Applications of Mathematics 12 June 2004 Provincial Examination

Transcription

1 Applications of Mathematics June 004 Provincial Examination ANSWER KEY / SCORING GUIDE Part A: Multiple Choice Organizers CURRICULUM: Sub-Organizers A. Problem Solving A Integrated Throughout B. Number B Number Operations I C. Number C Number Operations II D. Patterns and Relations D Patterns E. Shape and Space E Measurement F. Shape and Space F 3-D Objects and -D Shapes G. Statistics and Probability G Chance and Uncertainty Q K C S CO PLO Q K C S CO PLO. D K.5 B B. A U.5 E E3. A U.5 B B. A U.5 E E 3. C U.5 B B 3. D U.5 E E 4. A U.5 B B 4. D U.5 E E 5. B H.5 B B 5. C H.5 E E 6. B H.5 B B 6. B H.5 E E, 4 7. D K.5 C C 7. D K.5 F F 8. D U.5 C C4 8. D U.5 F F 9. C U.5 C C3 9. A U.5 F F, F3 0. B U.5 C C4 30. B H.5 F F4. C U.5 C C4 3. A H.5 F F4. D H.5 C C 3. A K.5 G G5 3. C K.5 D D4 33. C K.5 G G 4. D K.5 D D 34. A U.5 G G 5. D U.5 D D4 35. A U.5 G G8 6. C U.5 D D4 36. B U.5 G G4 7. B U.5 D D 37. C U.5 G G5 8. C U.5 D D6 38. B H.5 G G8 9. B H.5 D D4 39. D H.5 G G8 0. B H.5 D D5, D6 40. B H.5 G G Multiple Choice = 60 s 046amak - - July 3, 004

2 Part B: Written Response Q B C S CO PLO a. U C C, C b. U C C, C c. 3 U C C, C a. 4 U G G b. 5 U G G c. 6 U G G 3a. 7 U 3 B B, B 3b. 8 U B B, B 4a. 9 U D D 4b. 0 U 3 D D 5a. U E E, E 5b. U E E, E 5c. 3 U E E, E 6a. 4 U 3 F F3 6b. 5 U F F3 Written Response = 30 s Multiple Choice = 60 (40 questions) Written Response = 30 (6 questions) EXAMINATION TOTAL = 90 s LEGEND: Q = Question Number K = Keyed Response C = Cognitive Level B = Score Box Number S = Score CO = Curriculum Organizer PLO = Prescribed Learning Outcome 046amak - - July 3, 004

3 PART B: WRITTEN RESPONSE Value: 30 s Suggested Time: 45 minutes. A couple plans to purchase a townhouse that will cost $0 000 after all fees and taxes are applied. They plan to provide a down payment of $ A financial institution is offering a mortgage rate of 4.5% per annum compounded semi-annually. a) What will the monthly payments be if they choose a 0-year amortization period for the mortgage? ( s) Using TVM Solver: 0-year mortgage: N = 40 I = 45. % PV = FV = 0 PY = CY = PMT : END Solve for PMT = amak July 3, 004

4 b) What will the monthly payments be if they choose a 0-year amortization period for the mortgage? ( ) Using TVM Solver: 0-year mortgage: N = 0 I = 45. % PV = FV = 0 PY = CY = PMT : END Solve for PMT = c) How much would they save if the mortgage is paid off in 0 years instead of 0 years? ( s) Total amount paid for a 0-year mortgage: $ ( ) = Total amount paid for a 0-year mortgage: $ ( ) = Difference: = = $ s was deducted if only the difference of the mortgage was found (ie ). 046amak July 3, 004

5 . The weights of males between the ages of 8 and 35 years in a certain city is normally distributed with a mean of 75.8 kg and a standard deviation of 3.8 kg. a) What is the z-score for a weight of 85 kg? ( ) x z = - m s = 38. =.4 b) What percent of this age group of males weighs more than 80 kg? ( s) ShadeNorm normalcdf ( ) = 80, E99, 75. 8, OR ( ) = 80, E99, 75. 8, ª 3. 45% c) What percentage of this group weighs between 60 and 70 kg? ( s) ShadeNorm normalcdf ( ) = 60, 70, 75. 8, OR ( ) = 60, 70, 75. 8, ª 634. % 046amak July 3, 004

6 3. Businesses in a certain community use either the postal service or a private courier to send their parcels. At present, 60% use the postal service and 40% use private couriers. A survey shows that 8% of those that use the postal service will continue to do so next year, and that 9% of those that use private couriers will continue using this method. a) If this trend continues, what percentage of businesses will be using private couriers after years? (Clearly show the initial state matrix and transition matrix used.) (3 s) PS PC I = [ ] From T = PS PC To PS PC È Í ÎÍ Ø PS PC K IT = [ ] \ approximately 5.46% b) Given that this trend continues for the long term, what percent of businesses will eventually be using the postal service? ( s) for any higher power concept. Ø PS PC IT 00 = [ K K] \ eventually 33.33% will be using the postal service. 046amak July 6, 004

7 4. On a certain day, the angle of elevation of the sun at the equator is measured at specific times on a 4-hour clock. Data for these measurements is as follows. Time (h) Elevation ( ) 06: : : : : : : a) Determine the sine regression equation for this data. ( s) Enter data into the calculator: Time as List and Elevation as List SinReg L, L: a = K b = K c = K d = K OR ( ) + y = 37. 0sin 0. 40x ( ) + y = Ksin 0. 40Kx K K 046amak July 3, 004

8 b) On this day, how long was the angle of elevation of the sun greater than 45? (3 s) Enter into the graphing calculator: Y Y ( ) + = 37. 0sin x = 45 Determine the points of intersection: x [ 0, 4] y [-0, 70] 930. h h 930. hto477. h = difference of 5.47 h \ the altitude of the sun is greater than 45 for 5.47 h (or 5h and 8 m). for concept subtraction Note: If intersection points time coordinates are not rounded to decimal places then the correct answer would be 5.46 h. 046amak July 3, 004

9 5. Some landscaping needs to be done to the front of an office building with dimensions as shown in the diagram. The cost of each item to be used is listed in the following table. Item Paving Stones ( m by m) Top Soil $0 Lawn Seed $5 Price $7.50 each 3 m 00 m 0 m Paving Stones Flower beds 0 m Lawn Building 0 m m 0 m 046amak July 3, 004

10 a) Determine the cost of the paving stones. ( ) 0 = 40 m Need $7.50 each = 40( 7. 50) = $ 300 Full s for approximate answers. b) Determine the cost of the flower beds if the top soil is 40 cm deep and the flower beds are 50 cm wide. ( s) Area of flower beds can be found in various ways = 9 m 3 ( ) = Volume of topsoil is m ( ) = Cost of topsoil in flower beds is 76. $ 0 $ 5 c) Determine the cost of preparing the lawn if it requires 0 cm of topsoil plus the seeding. ( s) ( ) + ( ) Area of lawn = = 446 m Cost of seeding = ( ) = $ Volume of topsoil = (. ) = m Cost of topsoil = 44. 6( 0) = $ 89 Cost of preparing lawns = $ 89 + $ = $ Ô Ô Ô Ô 046amak July 3, 004

11 6. An aircraft is cruising at 80 km h at [ 35 ]. The wind is blowing towards 070 at a speed of 50 km h. [ ] a) Use the axes provided to sketch the vectors and solve for the magnitude of the resultant velocity vector. (Answer to the nearest km h.) (3 s) R Æ N ( R) = + - ( )( ) cos 65 = R = 65. W 80 for E Answer = 65 km h for vectors without 65 Full s for precise scale diagram. S b) Determine the direction of the resultant vector. (Answer to the nearest degree.) ( s) q sin65 sin q = sin65 sin q = = K 65 - q = sin ( K) = 5. 9K \ the direction is = 33 [ ] Full s for students that correctly broke the vectors into components. END OF KEY 046amak - - July 3, 004