1-2 Cumulative Review

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1 1-2 Cumulative Review 1. Determine the future value and the total interest earned for each simple interest investment. a) $2, invested for a 5-year term at an interest rate of.2% b) $55, invested for a 3-year term at an interest rate of 2.% 2. Cam has been saving for a car. He has $25 that he wants to invest, hoping that he will end up with $3 to use as a down payment. His bank offers a savings account that earns 5.5% simple interest, paid annually. How long will it take Cam to reach his goal? 3. Determine the future value and the total interest earned for each compound interest investment. a) Principal of $5, invested for a -year term at an interest rate of 3.%, compounded monthly b) Principal of $2, invested for a 1-year term at an interest rate of.%, compounded quarterly. Suppose that you have $5 to invest for 1 years. Which of these investment options would you choose? Eplain. A. A rate of 5%, compounded annually B. A rate of 5%, compounded semi-annually 5. Estimate the value of an investment of $1 at 9%, compounded annually, for a term of 1 years. How close is your estimate to the actual future value?. Connie wants a down payment of $15 to buy new furniture for her apartment 2 years from now. Her bank offers a savings account that earns.% interest, compounded monthly. What amount does she need to invest now? 7. Hans invested money at %, compounded quarterly. Emma invested money at 3.%, compounded monthly. After 1 years, each investment was worth $5. a) Who made the greater original investment? b) Who had the greater rate of return?. Jaymee will deposit $5 into a savings account at the end of every months for 5 years. The account earns 3.%, compounded semi-annually. a) What amount will accumulate by the end of 5 years? b) How much of this amount will be interest? 9. Darka would like to accumulate $1 in savings before she retires 2 years from now. At the end of each month, she intends to make the same deposit in an RRSP. She hopes that the RRSP will grow at 5%, compounded monthly. What regular payment will enable Darka to reach her goal? 1 Chapter 2 Financial Mathematics: Borrowing Money NEL

2 1. Deb inherited $2 and is thinking about investing the entire amount. Which of the following two portfolio options would you advise her to choose for the net 1 years? Eplain. A. A 1-year $15 GIC at 3.5%, compounded annually, and a 1-year $5 CSB at.%, compounded monthly B. A high-interest savings account at 3.%, compounded daily, for the entire $2 11. In May, Stan borrowed $15 at 7.2%, compounded monthly, to buy a riding lawn mower for his summer business. He arranged to pay off the loan in months, with a single payment. a) What amount did Stan need to pay? b) What amount of interest did Stan pay? 12. Marlene has a student loan of $25 at.2%, compounded monthly. She plans to repay the loan over the net 5 years. The first payment will be due 1 month from now. a) Determine Marlene s monthly loan payment. b) What interest will she pay over the term of the loan? 13. Monty has a credit card balance of $52. The credit card company charges 19.5% interest, compounded daily. Monty decides to stop using his credit card and to make monthly payments so he can pay off his debt. a) How long will it take Monty to reduce his current credit card balance to zero if he pays $25 a month? b) If he doubles his monthly payment to $5, how much sooner will his debt be paid off? c) How much interest will he save if his monthly payment is $5 rather than $25? 1. Misty and Danielle are going to be starting their first year of university, and they need somewhere to live. Misty s parents decide to rent a one-bedroom apartment for her, at $ per month. Danielle s parents buy a four-bedroom house for $25, with a down payment of $5. They negotiate a 5-year mortgage at %, compounded semi-annually, with payments every month. They rent the other three bedrooms to students. Each student pays $75 per month. Misty and Danielle will both move after 5 years. a) Compare the housing costs for Misty s parents and Danielle s parents. b) Who made the wiser housing decision for their daughter? Eplain. NEL Cumulative Review 11

3 3-5 Cumulative Review 1. List five elements in each of the following sets. a) A 5 {provinces east of Alberta} b) B 5 {b b 5 1, [ I } 2. a) Represent these sets in a Venn diagram. U 5 {the days of the week} A 5 {the days you go to school} B 5 {the days of the weekend} C 5 {the days of the week that begin with the letter T} b) List a pair of disjoint subsets in part a), if any. c) Is each statement true or false? Eplain. i) A ( B iii) Ar 5 B ii) C ( A iv) n(a) 1 n(b) 1 n(c ) 5 n(u ) 3. Erynn surveyed 35 students about their favourite fruit. She recorded her results. a) Draw and label a Venn diagram to represent the data. b) Determine how many students like oranges or apples. c) Determine how many students like only oranges or only apples.. Describe two sets A and B to which each of the following formulas would apply. a) n1a c B2 5 n1a2 1 n1b2 b) n1a c B2 5 n1a2 1 n1b2 2 n1a d B2 5. Le Centre de développement musical, La Girandole, and L UniThéâtre offer a summer camp every year in Edmonton. At the Camp Multi-Arts, students can receive instruction in at least one of the following: singing, dancing, and acting. Of the 5 campers: 35 were instructed in singing, 3 in dancing, and 33 in acting. 21 were instructed in singing and dancing. 1 were instructed in singing and acting. 23 were instructed in acting and dancing. How many campers received instruction in all three art forms?. Consider this conditional statement: If a number is negative, then it is less than zero. a) Determine if the statement is true. b) Write the converse. Is it true? Eplain. c) Write the inverse. Is it true? Eplain. d) Write the contrapositive. Is it true? Eplain. e) Can this statement be written as a biconditional? Eplain. Favourite Fruit Number of Students oranges 1 apples 2 neither oranges nor apples NEL Cumulative Review 373

4 7. You roll a standard die and flip a coin. a) Draw a tree diagram to show all possible outcomes. b) Verify that you have listed all possible outcomes using the Fundamental Counting Principle.. Since 1991, Yukon licence plates have consisted of three letters followed by two numbers. a) How many different Yukon licence plates are possible? b) Of these different plates, how many use the same letter or the same number more than once? 9. Evaluate. a) 12! c) 1!!3! e) 9 C 5 b) P d) 9 P 5 f ) a 12 b 1. The camera club has 15 members. In how many ways can: a) All the members be arranged in a row for a yearbook photo? b) The club choose a president, vice-president, and treasurer? c) Jill and two other members be chosen to take pictures of the net football game? 11. A softball team has a record of 13 wins and 5 losses. How many different ways could this record have occurred? 12. Determine the number of possible routes to get from point A to point B, if you travel only south or east. a) A b) A B 13. Solve for n: nc 5 n P 3 1. There are 7 parents and 9 students who volunteer to be on a school fundraising committee. How many different si-person committees are possible if the committee must consist of: a) 3 parents and 3 students? b) At least 1 student? c) More parents than students? B 37 Chapter 5 Probability NEL

5 15. How many different three-card hands with one ace and one ten can be dealt from a standard deck of 52 playing cards? 1. Amber and Jackson take turns rolling two standard dice. If both numbers are even on her roll, Amber wins a point. If both numbers are odd on his roll, Jackson wins a point. If one number is even and the other is odd, neither wins a point on their roll, but they do roll again. The first player to 1 points wins. Is this a fair game? Eplain. 17. The odds against a post-secondary graduate being employed one year after graduation are estimated to be 1 : 25. a) Estimate the odds in favour of a post-secondary graduate being employed one year after graduation. b) Estimate the probability that a post-secondary graduate will be employed one year after graduation. 1. Si students are running for student council president. Suppose that each student is equally likely to win the election. What is the probability that Sonya will finish in the top three on election day? 19. In the game of bridge, four players are dealt 13 cards from a well-shuffled standard deck of 52 playing cards. Determine the probability that a player: a) Is dealt all the hearts. b) Is dealt all the kings. 2. Describe a pair of events that can be classified by each of the following descriptions: a) Mutually eclusive b) Dependent c) Independent 21. Enzo rolls two standard dice. Determine the probability of each event. a) The sum is 3 or 12. b) The sum is odd, or the sum is greater than 7. c) The sum is even, given that the first die is 1. d) The first die is less than 3, and the second die is greater than Jan and Stan are evenly matched table tennis players. However, each time Jan loses a game, her probability of winning the net game decreases by 1. When she wins, her probability of winning the net game increases by 1. If they play two games, determine the probability that: a) Jan wins at least one game. b) Stan wins both games. 23. A euchre deck consists of the 9, 1, jack, queen, king, and ace of all four suits. This results in a euchre deck of 2 cards. Audrey draws two cards from a well-shuffled euchre deck. Determine the probability that both cards are queens: a) If the first card drawn is replaced. b) If the first card drawn is not replaced. NEL Cumulative Review 375

6 - Cumulative Review Energy Consumed Month (degree days) Determine the -intercepts, the y-intercept, the end behaviour, the domain, and the range for each function shown. a) y b) y Determine the degree, the sign of the leading coefficient, and the constant term of the equation of each polynomial function shown in question Determine the possible number of -intercepts, the y-intercept, the end behaviour, the possible number of turning points, the domain, and the range of each function. a) f c) u b) v d) w The amount of energy consumed to heat a building on the first day of each month over 15 months is shown in the table. a) Create a scatter plot and determine the equation of the cubic regression function that models the data. b) Use your graph to estimate the time period when the energy consumed to heat the building is less than 15 degree days c) Use your cubic regression equation to determine the degree days needed to heat the building on the day that corresponds to the end of the seventh month. State any assumptions you made. 5. Predict the characteristics of each eponential function. Record your predictions in a table like the one shown on the top of the net page. Verify your predictions using graphing technology. a) y b) y Chapter Sinusoidal Functions NEL

7 Number of -Intercepts y-intercept End Behaviour Domain Range Increasing or Decreasing?. The population of the regional municipality of Wood Buffalo, Alberta, has grown rapidly since the discovery of one of the world s largest oil deposits. Its population in 199 was 35 and has grown since at an annual rate of %. a) Create a table of values that shows the population of Wood Buffalo from 199 to 2. b) How do you know that an eponential function can be used to model this contet? c) The eponential function that models the population growth of Wood Buffalo is y where represents the time in years and y represents the population. Determine the y-intercept of this function. What does it represent in this contet? d) Graph the function. Use the graph to estimate i) The population in 211. ii) When the population eceeded A 2 g sample of radioactive polonium-21 has a half-life of 13 days. The mass of polonium, in grams, that remains after t days can be modelled by the eponential function M1t2 5 2a 1 t 2 b 13 a) Create a graph of the function. b) Determine the mass that remains after 5 years. c) Determine the length of time needed for the sample to decay to 5 g.. A biologist tracks the population of an insect in a controlled environment over several years. Year (t) Population P(t) a) Create a scatter plot for the data. b) Determine the equation of the eponential regression function that models the insect s population growth. c) Eplain how the parameters in the equation are related to the contet. d) Determine when the population will eceed 1. NEL Cumulative Review 57

8 32 y Shown is the graph of y 5 25 log. State the -intercept, the number of y-intercepts, the end behaviour, the domain and the range. 1. Predict the characteristics of each logarithmic function by stating the -intercept the y-intercept the end behaviour the domain the range whether the function increases or decreases. Verify your predictions by graphing each function. a) y 5 3 log b) y 5 22 ln 11. Martin is a fruit grower in the Okanagan Valley. He has planted and tracked the growth of a new variety of cherry tree he is considering planting on 1 acres of his farm. Age of Tree (years) Height (feet) Age of Tree (years) Height (feet) a) Create a scatter plot for the data. b) Determine the equation of the logarithmic regression function that models the tree s growth. c) Determine the height of a tree of this variety when it is 15 years old. d) Determine the age of a tree of this variety when it is 12 feet tall. 12. a) Name these common characteristics of the graphs of y 5 sin and y 5 cos : the number of -intercepts the y-intercept the domain the range the period the amplitude the equation of the midline b) Sketch the graphs of both functions on the same ais where 5 # # p, [ R. 5 Chapter Sinusoidal Functions NEL

9 13. A group of students is tracking a friend, John, as he rides a ferris wheel. They created a graph that shows his height above the ground at various times on his ride. a) Describe this graph by determining its range, the equation of its midline, its amplitude, and its period. b) Discuss how these characteristics of the graph are related to the ferris wheel. 1. For each of the following sinusoidal functions, state the amplitude, the period, the equation of the midline, the domain, the range, and the horizontal translation. a) y 5 3 sin b) y 5 5 cos The average high temperature for each month in Vancouver, British Columbia, is shown in the following table. Month 1 represents January and Month 12 represents December. Average High Month Number Temperature ( o F) a) Create a scatter plot for the data. Why does assuming that this pattern repeats year after year make sense? b) Determine the equation of the sinusoidal regression function that models the relationship between the month of the year and average high temperature. c) Determine the average high temperature in Vancouver in the middle of July. d) Estimate when the average high temperature in Vancouver will be greater than o F. NEL Cumulative Review 59 Height (m) 12 f (t) 1 2 John s Height above the Ground 3 t Time (s)

10 3. a) Cumulative Review, Chapters 1 2, page 1 1. a) $2 2, $2 b) $59, $39 2. years 3. a) $5773.1, $773.1 b) $3 53., $ B. e.g., It has a more frequent compounding period that earns more interest. 5. estimate: $; actual: $ $ a) Emma b) Hans. a) $59.9 b) $ $ Option B. e.g., Option A is worth $ and option B is worth $ a) $ b) $ a) $2.7 b) $ a) 2 months b) 1 months sooner c) $ a) Misty s parents: $ ; Danielle s parents: $ 135. b) e.g., Danielle s parents, because if they resell the house they should make a profit. Cumulative Review, Chapters 3 5, page a) e.g., Manitoba, Québec, PEI, New Brunswick, Nova Scotia b) e.g., 1, 2, 3,, 5 2. a) U A Monday Wednesday Friday Tuesday C Thursday Saturday Sunday B b) 27 c) 1. a) e.g., odd whole numbers less than 1 and even whole numbers less than 1 b) e.g., odd whole numbers less than 1 and prime numbers less than a) Yes b) If a number is less than zero, then it is negative. Yes. e.g., All numbers less than zero must be negative. c) If a number is not negative, then it is not less than zero. Yes. e.g., All numbers that are not negative are either zero or positive, and all of these numbers are not less than zero. d) If a number is not less than zero, then it is not negative. Yes. e.g., All numbers not less than zero are either zero or positive, so they are all not negative. e) Yes. e.g., A number is negative if, and only if, it is less than zero. 7. a) Die Coin 1 tails 2 3 tails tails Answers b) A and B c) i) False; e.g., A and B are disjoint sets. ii) True; e.g., C is a subset of A. iii) True; e.g., A is the inverse of B. iv) False; e.g., A includes C. 5 tails tails tails b) outcomes from rolling a die and 2 outcomes from tossing a coin; # combined outcomes. a) b) a) 79 1 c) e) 12 b) 32 d) f ) a) b) 273 c) a) 12 b) 13. n Answers NEL

11 1. a) 29 b) 1 c) Yes. e.g., Both players have an equal chance of winning. 17. a) 25 : 1 b) 25 or about a) or about b) or about a) e.g., choosing an apple or a pear from a bowl of fruit b) e.g., choosing 2 blue marbles from a bag containing 7 blue and 3 red marbles, without replacement c) e.g., rolling a standard die and getting, tossing a coin and getting 21. a) 3 or about.33 3 b) 27 or.75 3 c) 3 or.5 d) 2 or about a) 1 1 or.25 b) 3 or a) 1 or about.27 3 b) 12 or about Cumulative Review, page 5 1. a) -intercepts: 23, 22, 2; y-intercept: 212; end behaviour: QIII to QI; domain: 5 [ R; range: 5 y y [ R b) -intercepts: 23, 1, 3; y-intercept: 29; end behaviour: QII to QIV; domain: 5 [ R; range: 5 y y [ R 2. a) 3, positive, 212 b) 3, negative, a) -intercepts: 1; y-intercept: 22; end behaviour: QIII to QI; turning points: ; domain: 5 [ R; range: 5 y y [ R b) -intercepts:, 1 or 2; y-intercept: 2; end behaviour: QIII to QIV; turning points: 1; domain: 5 [ R; range: 5 y y # 2, y [ R c) -intercepts: 1, 2 or 3; y-intercept: 22; end behaviour: QIII to QI; turning points: 2; domain: 5 [ R; range: 5 y y [ R d) -intercepts: 3; y-intercept: ; end behaviour: QII to QIV; turning points: 2; domain: 5 [ R; range: 5 y y [ R. a) Energy Consumed to Heat a Building 12 y Month Energy consumed (degree days) Answers y b) e.g., from halfway through month to three quarters of the way through month 13 c) e.g., 27 degree days; there are 31 days in the seventh month, so NEL Answers 21

12 5. Number of End Increasing or -Intercepts y-intercept Behaviour Domain Range Decreasing? 5 QII to QI 5 [ R 5 y y., y [ R increasing 3 QII to QI 5 [ R 5 y y., y [ R decreasing a) y b). a) 1 y Year Population d) 7. a) Population Population of Wood Buffalo, Alberta 12 y Years since 199 i) ii) during 29 Mass (g) Polonium-21 Half-Life 25 M(t) Time (days) b).2 g c) 27 days. a) Insect Population P(t) Population Year b) P1t t c) e.g., a represents the initial population, , b represents the growth factor, d) about years 9. -intercept: 1, y-intercept: none, end behaviour: QIV to QI, domain5., [ R, range: 5y y [ R t t Answers b) e.g., The rate of growth is constant, and growth occurs rapidly. c) 35 ; The y-intercept represents the initial population (in 199). NEL Answers 7

13 1. a) -intercept: 1, y-intercept: none, end behaviour: QIV to QI, domain: 5., [ R, range: 5 y y [ R, function: increasing b) -intercept: 1, y-intercept: none, end behaviour: QI to QIV, domain: 5., [ R, range: 5 y y [ R, function: decreasing y y a) range: 5 y 1 # y # 11, y [ R, midline: y 5, amplitude: 5 m, period: 9 s b) e.g., The Ferris wheel starts 1 m off the ground and over 9 s rotates to 11 m and back. The ais of the wheel is m off the ground and the wheel has a radius of 5 m. 1. Equation of the Midline Domain Range Horizontal Translation Amplitude Period a) 3 9 y [ R 5 y 21 # y # 5, y [ R 3 right b) 5.2 y [ R 5 y 27 # y # 3, y [ R. left 15. a) Average High Temperature in Vancouver, BC y Month e.g., The average monthly temperatures should be relatively consistent from year to year. b) y sin c) 7. F d) e.g., from early May to early October Average high temperature ( F) 11. a) b) y ln c) about 22. ft d) about 2. years old 12. a) number of -intercepts: multiple; y-intercept: sin, ; cos, 1; domain: 5 [ R; range: 5 y 21 # y # 1, y [ R, period: 2p; amplitude: 1; equation of the midline: y 5 b) Height (ft) y y Cherry Tree Height Age of tree (years) sin Sin vs. Cos cos 2 3 Answers NEL