11-09: The normal distribution and correlation 1. A machine produces steel pins with length approximately normally distributed with a mean of 2 cm and a standard deviation of 0.05 cm. (a) What percentage of pins will have a length: (i) greater than 2.05 cm? (ii) less than 2.05 cm? (iii) between 1.95 cm and 2.05 cm? (iv) between 2 cm and 2.1 cm? (b) If pins are produced per year and those with a length more than 3 standard deviations from the mean are rejected, how many pins are rejected? 2. Brightalonga Light Company guarantees that its lightbulbs last longer than 1000 hours. The lifetime of the bulbs is normally distributed with a mean of 1500 hours and a standard deviation of 250 hours. What percentage of bulbs will not satisfy the guarantee? REVISION 437

26 !NNC Maths Year 12 Ch 11 Page 438 Wednesday, October 4, :33 AM 3. Woody s timber yard wanted to cut down on wastage. It investigated the length of offcuts of skirting board and found the lengths to be approximately normally distributed with a mean of 30 cm and standard deviation of 10 cm. What percentage of offcuts were: (a) greater than 50 cm long? (b) between 10 cm and 20 cm long? (c) between 20 cm and 50 cm long? (d) less than 5 cm long? 4. The table shows the net weight for 1 kg packets of washing powder. Net weight (g) z-score 0 (a) Complete the table if the distribution of net weight has a mean of 1000 g and a standard deviation of 20 g. (b) What percentage of packets will have a net weight: (i) greater than 1000 g? (ii) between 940 g and 980 g? (iii) between 980 g and 1020 g? 5. In a History test, Matilda was given a z-score of 1.8 and Blinky was given a z-score of 2.5. (a) Who did better in the test? Justify your answer. (b) If the mean mark was 65 and the standard deviation was 12, calculate Matilda and Blinky s marks (to the nearest whole percentage). 6. Sweets are put into 50 g packets. The machine that performs this task is set to a mean mass of 51 g with a standard deviation of 1.5 g. (a) Complete the table. Mass (g) z-score 0 (b) What percentage of packets will have a mass: (i) more than 51 g? (ii) less than 54 g? (iii) between 48 g and 54 g? (iv) more than 46.5 g? (v) between 52.5 g and 55.5 g? 7. The heights of a group of primary school children are normally distributed with a mean of 120 cm and a standard deviation of 10 cm. (a) Find the z-scores (correct to 1 decimal place if necessary) for the following heights: (i) Kip 110 cm (ii) Trish 1.4 m (iii) Pippi 960 mm (iv) Jimbo 128 cm (b) Sketch a standard normal curve and mark the position of each z-score. 8. Batteries were tested and the life of each was recorded. They were found to be approximately normally distributed with a mean of 30 hours and a standard deviation of 6 hours. What percentage of batteries would you expect to last: (a) greater than 42 hours? (b) between 18 and 48 hours? (c) less than 24 hours? 438 NEW CENTURY MATHS GENERAL: HSC

27 !NNC Maths Year 12 Ch 11 Page 439 Wednesday, October 4, :33 AM 9. The table gives statistics for the red blood cell counts for males and females. Mean Standard deviation Male Female (a) What percentage of males have a red blood cell count: (i) greater than 6.1? (ii) between 4.7 and 6.1? (iii) less than 5.05? (b) What percentage of females have a red blood cell count: (i) greater than 5.7? (ii) between 4.2 and 5.4? (iii) less than 4.2? 10. In a university study a correlation was found between students assignment marks and their test marks. If the correlation coefficient was calculated to be 0.82, what conclusion can you draw? 11. The correlation coefficient between the daily maximum temperature and the number of ice-creams sold was found to be 0.1. What conclusion can you draw? 12. Give an example of two variables with a strong positive correlation but where the relationship: (a) is a causal one (b) is not a causal one 13. The heights and weights of 12 students are Height, h (cm) Weight, w (kg) shown in the table (a) Plot each pair of values on a scatterplot. (b) Is there a linear pattern to the data? (c) What conclusion can you draw about people s height and weight? The cost of advertising and profit at Connie s Cafe were recorded for 6 months. Month Cost of advertising ($) Profit ($) January February March April May June (a) Plot the data on a scatterplot and construct the median regression line. (b) Find the equation of this line. REVISION 439

28 !NNC Maths Year 12 Ch 11 Page 440 Wednesday, October 4, :33 AM 15. The number of employees (N) in a small factory and the output of palm computers (C) per month were found to have a high correlation. Data was plotted and the equation of a line of fit was found to be C = 6.7N (a) How many palm computers were made when there were: (i) 20 employees? (ii) 10 employees? (b) Is it possible to predict the output for 1000 employees from this equation? Give reasons for your answer. PROBABILITY PROBABILITY Preliminary Course The language of probability Outcomes and sample spaces Multistage events Experimental probability Theoretical probability: number of favourable outcomes P(event) = total number of outcomes Complementary events: P( ) = 1 P(E) E ~ HSC Course The multiplication rule for counting Tree diagrams Counting arrangements Counting unordered selections Counting ordered selections Probability tree diagrams Expectation Probability simulations Probability in testing Exercise 11-10: Probability 1. An Internet password is made up of 5 alphanumeric characters (letters or numbers): for example, LUV17. How many 5-character passwords are possible? 2. A jar of jellybeans contains 4 pink, 7 green, 6 yellow, 4 white and 4 black. If 1 jellybean is selected at random, what is the probability that it is: (a) pink or yellow? (b) not white? (c) black, yellow or white? (d) not red? (e) blue? 3. A matchbox is tossed 20 times. There are three possible sides that can face upwards: a flat side (F), a long side (L) and a short side (S). The following results were obtained: Outcome Frequency Flat 16 Long 4 Short 0 L F S (a) What is the experimental probability that a matchbox lands with a flat side up? (b) Why do you think the probability of the matchbox landing on the flat side is greatest? (c) What is the experimental probability that a matchbox lands with a short side up? (d) Does this mean that it is impossible for the matchbox to land with a short side up? Justify your answer. (e) What is the experimental probability that a matchbox does not land flat side up? 440 NEW CENTURY MATHS GENERAL: HSC