Applications of Mathematics 12 January 2001 Provincial Examination

Transcription

1 Applications of Mathematics January 00 Provincial Examination ANSWER KEY / SCORING GUIDE Part A: Multiple Choice CURRICULUM: Organizers Sub-Organizers. Problem Solving A Problem Set. Number B C Matrices Financial Decision-Making 3. Patterns and Relations D E F Fractals Linear Programming Non-Linear Functions 4. Shape and Space G H Periodic Functions Geometry Applications 5. Statistics and Probability I J Data Analysis Applications of Probability Q K C S CO PLO Q K C S CO PLO. D K B 4. C U 4 G3. A U B3 5. B H 4 G, G 3. A U B 6. A K 4 H 4. C U B 7. D U 4 H 5. A U B 8. D U 4 H 6. D U B3 9. C U 4 H 7. C H B3 30. D H 4 H 8. D U C 3. A H 4 H 9. D U C 3. B K 5 I 0. C H C 33. A U 5 I. D K 3 D 34. B H 5 I. B U 3 D 35. B K 5 J5 3. A U 3 D4 36. B U 5 J4 4. A U 3 D4 37. A U 5 J3 5. C H 3 D3 38. C U 5 J6 6. C K 3 E 39. C U 5 J7 7. D H 3 E3 40. B H 5 J, J 8. D K 3 F 4. B H 5 J5 9. B U 3 F3 4. C U A 0. B U 3 F3 43. C U A. B H 3 F 44. B H A. C U 4 G4 45. B H A 3. A U 4 G4 Multiple Choice = 45 s 0amak - - February 3, 00

2 Part B: Written Response Q B C S CO PLO a. U 5 I b. U 5 I a. 3 U B3 b. 4 H B U 3 C 4. 6 U 3 5 J U 3 A 6. 8 U 4 3 E U 3 F3 8a. 0 U 3 4 H 8b. U 4 H Written Response = 5 s Multiple Choice = 45 (45 questions) Written Response = 5 (8 questions) EXAMINATION TOTAL = 70 s LEGEND: Q = Question Number K = Keyed Response C = Cognitive Level B = Score Box Number S = Score CO = Curriculum Organizer PLO = Prescribed Learning Outcome 0amak - - February 3, 00

3 PART B: WRITTEN RESPONSE Value: 5 s INSTRUCTIONS: Suggested Time: 45 minutes Rough-work space has been incorporated into the space allowed for answering each question. You may not need all the space provided to answer each question. Where required, place the final answer for each question in the space provided. If, in a justification, you refer to information produced by the calculator, this information must be presented clearly in the response. For example, if a graph is used in the solution of the problem, it is important to sketch the graph, showing its general shape and indicating the appropriate window dimensions. When using the calculator, you should provide a decimal answer that is correct to at least two decimal places (unless otherwise indicated). Such rounding should occur only in the final step of the solution. Full s will NOT be given for the final answer only.. A restaurant chain gathered data to relate advertising costs and sales. This is summarized in the following table: Advertising Costs ($) Sales ($) a) What is the correlation coefficient for the least squares line of best fit? ( ) Enter data in graphing calculator r = amak February 3, 00

4 b) Determine the least squares linear regression equation and predict the expected sales if $0 000 was spent on advertising. (Give your answer to the nearest $ 000.) ( s) Using linear regression: y =. 36K x K if x = y = $ amak February 3, 00

5 . A survey revealed that 30% of a certain population smokes. Each year, 0% of the smokers quit and 5% of the non-smokers start. a) Determine the percentage of smokers after years. ( s) Transition matrix: From S NS To S NS [. 3. 7] = [ ] = 6% 0amak February 3, 00

6 b) Determine the percentage of smokers in the long term. ( ) = = 0%. 8. x y x y [ ] = [ ]. 8x+. 05y = x. x =. 05y 0x = 5y 4x = y x+ y = x+ 4x = 5x = x = = 0% 5 0amak February 3, 00

7 3. Amy deposited $ 000 into a savings fund earning 9% compounded annually on each of her 4 th, 5 th and 6 th birthdays. If she makes no additional deposits after these three, but leaves the accumulated amount in the account earning 9% compounded annually, how much will she have in the account when she retires on her 60 th birthday? (3 s) 000(.09) (.09) (.09) 34 = $ 787. Amy will have $ 787. in the account when she retires at age 60. First find the amount on deposit after three years. $ 000 $ $ $ ( )+ ( ) = Then find the amount when $ is left for an additional 34 years. 34 $ $ 787. ( ) = Amy will have $ 787. in the account when she retires at age 60. 0amak February 3, 00

8 3. Amy deposited $ 000 into a savings fund earning 9% compounded annually on each of her 4 th, 5 th and 6 th birthdays. If she makes no additional deposits after these three, but leaves the accumulated amount in the account earning 9% compounded annually, how much will she have in the account when she retires on her 60 th birthday? (3 s) TVM Solver: Enter: N = 3 I% = 9 PV = 0 PMT = 000 n FV = PY= CY= After three years Amy has $ on deposit. Enter: N = 34 I% = 9 PV = PMT = 0 n FV = PY= CY= at age 60 Amy will have $ 787. on deposit. 0amak February 3, 00

9 4. A newly developed antibiotic causes harmful side effects in 5% of the patients treated with the drug. If this medication is given to 00 randomly selected patients, what is the probability that 30 or more of them will suffer from these side effects? (3 s) n = 00 p = 05. q = 075. µ σ = 00( 0. 5) = 5 = 00( 0. 5)( 0. 75) = Using continuity correction z = = Area under curve is = Required probability is = Using the binomial cdf on a graphing calculator, binomcdf ( 00, 0. 5, 9) = s 0amak February 3, 00

10 5. A car rental company has 00 cars. The company can rent out all the cars if the price is $36 per day. With each $ increase per day, 5 fewer cars are rented out. What is the maximum net revenue? (3Ês) Revenue = ( no. of cars)( price per car) Let x represent the number of $ increases Revenue = 00 5x 36 x ( )( + ) R = x 0x ( ) ( ) Graph produces maximum at, 8 40 or completing the square shows vertex at, 8 40 Maximum net revenue is $8 40 No. of cars Cost $ Revenue $ s for chart By symmetry of quadratic function Maximum revenue occurs at $8 40 0amak February 3, 00

11 6. Fertilizer requirements and costs for maintaining the grass at a golf course are outlined in the table below. Type A Type B kilograms per m 3 Minimum required Phosphoric acid Nitrogen Potash Cost per m 3 $30 $35 Let x represent the number of m 3 of Type A fertilizer and y represent the number of m 3 of Type B fertilizer. List the constraints and objective function, then solve the linear programming problem to determine the minimum cost of fertilizer for the golf course. (4 s) 0x+ 0y x+ 30y 440 5x+ 0y 330 x 0 y 0 C = 30x+ 35y for constraints y 70 (0, 69) Feasible region Test corner points 30 (, 7) ( ) = ( ) = ( ) = ( ) = 0, 69 C $ 45, 7 C $ , 8 C $ , 0 C $ 980 minimum cost is $ (30, 8) (66, 0) x x + 0y x + 30y 440 for values at corner points and answer 0x + 0y 690 0amak - - February 3, 00

12 7. The formula D = 0 log gives the decibel level, D, of sound at a rock concert at r r rows back from the stage. How many rows back can Jen sit and still hear the music at a level of at least 80 decibels? ( s) If providing a graphical solution, state the function(s) used, sketch the graph, indicate appropriate window dimensions and clearly explain how your solution is derived from the graph. (.36068, 80) Y Y 0 = 0 log 5 0 x = 80 for equations for graph x [ 0, 00] y [ 0, 00] Jen can sit rows back and still hear 80 decibels of sound. 0amak - - February 3, 00

13 7. The formula D = 0 log gives the decibel level, D, of sound at a rock concert at r r rows back from the stage. How many rows back can Jen sit and still hear the music at a level of at least 80 decibels? ( s) If providing a graphical solution, state the function(s) used, sketch the graph, indicate appropriate window dimensions and clearly explain how your solution is derived from the graph. 0 Y = 0 log 5 0 x 80 for equation for graph (.36068, 0) Zero occurs at x [ 0, 30] y 5, 5 [ ] Jen can sit rows back and still hear 80 decibels of sound. 0amak February 3, 00

14 7. The formula D = 0 log gives the decibel level, D, of sound at a rock concert at r r rows back from the stage. How many rows back can Jen sit and still hear the music at a level of at least 80 decibels? ( s) If providing a graphical solution, state the function(s) used, sketch the graph, indicate appropriate window dimensions and clearly explain how your solution is derived from the graph. Algebraically: 0 log 5 0 r log 5 0 r r = 80 = 8 0 r = 0 8 = 500 r = 500 =. 36 Jen can sit rows back and still hear 80 decibels of sound. 0amak February 3, 00

15 8. A solarium is semicircular in shape with a radius of 9 m. It consists of a triangular-tiled area and gardens in the non-tiled area, as shown in the diagram. Garden 6 m Garden 9 m 4 m a) Determine the area of the garden. (3 s) cos θ = = ( )( ) θ = θ 6 m 9 m 9 m 4 m A = ( 6)( 4) t sin θ = m A A = π 9 A s t t = m 0amak February 3, 00

16 b) If the topsoil in the garden is 0. m deep and costs $5 topsoil. 3 m, determine the cost of the ( ) V = = m Cost = = $ END OF KEY 0amak February 3, 00