Applied Math 40S Jan 2019

i G RADE 12 A PPLIED M ATHEMATICS A CHIEVEMENT T EST D ESCRIPTION Total Possible Marks: 64 Time: 3 hours Unit Marks Relations and Functions 14 Probability 17 Financial Mathematics 16 Design and Measurement 9 Logical Reasoning 8 D IRECTIONS Remember to • indicate your input values by writing them in your booklet or printing a copy if using a technology tool • state any assumptions you make • express your answers in decimal and percentage form to at least the nearest hundredth (two decimal places) when rounding, except for monetary values or when otherwise indicated Example: 15 0.52 or 51.72% 29 = Note: Rounding too soon in your solution may result in an inaccurate final answer for which full marks will not be awarded. A “graphic organizer” is a visual representation of information. Examples include a tree diagram, a chart, a list, a Venn diagram, a truth table, Pascal’s triangle, etc. A clearly communicated answer — is easily identified in the response space — includes the parameters in the equation, and “ y =”, “sin”, “ln”, or “ x ”, as applicable — includes the units of measure, where applicable — includes labels, units, scales for the axes on graphs, and key characteristics of functions (e.g., maximum, minimum, intercepts, and appropriate shape) — is expressed as an exact value or is appropriately rounded Marks may be deducted for errors relating to any of the above. Electronic communication between students through phones, email, or file sharing during the test is strictly prohibited.

2 Applied Mathematics: Student Booklet (January 2019) Question 2 Total: 1 mark Select the best answer. 102 Which of the following functions has a domain of ( ) 0, ? ∞ A) 3 x y = B) 2 –3 18 – 18 y x x = + C) 3 2 3 2 y x x x = + + D) 3ln y x =

Applied Mathematics: Student Booklet (January 2019) 5 Question 5 Total: 3 marks A bridge is designed to move up and down with the wind. The graph below represents the vertical position of a point at the top of the bridge on a windy day. 107 a) Determine a regression equation that represents this situation. Show your work. (2 marks) Time (s) Vertical Position (cm) 108 b) Determine how many times the point at the top of the bridge is at its maximum height during the first 60 seconds. (1 mark) Vertical Position (cm) Vertical Position as a Function of Time 40 Time (s) 0 – 40 15 30

8 Applied Mathematics: Student Booklet (January 2019) P ROBABILITY Question 7 Total: 1 mark Select the best answer. 111 A building has 8 doors that can be used to either enter or exit. How many ways can Cindy enter and exit the building if she must enter through one door and exit through another? A) 15 B) 16 C) 56 D) 64

Applied Mathematics: Student Booklet (January 2019) 11 Question 10 Total: 2 marks 115 Each housing area in Manitoba is assigned a postal code. • Each postal code is made up of 3 letters and 3 digits that alternate. • The postal code must begin with the letter R. • The letters D, F, I, O, Q, U are not used. • Repetition is allowed. Determine the number of postal codes that can be created in Manitoba. Show your work.

14 Applied Mathematics: Student Booklet (January 2019) Question 13 Total: 4 marks Of the 41 students in Grade 12, • 21 students are in drama club • 17 students are in chess club • 14 students are in environmental club • 8 students are in chess club and drama club • 10 students are in drama club and environmental club • 3 students are in chess club and environmental club only • 2 students are in all three clubs 120 a) Draw a Venn diagram to represent this situation. (3 marks) 121 b) What is the probability that a randomly selected student is not in any of these clubs? (1 mark)

Applied Mathematics: Student Booklet (January 2019) 17 Question 16 Total: 5 marks Himesh wants to retire in 35 years. After meeting with his financial advisor, he determines that his portfolio will consist of the following two investments: Investment 1: $15 000.00 in a mutual fund that earns an interest rate of 6.50%, compounded monthly. Investment 2: regular biweekly deposits of $180.00 in a tax-free savings account (TFSA), with an interest rate of 3.75%, compounded biweekly. (Assume the initial value of the TFSA is 0.) 124 a) What will be the value of each investment when Himesh retires? Show your work. (4 marks) 125 b) Himesh’s goal is to have $500 000.00 in his portfolio by the time he retires. Determine if he will meet his goal. Justify your answer by using your investment values in (a). (1 mark)

18 Applied Mathematics: Student Booklet (January 2019) Question 17 Total: 5 marks Kaia purchased a house for $400 000.00. At the time of purchase, she made a down payment of $100 000.00. The remaining balance of the mortgage was financed at an interest rate of 4.30%, compounded semi-annually, over 25 years. 126 a) How much is Kaia’s monthly mortgage payment? Show your work. (2 marks) 127 b) Kaia’s house appreciates in value at an average rate of 2.00% per year. What will the value of her house be after 15 years? (1 mark) 128 c) How much equity will Kaia have in the house after 15 years? Show your work. (2 marks)

Applied Mathematics: Student Booklet (January 2019) 19 Question 18 Total: 3 marks Dimitri owns a house with a mortgage of $175 000.00. He has $165 000.00 in other loans and his net worth is $300 000.00. 129 a) Determine his debt-to-equity ratio. (1 mark) 130 b) Determine his total assets. (1 mark) 131 c) Dimitri wants to borrow $10 000.00 to take his family on a vacation. Using his debt-to-equity ratio in (a), explain if the bank will lend him the money. (1 mark)

20 Applied Mathematics: Student Booklet (January 2019) D ESIGN AND M EASUREMENT Question 19 Total: 1 mark Select the best answer. 132 Maala needs to shovel a 1 m × 2.8 m walkway after a 30 cm snowfall. The maximum amount of snow each full shovel can remove is 0.04 m 3 per scoop. Assuming each scoop contains the maximum amount of snow, how many scoops will it take for her to remove all the snow from the walkway? A) 21 B) 70 C) 336 D) 2100 1 m 2.8 m

Applied Mathematics: Student Booklet (January 2019) 21 Question 20 Total: 2 marks 133 Hugo creates a scale model of a planet for art class. He wants to paint it. • The radius of the model is 20 cm. • He has one bottle of paint that will cover 12 000 cm 2 . • It is assumed that each coat applied requires the same amount of paint. How many full coats of paint can be applied using the one bottle? Show your work.

22 Applied Mathematics: Student Booklet (January 2019) Question 21 Total: 6 marks You are entered in a sandcastle building contest. Pails in the following shapes are available for use: Shapes are not drawn to scale. Your sandcastle design must: • be built on a 2 ft. × 1 ft. area • use each shape at least once • include a minimum of 5 shapes • be no more than 3 shapes high 134 a) Draw or describe a sandcastle that meets the above criteria. (2 marks) Cylinder 6 in. 9 in. Cube 12 in. 12 in. 12 in. Cone 6 in. 11 in.

Applied Mathematics: Student Booklet (January 2019) 23 135 b) Calculate the volume of sand required, in cubic feet, to build your sandcastle. Show your work. (3 marks) 136 c) If the organizers of the contest purchased sand for $1.11/ft 3 , what is the cost of the sand needed to build your design? Assume whole units do not need to be used since the sand has already been purchased. (1 mark)

24 Applied Mathematics: Student Booklet (January 2019) L OGICAL R EASONING Question 22 Total: 2 marks 137 An online app awards 0 points, 3 points, or 7 points for performing tasks. Once your score adds up to 25 points or more, you pass the level. State the scores that are not possible at any time during any level. Show your work using the list of numbers below. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Scores that are not possible: ____________________________________________________

Applied Mathematics: Student Booklet (January 2019) 25 Question 23 Total: 2 marks The universal set of living things includes: • the set of all birds ( B ) • the set of all plants ( P ) • the set of all owls ( O ) • the set of all dogs ( D ) • the set of all mammals ( M ) • the set of all roses ( R ) 138 a) Give one example of two disjoint subsets. (1 mark) 139 b) Using set notation, give one example of a set that is a subset of another. (1 mark)

26 Applied Mathematics: Student Booklet (January 2019) Question 24 Total: 2 marks A movie theatre employee tracked the number of people who purchased popcorn ( P ) and chocolate ( C ) in one month. The results are given below. 140 a) What percentage of people purchased popcorn? (1 mark) 141 b) Describe what ( ) P C ′  represents in this situation. (1 mark) P C 57% 16% 9% 18%

Applied Mathematics: Student Booklet (January 2019) 27 Question 25 Total: 2 marks Given the following conditional statement: “If you live in Manitoba, then you live in Canada.” 142 a) Identify what type of statement is logically equivalent to the above conditional statement. (1 mark) Select the best answer. A) converse B) inverse C) contrapositive 143 b) Write the statement you identified in (a). (1 mark) END OF TEST

28 Applied Mathematics: Student Booklet (January 2019) NO MARKS WILL BE AWARDED FOR WORK DONE ON THIS PAGE.

Applied Mathematics: Student Booklet (January 2019) 29 Formula Sheet: Applied Mathematics R ELATIONS AND F UNCTIONS F INANCIAL M ATHEMATICS y ax b = + 2 y ax bx c = + + 3 2 y ax bx cx d = + + + x y ab = ( ) ln y a b x = + ( ) sin y a bx c d = + + 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       = × P ROBABILITY ( ) ( ) ( ) ( ) – P A B P A P B P A B = +   ( ) ( ) ( ) P A B P A P B = ×  ( ) ( ) ( ) P A B P A P B A = ×  ( ) ! – ! n r n P n r = ( ) ! ! – ! n r n C r n r = D ESIGN AND M EASUREMENT Prism: Surface area 2 Volume Ph B Bh = + = 1 Pyramid: Surface area 2 1 Volume 3 B Ps Bh = + = 2 3 Cube: Surface area 6 Volume l l = = Rectangular prism: Surface area 2 2 2 Volume lw lh wh lwh = + + = ( ) Triangular prism: Surface area 1 Volume 2 bh l a b h bhl = + + + = 2 2 Square-based pyramid: Surface area 2 1 Volume 3 b bs b h = + = 2 3 Sphere: Surface area 4 4 Volume 3 r r π π = = 2 2 Cylinder: Surface area 2 2 Volume r rh r h π π π = + = 2 2 Cone: Surface area 1 Volume 3 r rs r h π π π = + =

30 Applied Mathematics: Student Booklet (January 2019)