The numerator of the fraction is the central angle; the denominator is the whole circle (360): Area a of of sector sector == 8 A 5 This can be written as: A ector = 36 7T 2 where r is the radius of the circle and is the measure of the sectors central angle in degrees. The total cost charged by a different taxi driver can be modelled by G 1d 2 (c) Explain what the gradient of the line C = d + 2 represents. = (300 02 + 50% = Dg = /(300 0* + 50* Using the general distance-time model, we can now write: = + 502 _ /300 s 10060 = = %(300 =B Substituting the expressions for Ty and T into our total-time model, we obtain an equation in a single variable: = Ty 2 Ti==C 100 PO ramrsosz i) 2 2yS\(300 o)t 50; Now we are ready to use calculus to find the value of which will minimise the total time it takes the lifeguard to reach the swimmer. We are given that the radius of the Earth is 6370 km, so the distance along the Earth's surface is 107.5 60 @m(6370) ~ 11952 km = 12000 km (3 5.) (b) The new radius will be 6370 + 10 = 6380 km. (Ignore any currents in the water.) What shape will this distribution take? (b) Write down an equation for the volume of the lobster trap in terms of r,land . local time (the usual start time is 8:30 AM or 9:30 AM). Next, we have to consider how to find algebraic relationships for T, and T in terms of c. Since both of these have to do with rate and time, we can distance use the model rate = , time . Learn more about mathematics in a DP workshop for teachers. (d) In words, accurately describe the motion of the object during the interval -1 0 so production costs are increasing forallx >0 (d) Here we need a new model. (b) Find them in a + bi form. A geometric sequence has a third term of 8 and a sixth term of 27. At her daughter Janes birth, Charlotte put $500 into a savings account. AI SL. What problems does it solve or what questions does it answer? If not, state why not. The mathematician Reuben Hersh argues for a type of constructivism that he calls Humanism. The other chapters in this book are all about establishing and using this very special vocabulary and becoming fluent in the methods that connect mathematical concepts into meaningful mathematical sentences. 31. Give a reason for your answer. R 5 S 1) s (1 25 -3) 1 ) ) . + 4110900 200d + d? (@) flry=x+2 (b) fiy=322t+1 g =%-20 (d) fiH) = (t VRt +3) (&) gu) = u 4u? WebFeaturing a wealth of digital content, this concept-based Print and Enhanced Online Course Book Pack has been developed in cooperation with the IB to provide the most comprehensive support for the new DP Mathematics: applications and interpretation SL syllabus, for first teaching in September 2019. @) 3= (f) 33 G -2 @ @ (b) 156 8. (a) Generate a scatter diagram for these data and describe it. Solution The gradient of AB is 76;225 = 1, so the gradient of the perpendicular to - _ [AB] is 1. However, here we can use one of two approaches, introduce the factor 6, as we have done before, i.e., [ FiTde=1 6 [VexF 1T 6dx _1 _if 7Efdu76fu _lui 5 du 1 teTguite =g(6x+1Di+c 2 orsinceu=6x+ [T 11 = du=6dx=dx= dG,then iTde= [Vadt= Jutdu and we follow the same steps as before. It is also called a center. Find the solution to the initial value problem. For example, in the second sequence, each term is 3 times the previous term. 8 3. As we've emphasised before in this textbook, extrapolating assumes that the pattern you've observed will continue. What is the difference between Mass and Weight? The horizontal axis represents time, not distance. Model Answers. (a) 2 2i (d) 2 + 2i (b) 22i (e) 2+ 2/3i (g) 2+2/3i (h) 2 2/3i (c) 2+2i OP 2B 157 Find the products of these complex numbers. 2 Period ==~ 12 = 4=2 6] = 27 _ |h|==~== 4 2 Sob:gorb:f;I Finally, we need to decide if a sine or cosine model is better suited for this situation. IB Applications and Interpretations SL and HL Resources. The edge of the flysheet is to be 1.4 m off the ground. Basic definitions Given a right-angled triangle ABC, the sides opposite angles A, B, and C are designated a, b, and c respectively by mathematical convention (Figure 4.7). Printed in Slovakia by Neografia Acknowledgements The authors and publisher would like to thank the following individuals and organisations for their kind permission to reproduce copyright material. The differential equation (1) is separable and can be easily solved. (a) B 04195 C 02634 C 02015 (0)3 _0 _ (08 03)/ oy (0.2 0.7)(1) (0.7) - (@) Xo=TXs _ oy (08 03)(03) _ (045 sin0 =L and sing = Lat the same time! 143 Geometry and trigonometry 2 Solution Since cos? + sin?f = 1 cos?f + el N2 are 1. The IB Math AI Formula Booklet is useful for two reasons: 1) to easily refer to formulas when solving questions, and 2) to get a high level view of the full scope of the AI SL & HL courses. RPM Torque (Nm) RPM Torque (Nm) 1500 2000 2500 3000 3500 380 461 475 508 529 4000 4500 5000 5500 6000 515 488 475 420 339 (a) Generate a scatter diagram for the data and describe it. Find a formula for the nth term. Solution (a) Substituting into the formula gives: 232 X110 10712 = 10log,(3.2 X 107) = 75.1dB n(3.2 X 1079= = 1olog,,,( 5 . Also, recall that y = fix) 38 flx)=4x8 =x =8 writey = fix) x=4y8 interchange x and y (i.e. Sequences formed in this manner are called arithmetic sequences. 905 Theory of knowledge Platonic view of mathematics One way to explain why mathematics applies so well to things like bridges and planets is simply to take mathematics as being out there in the world, independent of human beings. is critical. (a) @ 5 x+6 x+ 3n 4x 2 ) () () () ( 1 (iii) 7 (i) 79 (i) 12x 422 =7 (iv) dx2xt2 (b) 42+ 1,xeR (@2x+3 ( 16 _ (e) $1165396 2.4 3.6 4@ 22 )6 aer (b) 4x7,xR ( x%x=0 @L-2x%0 (& T x,x=4 f) 920 x2+5x=0 (0,4) -1 x+12 (b) St 1. In this case, we can assume that air resistance is acting against the model rocket and causing its net acceleration to be less than g. (e) The coefficient of determination of R? =1 Figure 2.1 Graphof x + y* = 1 Avertical line intersects the graph of a function at no more than one point (this is known as the vertical line test). From August 2019 the following courses, with first assessment in May 2021, are available: Students can only study one course in mathematics as part of their diploma. 17. If we write this out in full we get 5 S2=2+2+2+2+2=5X2=10 = In general, if we sum a constant # times then we can write " Sk=k+k+..+k=nXk=nk = Suppose we have the sum of a constant times i. If C > 2, 2 sinx + C will always be positive. Find two angles - one positive and one negative - that are coterminal with the given angle. () 12 10 0 15, 16. Developing your ability to read and understand mathematical explanations will prove to be valuable to your long-term intellectual development, while also helping you to comprehend mathematical ideas and acquire vital skills to be successful in the Applications and Interpretation HL course. (d) Atwhat time is the velocity of the tennis ball equal to zero? (vi) Give a reason why this model should not be used to predict the CO; concentration in the year 2050. ), then we can calculate the relative loudness in decibels, 1, using the formula nes) = lologm( 10:2) The number 10~!2in the formula represents the threshold of human hearing. That is, the angular velocity of each wheel is 1.5 revolutions per second. We will factorise the derivative function and solve algebraically: \u_p/ flo=x2~x6=(x3)x+2) sof'c=0=>x=30rx=2 versus concave down R Concave down Figure 14.9 Concave up Ifyou have trouble Now find the second derivative: remembering what floo=2x1 concave down look like, concave up and f"B3)=5andf"(=2)= -5 remember that up is happy, like a smile @, Therefore: since f'(3) = 0 and f"(3) > 0, there is a local minimum at x = 3 since f'(2) = 0 and f"(2) < 0, there is a local maximum at x = 2 The graph of fix) is and evaluate at x = 3and x = 2 and down is sad, like a frown . Students have the option to choose between two different courses: IB Math Analysis and Approaches (AA) and IB Math Applications and Interpretation (AI). Thus, the amount of the gift should be 1 Present value = | o 20000 095 dt ~ 134726 19. We can use the logic from section 5.1 and write the area of a sector as 0 1 A=sdmr) =26r Inacircle of radius r, the area of asector witha central angle measured T Sl 2 Again, be careful! However, each of the payments earns compound interest at the rate of 2% per year. There are those who go as far as saying that it is present in the proportions of the perfect human figure and that we have a predisposition towards this ratio. Find the product of each complex number and its conjugate. You donit need to memorise this, because you already know where yand x are positive and negative, which tells you when sine and cosine are positive and negative, And, if you remember tha tangent is 2, iie. 50 4 3 =2 1 1 3 2 2X3 =)= | 14 11 13 3X2 2ixa2 When matrices are the same size, the product is the same size. =2 ,.2 650 " 65 15. This method requires us to be more careful about considering the domain 0 < x < 25 and setting the viewing window appropriately to obtain an appropriate display. Annuities are equal regular payments that earn an annual interest rate. Mathematics Applications and Interpretation for the IB DiplomaHigher Level provides comprehensive coverage of the new c, Mathematics Applications and Interpretation for the IB Diploma Standard Level provides comprehensive coverage of the new, Worked solutions for the Mathematics Analysis and Approaches for the IB Diploma Higher Level Pearson book, Mathematics Analysis and Approaches for the IB Diploma Higher Level provides comprehensive coverage of the new curriculu, Mathematics Analysis and Approaches for the IB Diploma Standard Level provides comprehensive coverage of the new curricu, R ==} ] Mathematics Applications and Interpretation for the IB Dlploma (UYL R ety IBRAHIM WAZIR GAI 3338 JIM NAKAMOTO EEEEEEEEEEEEEE e Published by Pearson Education Limited, 80 Strand, London, WC2R ORL wwwpearsonglobalschaols.com Text Pearson Education Limited 2019 Theory of Knowledge chapter authored by Ric Sims Development edited by Jim Newall Copy edited by Linnet Bruce Proofread by Eric Pradel and Linnet Bruce Indexed by Georgie Bowden Designed by Pearson Education Limited 2019 Typeset by Tech-Set Ltd, Gateshead, UK Original llustrations Pearson Education Limited 2019 lustrated by Tech-Set Ltd, Gateshead, UK Cover design by Pearson Education Limited 2019 Coverimages: Front: Getty Images: Busi Photography Inside front cover: Shutterstock.com: Dimitry Lobanov Weare grateful to the following for permission to reproduce copyright material: Text pages 904-905, Edge Foundation Inc.: What Kind of Thing Is a Number? 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