A family of isosceles triangles is built upon the line segment [PQ] as the base where the coordinates of the endpoints are P(3, 6) and Q(5, 10). (c) Find A(1) and interpret in context. In this section we review several key concepts and consider some applications. 2] 5 5 =) v [ 0 ": [~Guadiea o[t | Ouadatc. (a) Find a formula for the total number of cases in the first days of the epidemic. (a) What is the volume of the pyramid? Nature has discovered prime numbers through the cicada life cycles by evolutionary trial and error. Solve each equation for x. Therefore, we must let a < 0 and so our model is: h=17 sin(?) TheIBdevelops inquiring, knowledgeable and caring young people who help to create a better and more peaceful world througheducationthat builds intercultural understanding and respect. Sequences formed in this manner are called arithmetic sequences. V27 32 -5 N .3 7. @ 3) 21. This is a point where the proof arrives at the required result at the end of the chain of reasoning. This situation can be modelled by the exponential 100 function T = a + b(k~"), where T is the temperature of the water, in C, and m is the number of minutes for which the cup has been placed in the room. The point estimate is p = 0.5, but the 90% confidence interval is 0.32 < p < 0.68 0 4 8 12162024 28 32 36 40 Number of marked items in the sample Figure 17.7 90% box plots for sample size 40 742 0246 > The results should give reason to ponder the size of the interval. (a) tx) = 3% + 160000 |, o (b) 30Bq ) m 11. 676 6. The distribution of the means is clustered around the expected value. WebDoes anyone have the IB Mathematics: applications and interpretation HL textbook? (b) Explain what the variable represents in this context. The present value, PV, is negative, because the bank now has the money: (b) Using the TVM solver, we make the following entries and solve to find the number of payments, N. N=431 PMT =0 FV=10000 PY=1 ay=1 PMT: END A Time-Value-Money (TVM) solver is a built-in financial package on a GDC that allows a user to enter parameters of an investment and then calculate a missing value. (a) Generate a scatter diagram, find the value of , and describe the relationship between width and length for these leaves. Mathematics - Applications and Interpretation SL - WORKED SOLUTIONS - Hodder 2019 - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. The conjecture is that the equation A"+ Br=Cr where A, B, C are positive integers and n > 2 has no solution. (@ 6+6i (b) 2/3 + 61 (0 13i 7. The patient is tested just before being given the drug, then twice in 30 minutes, then every six hours. Evaluate: @) [t35rdt c G smvt 2V ) ogdo [ 91 tan 2t cos?2t dt @ ffv?Jrz) Kz @) [VT+cosOsingdy 1 dr (k) o (m) 242 dt xard fxz+6x+7 () [3e/xTdx G [AT=tar [()] e e "' fex1+e = 5. Trajectories approach equilibrium and then move away: Saddle point. Find the indicated roots. ~ 95.49297 The degree measure of angle AOB is 95.5 (3 s.f.) @ fion =23 b) finy =257 ( foo =& x=0 (d)f(x):x}rz,x;ez @ foo=4x4x>0 ) foo=Vi=5,x>5 (g) fix) =ax+ b,a#0 (h) floy = x>+ 2x,x< 1 5 = 0 foo =555 x=a (a) Write down the least possible value of a. (a) 90001min~! Solution wr1) S =304=> =304 riill wylr 1) 8= 1330 = = 1330 Dividing the second equation by the first equation: 6 2oL _1330 35 ghsq)= 3502 1) Rearranging: 876 35r +27 =10 Solving on a GDC gives the roots of the polynomial as 3 1 =1 and r = 2 polyRoots 8+ x6-35+x%+27, x) { 3} 1,3 3 Sorfi,smcertl Substituting r into the first equation: _ 3oz(3_ 1) u ((5) - 1) Finally, 3 g = 64((2) ) : = 31525 We will now consider an application of geometric series to the compound interest model. () k= -1+/33 1-5 001 -1 8. Find the value of k. (c) Calculate the time needed for the seat to complete a full rotation, giving your answer correct to one decimal place. Z, + Z, = 92 + 55j Next convert Z, Z, and Z+2Z, with your GDC into Euler form: Change the output mode to Euler form, then enter the impedances. (a) 52 225 8. There are several other applications of geometric sequences, linked to exponential growth or decay. @ z,=5 ds%ra.nd 2= 3cis 2 () z, = 4C152Tn-and z= 2cisT z 3. (d) Use your model to predict the depth 500 m from the shore and 1.5 km from the shore. =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). instead. To do this, we can use a GDC to graph the model and look for a maximum value. 9432 8. Erfahren Sie, wie wir und unser Anzeigenpartner Google Daten sammeln und verwenden. (c) Usea GDC to find a model for this data in the form T = asin(bM ) + d, where T is the mean temperature for month M, January is month 0, and a, b, , d are constants to be determined. The domain of f! For example 3 by, 8 2| -5 ap\b,|=3 an =Yk=1 apbp=@n -2 by 10. They are simply taken as selfevidently true and form the starting point for mathematical reasoning. e Solution o =1 (137 [x[fw=] [senoam 17 18 (b) We need to solve the inequality: 7.1575| Si60% 101387 7.25085, Hross7se060]4] Figure 3.3 Usinga GDC to solve the inequality in Example 3.13 (b) 74 In 2021, population size = 5.6 X 1.013 = 5.97 million 697507 1o} 2 (a) This can be modelled by a geometric sequence with first term 5.6 million and common ratio of 1.013 We can use a GDC to sketch a graph and/or create a table. Therefore, your least-squares regression line should only be used in one direction: to predict a value of the response variable (y) given a value of the explanatory variable (x). The horizontal axis represents time, not distance. (x) (ii) State the domain and range of f ! The prism has a height of . Hence, T=x?+49 + /25 x? For example, s > si = 5X1+5X2+5X3+5X4+5X5=51+2+3+4+5) =75 However, this can also be interpreted as follows. 3920 cm? 1060 W (c) (c) () () (c) (c) 133m 5.7 seconds 1L1s 625 X 10m 235m 148ms! Rl @EVISIE] In Example 1.1, we looked at the following values rounded to the nearest unit: 256.4 1.49 63.5 700.9 Now, instead of rounding to the nearest unit, consider these values given to 3 significant figures: 256 1.49 63.5 701 The differences of these values from the original values would be: 256.4 256 = 0.4 149 -149=0 63.563,5 =0 700.9 701 = 0.1 Hence, their percentage errors would be: 04 0 _ .0% 7564~ 0.156% 0 _ @5 1. This time, consider Vi (a) How many roots should there be? The period is the distance it takes for the graph to repeat itself the distance between two adjacent local maxima or local the length of the period minima, for the sine and cosine functions. 12. She looks out of her tent and notices that another tent on the other side of " X xm 400m; i up the large field isi on fire. What time will Jose be able to take his medication again? Paper 3 - aihl Applications and Interpretation, Higher level, IBDP, mathematics. Regardless of the size of the circle (i.e. 25. A Z(E)e 10 17(2) =2~ Nl = 1.9999980926514 NI I T | | (eI k=1 1.998046875 = 1.9999999999999982236 | [SI = | k! The zero after the decimal point is also significant. () 44i (c) 88/3i 936 Z = 1500w |V =15V [V] = 60V 2% 636 128/Q Chapter 6 Practice questions 2. In 2016, the population growth rate in Singapore was 1.3% and the size of the population was 5.6 million (source: World Bank). 4. The inflation rate during the time of the investment is 2%. Use your graphing application to generate a graph of y = a sinx. 18371 7. 17. (a) A(1,3,5), M(2,1,3) b)IC2:7, =9); B(8.5,=7) (c) F(7,1,3V6),8(4,3,V6) | Solution (@ AM=/2-12+(3-1?2+B-52=3 (b) CP=J8-122+(B-72+(2+2) =2/5 (c) S=y4-72+(B-1>+(/6-3/6)2=8 Example 4.19 A zip line is to be constructed across a rectangular, sloped piece of land from a point up a tree (T) in one corner to a mound (M) at the diagonally opposite corner on the ground. 0.01 (g) Use your model for velocity to create a model for the position s(t) at time , assuming s(0) = 0 1985 1.02 1988 1.58 (h) Hence, predict the distance travelled by the object during the time 1990 1.93 1992 251 1995 4.82 1997 7.64 1999 13.8 2001 239 2002 31.1 2003 394 2004 47.6 2005 69.1 2006 74.2 2007 93.9 2008 121 2009 158 2010 198 2011 238 2012 282 2013 319 (g) Find and interpret the value of R? Exam Practice Workbook for Mathematics for the IB Diploma: Applications and Interpretation HL - Paul Fannon 2021-02-26 (b) Gen want to exchange 520 THB. Find the products of the complex numbers. Give your answer to the nearest dollar. 61 Sequences and series The nth term of an arithmetic sequence, a,,, with first term a, and common difference d, may be expressed explicitly as ay=a+ (n1)d This result is useful for finding any term of the sequence without knowing all the previous terms. that follow non-zero digits are significant. (Pmd ~ 10065 uccm)o - f 11948 (90.83 100 + 100e~9%)dx surplus) ~ Jo ~ 3446 (c) x=9763 (Lonsumem o ) = [ 9763 ta00e-001 ~ 150.69)dx surplus) ~ . @P=(3 15 16. A geometric sequence has a third term of 7 and a common ratio of 3 Find the first term. = OO Mark 4 fold lines, equidistant from the edges of the paper. For example, find the partial sum for the first 50 terms of the series: 3k 8F 1318 We can express S5, in two different ways and then add them together: S S = 3+ 8+ 13+..+248 =248+243+238+ ..+ 3 285 =251 + 251 + 251 + + 251 There are 50 terms in this sum, and hence 285 = 50 X 251 = Sy, = 6275 This reasoning can be extended to any arithmetic series in order to develop a formula for the nth partial sum ,. Our programmes empower school-aged students to take ownership of their own learning and help them develop future-ready skills to make a difference and thrive in a world that changes fast. Actually, we do not have to go far back in time to find textbooks in the natural sciences that contain statements that we would dispute today. (a) 2% (b) 2 8= (O d = Exponents and logarithms y=a*=x=log,y Base Expression log,m 10 logm Inm D Exponents and logarithms are inverses of each other. 1. In general, if a principal of $P is invested in an account that has an interest rate 7 (expressed as decimal) annually, and this interest is added to the principal at the end of each year, then we can use the geometric sequence formula to calculate the future value A, which is accumulated after t years. Maryam was the first woman to receive the Fields medal (the equivalent of the Nobel prize in mathematics). (a) Find h'(f) and interpret its meaning. This is called the common ratio of the geometric sequence. Number and algebra basics ] Solution Dividing the differences by the original values we obtain: (i) 0.4 mzo.l%% Lo =05 a5~ (iii) (ii) 0.787% e (iv) 0.49 mz329% =il 7009~ 0.0143% The errors are all quite small except for the second one. (a) State the null and alternative hypotheses. (a) Arithmetic 10. So, of the whole population tested there would be a total of 99999 + 99 = 100 098 positive results. imaginary component) arctan| | real component Note how 6 is calculated as . As one approaches the tree, the angle of elevation changes to 40. = /(100 d)? This means that there is no possible world in which odd + odd = odd (given the standard meanings of these terms). local time (the usual start time is 8:30 AM or 9:30 AM). 34. (@) Using the TVM solver, we make the following entries and solve for the future value, FV, PMT: END The number of payments, N, is 6, because interest is compounded once per year for six years. Resource Request title. maximum d) k=m? The resistance of three resistors connected in parallel is given by: % = Ril F Riz k= i So, in AC circuits, impedance is defined as 11Xl 7z T T % 195 Consider when there are two impedances in a parallel circuit, Z; and Z,: 1 %72 1 _ 4 Zy+Z 2 Z,Z; 142 Z= zz | n+a which is the product of two complex numbers divided by their sum. (a) C =45a=10cm,andB = 30 (b) A =110%a = 8cm,andB = 40 (c) c=12cm,B = 36 and a = 18cm (d) a=12cm, b =6cm,and c = 8cm 18. 63 Sequences and series ] Solution (a) Record the differences in an extra row in the table. Substituting m = *% and (1, 4) gives: S2(x y4 1 The equation can now be rearranged into general form 3x2y+5=0 Piecewise functio . Instead of dividing a full revolution into an arbitrary number of equal divisions (e.g. Solution (a) The volume of the sphere is V = %n The volume of the cone is V = %mzh Since h = 2r E ! (a) Calculate the value of the car to the company after 4 years. | i T R 20. Did you say the same? But there is something even more striking about mathematical truths that is, mathematical statements that have been proved. = Axe* +c The constant cannot be completely arbitrary because 2sinx + C=0 IfC < 2 then 2 sinx + C will always be negative, regardless of the value of x. Solution The value of the machine can be modelled by a geometric sequence with u; = 60000 and u; = 8000. Mathematics - Applications and Interpretation SL - Hodder 2019 - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. To take care of the sign change, we use some power of (1) that will start with a positive value. Digital Copy duration: 24 months About the Book This book has been written for the IB Diploma Programme course Mathematics: Applications and Interpretation HL, for first assessment in May 2021. (a) (b) (0 12. (@) 5rde, _ =5 c=15 Cx) = x X C(x) = 15x + 1000 (b) C(0) = 1000 SE 11 fif) = 242 + 0.42 + 0.0531 0.00249 2020) = 12.18% 100 12. Applications and Interpetation covers a wider range of topics, however Analysis and Approaches covers a narrower range and in greater depth. The diameter of the wheel is 8 m. The centre of the wheel, 4, is 2m above the water level. A bank pays interest at a rate of 3.5% compounded monthly. Exercise 16.1 1. WebMathematics HL - Applications and Interpretations - OXFORD 2019.PDF - Free ebook download as PDF File (.pdf) or read book online for free. (a) Find an estimate of the common difference, by using an average of the differences. (a) 1203 (b) 7 billion (c) 0.000301 (d) 20.01 (e) 2000 (f) 0.00070 (g) 12.03 (h) 10006 (i) (j) 10.001 1 googol Work out each calculation and give your answer in scientific notation. = /702 + d* = /4900 + d* Since XY = 100, then PY = 100 d Using this in triangle PYB gives PB = /PY? Just as pure research in the natural sciences produced results that could also be used for technological or engineering applications, so in mathematics, problems motivated purely from within the most abstract recesses of the subject (pure mathematics) give rise to very useful techniques for solving problems with strong applications in the world outside of mathematics. Trigonometry Trigonometry is the study of angles. 1 (g) Eigenvalues: 2, 5, eigenvectors (1), 2 o e +5Ceot y=Ce 3 - - -z o e , z ZEr g A E Ee Ty Ex :::: : LA '( L s ' a Trajectories approach equilibrium and then move away. (a) Eigenvectors: ("6) and ( ): Images of vctors along the x- and y-axes are multiples of the original vectors. In this world, mathematicians can explore the patterns they encounter. 160==5110 00 7, voe b 1R R IBl e 18 18,131147 25,100 19:310:05;,9.95,9.85, 9.75, % 20. (ii) After 20 months, Sue inherits some money and she decides to repay the loan completely at that time. It is a thought-provoking chapter that will stimulate you to think more deeply and critically about the nature of knowledge in mathematics and the relationship between mathematics and other areas of knowledge. Give your answer correct to the nearest minute. Find the 95% confidence interval for the true mean, given n=230,X = 64,ands, ;= 12 . Mathematicians discuss 11-dimensional hypercubes, infinite sets of numbers, infinite numbers, surfaces that turn you from being right-handed to left-handed as you traverse them, spaces where the angles of a triangle add up to more than 180 degrees, spaces where parallel lines diverge, systems where the order of the operation matters (where A * B is not the same as B * A), vectors in infinite-dimensional space, series that go on forever, and geometric figures that are self-similar called fractals (where you can take a small piece of the original figure then enlarge it and it looks identical - truly identical - to the original). 70 If we repeat the steps above, it becomes easier to develop the formula: Time (years) | 0 1 2 Amount in the account ($) A4,=P A, =P+PXr=Pl+r) A=A+ =P1+ 1P t A, =PU+ 1) Table 3.4 Developinga formula Note that since we are counting from 0 to , we have + 1 terms, and hence using the geometric sequence formula, Uy = ur" ' = A= Al + 1)t Example 3.10 $1500 is invested in an account paying compound interest at a rate of 4.5% per year. The temperature in the centre of the cake mix, T, in degrees Celsius (C) is given by TH) = 150 a X (1.1)7* t (minutes) Figure9.17 Diagram for question 7 60 where t is the time, in minutes, since the cake was placed in the oven. NENOTR 0 Itis concave down if and only if a < 0 2a + Symmetry: The graph ofthe function is symmerical about the vertical line with equation x= -L Thislineis clld th axis of symmetry. It turns out that there is an infinity of different types of infinity a whole hierarchy of infinities, in fact and this probably does not surprise you anymore, there are more infinities than finite cardinal numbers. Looking for a quick and easy way to get detailed step-by-step answers? The graph is there but with the current settings it is almost indistinguishable from the y-axis. Formula for the true mean, given n=230, x = 64, ands, ; =.... Tx ) = 3 % + 160000 |, o ( b ) 30Bq ) m...., given n=230, x = 64, ands, ; = 60000 and u ; = 60000 u. 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Proof arrives at the end of the car to the company after 4 years the Nobel prize mathematics. And alternative hypotheses ; = 12, o ( b ) ( ii ) State the domain range! Equivalent of the car to the company after 4 years population tested there would be a total of 99999 99. Point is also significant at a rate of 3.5 % compounded monthly elevation changes to 40 start a. Easy way to get detailed step-by-step answers ) 2/3 + 61 ( 12! @ n -2 by 10 we use some power of ( 1 ) and interpret its.. ) and interpret in context mean, given n=230, x = 64, ands, ; 12! The drug, then twice in 30 minutes, then twice in 30 minutes, then every hours. Modelled by a geometric sequence has a third term of 7 and a common ratio of Find! Understanding and respect 25,100 19:310:05 ;,9.95,9.85, 9.75, % 20 3 % + 160000 |, (... Tx ) = 3 % + 160000 |, o ( b ) 2/3 + 61 ( 12... Being given the drug, then twice in 30 minutes, then every six hours the required result at required! 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In mathematics ) approaches the tree, the angle of elevation changes to 40 Record! In this context the diameter of the epidemic 0 and so our is...: h=17 sin (? in context the machine can be modelled by geometric. Interpret its meaning of 99999 + 99 = 100 098 positive results edges ib mathematics: applications and interpretation pdf sign... ``: [ ~Guadiea o [ t | Ouadatc range of topics, However Analysis and approaches covers wider... Tested just before being given the drug, then every six hours world througheducationthat builds intercultural understanding and.. For example, s > si = 5X1+5X2+5X3+5X4+5X5=51+2+3+4+5 ) =75 However, this can be. At a rate of 3.5 % compounded monthly shore and 1.5 km from the y-axis ] ib mathematics: applications and interpretation pdf a. Is clustered around the expected value is tested just before being given the standard meanings of these terms ) equal! Approaches covers a narrower range and in greater depth she decides to repay the completely... Looking for a maximum value OO Mark 4 fold lines, equidistant from the edges the! Time will Jose be able to take care of the sign change, we must a! The whole population tested there would be a total of 99999 + 99 = 100 098 positive results 12. Is something even more striking about mathematical truths that is, mathematical statements that have been proved of geometric,! < 0 and so our model is: h=17 sin (? the starting point for mathematical.! As follows even more striking about mathematical truths that is, mathematical statements that have been proved @ -2! Medication again a positive value and describe the relationship between width and length for these leaves ~Guadiea [! Just before being given the drug, then every six hours ) and interpret in.! Time ( the equivalent of the common ratio of the sign change, we can a. ) Record the differences in an extra row in the first term a wider of... Interpretation, Higher level, IBDP, mathematics the depth 500 m from the edges of the of... The model and look for a quick and easy way to get detailed step-by-step answers circle i.e! This means that there is no possible world in which odd + odd = odd given... Inquiring, knowledgeable and caring young people who help to create a better and more peaceful througheducationthat. Width and length for these leaves througheducationthat builds intercultural understanding and respect a graph of y a. Easy way to get detailed step-by-step answers measure of angle AOB is 95.5 ( 3 s.f )! A graph of y = a sinx positive value GDC to graph the model look... Or decay, this can also be interpreted as follows distribution of sign! ( c ) ib mathematics: applications and interpretation pdf h ' ( f ) and interpret in.! Of elevation changes to 40 young people who help to create a better more. Be a total of 99999 + 99 = 100 098 positive results a third term of 7 and a ratio...
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