Applied Math 40S Jun 2016

Applied Mathematics: Student Booklet (June 2016) 1 G RADE 12 A PPLIED M ATHEMATICS A CHIEVEMENT T EST D ESCRIPTION Total Possible Marks: 62 Time: 3 hours Unit Description Marks A Relations and Functions 15 B Probability 16 C Financial Mathematics 17 D Design and Measurement 6 E Logical Reasoning 8 R ESOURCES You may use the following resources: • Formula Sheet (tear-out page at the back of this booklet) • one 8½ ″ × 11 ″ study sheet • ruler • graphing calculator, computer software, and/or app • Internet access for tools such as applets or mortgage payment calculators Use of the Internet to communicate or access other content, including but not limited to course notes, definitions, or conceptual information is not permitted during the test. Electronic communication between students through phones, email, or file sharing during the test is strictly prohibited.

4 Applied Mathematics: Student Booklet (June 2016)

Applied Mathematics: Student Booklet (June 2016) 7 Question 3 Total: 4 marks A certain medication destroys diseased cells in a person’s body. The number of diseased cells decreases for a short time after a dose of the medication is administered and then increases again. This situation varies sinusoidally and is modelled by the following equation: ( ) 350 sin 3.14 1.57 650 c t = + + where t represents the time (in weeks) and c represents the diseased cell count. 103 a) If the initial dose of medication is administered at 0, t = when is the second dose administered? (1 mark) 104 b) What is the range of this function? (1 mark) 105 c) A patient claims to feel well when the diseased cell count is below 500. At 2.5 t = weeks, will the patient be feeling well? Show your work. (2 marks) c 0 t

10 Applied Mathematics: Student Booklet (June 2016) Question 5 Total: 6 marks The ice thickness on Lake Mathitoba is measured weekly every winter. The data for one season is shown in the table below: Time (day) 7 14 21 28 35 42 49 Ice thickness (cm) 5.2 17.4 26.8 32.4 37.1 42.6 44.3 108 a) Determine the logarithmic regression equation that models this data. (1 mark) 109 b) Create a clearly labelled graph of the equation in (a). (3 marks)

Applied Mathematics: Student Booklet (June 2016) 13 Question 8 Total: 3 marks Evan is arranging 8 different bicycles in a rack at his school. 113 a) How many ways can the bicycles be arranged in the rack? (1 mark) 114 b) How many ways can the bicycles be arranged in the rack if 3 students want their bicycles placed together? Show your work. (2 marks)

16 Applied Mathematics: Student Booklet (June 2016) Question 11 Total: 3 marks Tim has a set of cards numbered 1 to 15. He randomly draws one card. Consider the following events: Event A: drawing a card that is a multiple of 2 Event B: drawing a card that is a multiple of 3 118 a) Are these events mutually exclusive? Justify your answer. (1 mark) 119 b) What is the probability of drawing a numbered card that is a multiple of 2 or a multiple of 3? Show your work. (2 marks)

Applied Mathematics: Student Booklet (June 2016) 19 F INANCIAL M ATHEMATICS Question 14 Total: 1 mark Select the best answer. 124 Which of the following graphs shows the amount of principal paid on each payment over a 25-year mortgage? A. C. B. D. Years Principal Paid Years Principal Paid Years Principal Paid Years Principal Paid

20 Applied Mathematics: Student Booklet (June 2016) Question 15 Total: 1 mark Select the best answer. 125 The price of a new car is $26 000 (taxes included). It depreciates at a rate of 20% per year. What is the approximate residual value of the car after 3 years? A. $10 400 B. $13 300 C. $15 600 D. $20 800 Question 16 Total: 1 mark Use the following information to answer this question and select the best answer. 126 Scenario 1: A loan of $30 000.00 at an interest rate of 7.00%, compounded monthly for 1 year Scenario 2: A loan of $30 000.00 at an interest rate of 7.00%, compounded _______ for 1 year Assuming no payments are made, which compounding period frequency in Scenario 2 would result in less interest than in Scenario 1? A. every two weeks B. daily C. semi-annually D. weekly

Applied Mathematics: Student Booklet (June 2016) 21 Question 17 Total: 2 marks 127 Mr. and Mrs. Murthy have a total monthly gross income of $6000.00. They are interested in purchasing a house with a mortgage payment of $1300.00 per month, annual heating costs of $2100.00, and annual property taxes of $3675.00. Calculate the gross debt service ratio (GDSR). Determine if a bank is likely to offer them a mortgage. Justify your answer.

22 Applied Mathematics: Student Booklet (June 2016) Question 18 Total: 2 marks 128 List two advantages of leasing a vehicle over buying.

Applied Mathematics: Student Booklet (June 2016) 23 Question 19 Total: 4 marks Gabrielle wants to buy a cabin at Lac du Bonnet that costs $165 000.00. She has saved $25 000.00 that she will use as a down payment. The bank will give her a 15-year mortgage for the balance at 3.49%, compounded semi-annually. 129 a) What will Gabrielle’s payment be every two weeks? Show your work. (2 marks) 130 b) If the cabin appreciates in value an average of 3.00% per year, calculate the appreciated value of the cabin after 10 years. (1 mark) 131 c) How much equity will Gabrielle have in the cabin after 10 years? (1 mark)

24 Applied Mathematics: Student Booklet (June 2016) Question 20 Total: 6 marks At the age of 30, Alfred began investing $350.00 monthly into an investment account at an interest rate of 7.00%, compounded monthly. When he turned 45, the interest rate on this investment decreased to 5.00%, compounded monthly. Alfred plans to continue making monthly investments until he retires at 58. 132 a) How much money will Alfred have in his account when he retires? Show your work. (3 marks) 133 b) Alfred withdraws $2000.00 per month from his account after he retires. If the interest rate remains at 5.00%, how many months can he withdraw $2000.00? (1 mark)

Applied Mathematics: Student Booklet (June 2016) 25 134 c) Alfred’s sister, Marianne, retires at the age of 60. Her portfolio is valued at $200 000.00, earning 5.00%, compounded monthly. If Marianne wants the money to last until she is 85 years old, what is the maximum she can withdraw each month? Show your work. (2 marks)

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Applied Mathematics: Student Booklet (June 2016) 27 D ESIGN AND M EASUREMENT Question 21 Total: 2 marks 135 Sheena is wrapping boxes using a roll of wrapping paper with a total area of 3.5 m 2 . Each box has a 20 cm width, 20 cm length, and 40 cm height. If Sheena uses 20% extra paper on each box, how many boxes can she wrap? Show your work.

28 Applied Mathematics: Student Booklet (June 2016) Question 22 Total: 4 marks A traditional plains tipi is built using a rectangular canvas that is twice as long as it is wide. The diagram below shows how a semicircle would be cut out of a canvas with dimensions of 30 ft. (length) and 15 ft. (width). (Diagram is not drawn to scale.) 136 a) Calculate how much canvas is left over after the semicircle has been cut out. (2 marks) 15 ft. 30 ft.

Applied Mathematics: Student Booklet (June 2016) 29 137 b) You would like to construct a tipi with both dimensions three times greater than those for the tipi in (a). Given that canvas is only sold in a rectangular shape, how much will the canvas cost if the price of canvas is $7.39/ft 2 , plus taxes? (Note: GST 5%, PST 8%) = = (2 marks)

30 Applied Mathematics: Student Booklet (June 2016) L OGICAL R EASONING Question 23 Total: 2 marks 138 Complete the truth table. p q ~ q (not q ) p ⇒ ~ q True True True False False True False False

Applied Mathematics: Student Booklet (June 2016) 31 Question 24 Total: 4 marks There are 160 Grade 12 students attending a high school. This semester, the science teacher noted that there are • 30 students in Biology • 54 students in Chemistry • 43 students in Physics • 11 students in both Biology and Physics • 8 students in both Biology and Chemistry • 9 students in both Chemistry and Physics • 3 students taking all three courses 139 a) Use a Venn diagram to represent this situation. (3 marks) 140 b) How many students are taking Biology or Chemistry? (1 mark)

32 Applied Mathematics: Student Booklet (June 2016) Question 25 Total: 2 marks 141 Paula is trying to solve the following puzzle. Each 2 × 2 box must contain the digits 1 to 4 only once. As well, each column and each row must contain the digits 1 to 4 only once. Paula indicates that the  must be a “4”. Explain why she is incorrect and solve the puzzle. 1 2 3 2 1 4 END OF TEST 

Applied Mathematics: Student Booklet (June 2016) 33 NO MARKS WILL BE AWARDED FOR WORK DONE ON THIS PAGE.

34 Applied Mathematics: Student Booklet (June 2016) NO MARKS WILL BE AWARDED FOR WORK DONE ON THIS PAGE.

Applied Mathematics: Student Booklet (June 2016) 35 Formula Sheet: Applied Mathematics Relations and Functions y ax b = + 2 y ax bx c = + + 3 2 y ax bx cx d = + + + x y ab = ( ) ln y a b x = + log b y a x = ( ) sin y a bx c d = + + ( ) cos y a bx c d = + + Financial Mathematics 72 t i = I Prt = 1 nt r A P n   = +     Net worth Total assets – Total liabilities = Debt-to-equity ratio (%) ( ) Total liabilities – Mortgage 100 Net worth = × Gross debt service ratio (%) Monthly Monthly Monthly mortgage property heating payment taxes costs 100 Gross monthly income   + +     = × Rate of return (%) Current value Previous value – of portfolio of portfolio 100 Previous value of portfolio       = × Probability ( ) ( ) ( ) ( ) or – and P A B P A P B P A B = + ( ) ( ) ( ) and P A B P A P B = × ( ) ( ) ( ) and P A B P A P B A = × ( ) ! – ! n r n P n r = ( ) ! ! – ! n r n C r n r = Design and Measurement Prism: Surface area 2 Volume Ph B Bh = + = Pyramid: Surface area ( slant height) 2 Volume 3 Ps B s Bh = + = = 2 3 Sphere: Surface area 4 4 Volume 3 r r π π = = 2 2 Cylinder: Surface area 2 2 Volume rh r r h π π π = + = 2 2 Cone: Surface area Volume 3 r rs r h π π π = + =

36 Applied Mathematics: Student Booklet (June 2016)