Applied Math 40S Exam June 2002
PERSONAL FINANCE Total: 10 marks
Bonnie has recently inherited $10 000 and has decided to invest it for her retirement. She will invest in a fixed income deposit paying 6% per annum, compounded annually.
She has narrowed her options down to these two:
Option A: Invest the money in an RRSP
By investing in a tax-sheltered plan, she will receive a tax rebate equal to 39% of her investment. The first year, she invests $10 000. Only in the second year, she adds the tax rebate of $3900 to her initial investment. Use TVM Solver or create a spreadsheet like the one shown below to calculate the value of the investment after 10 years.
(4 marks)
Year / Opening Balance / Interest (6%) / Closing Balance1 / $10 000.00 / $600.00 / $10 600.00
2 / $10 600.00 + $3900.00 = $14 500.00 / $870.00 / $15 370.00
3 / $15 370.00
Closing Balance after 10 years: ______
She withdraws the money from the RRSP after 10 years. The amount is taxed at a rate of 22%.
Income taxes paid: ______
How much money will remain after all taxes have been paid? ______
Show TVM Solver values or print spreadsheet
Option B: Compounded Canada's Savings Bond
Under this plan, there is no tax rebate. For the first 9 years, all interest earned is taxed at an income tax rate of 39% a year. In the 10th year, she retires and her tax rate drops to 22% a year. Use the TVM Solver or create a spreadsheet using the headings shown below to calculate the value of the investment after 10 years.
Year / Opening Balance / Interest (6%) / Closing Balance / Income tax on interest1 / $10 000.00 / $600.00 / $10 600.00 / $234.00
2 / $10 600.00 / $636.00 / $11 236.00 / $248.04
Closing Balance after 10 years: ______
Total income taxes paid: ______
How much money will remain after all taxes have been paid? ______
TVM solver values: (or print your spreadsheets)
Use your results to answer the following questions:
a) Give a reason why she would choose to invest in Option A.
b) Give a reason why she would choose to invest in Option B.
Design & Measurement
Many stores sell material to build shelving units. The design for one such unit is shown below. Your task is to determine the cost of building such a unit by answering the following questions.
Three dimensional view
COSTS:
The shelves, sides, and top are made from %-inch thick laminated particle board that is purchased in sheets. Each 48 inch x 96 inch sheet costs $20.00. Each 16 inch x 96 inch sheet costs $8.00. All edges of the sheets are unfinished. All visible edges of the shelving unit must be laminated. Laminated edging costs $0.15 per foot. The hardware cost of one shelving unit is $10.00. All prices are subject to PST and GST.
a) The backing for the shelves is cut from hardboard that costs $7.00 for a 4 foot x 8 foot (48 inch x 96 inch) sheet. Note that only full hardboard sheets can be purchased. How many backs can be cut from one sheet assuming no material is lost in cutting pieces? (Show how you will cut the sheet, with the use of a scale diagram. Indicate the scale of your diagram.)
b) Calculate the total cost of laminating a shelving unit.
c) Sketch two possible layouts for the sides, top, and shelves of the unit: one using the 48 inch x 96 inch sheet(s) and the other using the 16 inch x 96 inch sheet(s). Assume no material is lost in cutting pieces.
d) Calculate the total cost for building each shelving unit according to your layouts in c) by itemizing the materials needed for each unit. Include costs for laminating, backing, hardware as well as taxes.
Which one of your layouts would result in the lowest price? Support your answer.
SECTION A: MULTIPLE-CHOICE Value: 11 marks Suggested Time: 20 minutes
For questions 1 and 2, refer to the following matrices:
1.
What are the dimensions of matrix P?
A) B) 2x3 C) 3x2 D) 6
2. Which of the following matrix operations cannot be performed?
A) P+Q B) Q*R C) Q-P D) Q*P
3. Which of the following is a vector quantity?
A) a train traveling North from Winnipeg
B) a computer operating at a speed of 333 MHz
C) a motorcycle travelling at 50 km/hr
D) a car travelling NW at 100 km/hr
4. Which of the following vectors has a bearing of 220ºC?
A B C D
5. A bag contains 15 balls, numbered 1 to 15. One ball is picked randomly. What is the probability that the number picked is divisible by 2 or divisible by 3
A) B) C) D)
6. What percentage of the area under the standard normal curve lies between z-scores of -1.8 and 0.7?
a) 20.6 % b) 72.2 % c) 75.8 % d) 96.4%
7. As a part of a physical fitness test, 500 students were required to run a race and their times were recorded. The results were normally distributed with a mean of 5.8 minutes and a standard deviation of 0.3 minutes. The approximate number of students with a time less than 5.5 minutes was:
a) 421 b) 170 c) 79 d) 34
8. Which of the following graphs has a period of 2?
9. Which of the following sinusoidal equations has a vertical shift of 1 unit up when compared to y = 3sin x?
a) y = 3sin(x+1) –4 b) y = 3sin(x-4)-1
c) y = 3 sin(x+4) +1 d) y = 3sin (x+1)
10. Based on data from satellite observations, recent estimates report 775 million hectares of rainforest remaining. If on average, 1.3% of the rainforest is being destroyed every year, how many million hectares of rainforest will remain after two years?
A) 586.60 B) 754.98 C) 764.93 D)772.40
11. Equilateral triangles are formed with coins. Three triangles are illustrated below. What is the total number of coins necessary to form the 7th triangle?
12. Total: 4 marks: Some people lease a new vehicle every year. The switching behaviour of these people has been studied to determine future leasing trends. These switching behaviours are given below.
Leasers of North American made vehicles:
15% switch to a Japanese model
10% switch to a German model
10% switch to other brands
Leasers of Japanese made vehicles:
25% switch to a North American model
15% switch to a German model
5% switch to other brands
Leasers of German made vehicles:
40% switch to a North American model
10% switch to a Japanese model
10% switch to other brands
Leasers of other made vehicles:
20% switch to a North American model
30% switch to a Japanese model
10% switch to a German model
a) Complete the transition matrix :
b) The year 2001 market share held by each of these major manufacturers is given in the following table.
North American / Japanese / German / Others55% / 30 % / 10% / 5%
State the projected market share for each vehicle manufacturer for the year 2002. Express your answer as a labelled matrix.
c) Briefly describe how you would determine if the market stabilizes.
13. A clothing store had a 3-day sale and advertised pants for $19.50, shirts for $12.99 and hats for $8.50. The daily sales for each item are given in the following matrix:
day1 day2 day3
a) What does the value 150 represent in the matrix?
b) Carmen was asked to determine the store's income for each day. She multiplied the matrices as shown below: Explain why Carmen's strategy will not result in the correct answer. Illustrate a strategy to arrive at the correct solution.
14. Norman leaves for his morning bike ride. He travels 35° E of N at 16 km/hr for 45 minutes. He then turns and bikes in a direction of 47° W of N at 15 km/hr for 20 minutes. How far is he from his original starting point? Provide a clearly labelled vector diagram for this situation.
15. Manuella wishes to roller blade to school. Manuella's route to school is N 30° E for a distance of 4 km and then W 20° N for a distance of 8 km.a) Sketch a vector diagram of this situation and calculate the direct distance between Manuella's house and the school. Show how you arrived at your answer.
b) From her school, Manuella wishes to skate back home by skating between 13 km and 15 km. Manuella does not wish to intersect with her original route.Sketch a route she could take and clearly indicate all distances and directions.
16. Consider the map below. Bonnie leaves from Carman and travels in the general direction of south or east. Calculate how many different ways she can drive from Carman to Altona. Illustrate how you arrived at your answer.
17. In a given population, the probability of being infected with a known bacteria is 30%. When tested for the bacteria, 90% of those infected show the positive indicator that they are infected, while 20% of those who are not infected also show the positive indicator that they are infected.
a) Use a diagram to show the situation described above indicating all probabilities.
b) What is the probability that a person selected randomly will show the positive indicator when tested?
18. Mrs. Ducharme wishes to select one group of 5 students from a class of 20 to present a problem.
a) In how many different ways can the group be selected?
b) Each of the 5 students in the group will be responsible for presenting a solution. In how many different orders could the 5 students present their solution?
19. A vending machine has items priced at $0.25, $0.50, $0.75 and $1.00.
a) Show all the ways that a one-dollar coin can be spent if no change is to be left at the end of purchases. The order of purchases does not matter.
b) What is the probability that three or more items will be purchased with this amount?
20. The following data represent the marks of a Senior 4 Applied Mathematics class on their final test.
58 71 69 67 58 91 88 75
77 43 83 37 94 64 79
a) Find the population standard deviation for the class marks. Round your answer to two decimal places.
b) The mean of the set of data above is 70.27. What would be the z-score of a student with a mark of 43?
c) Explain how the data compares to a normal distribution.
21. A group of 200 people are taking the driver's licence test. Previously, the pass rate for the test was 60% of the population. The remaining people do not pass the test.
a) Determine the mean and population standard deviation for the people who pass the test.
b) Explain the type of distribution for this data.
22. The following chart gives the average monthly temperatures for Morris, Manitoba in °C.
Jan. / Feb. / Mar. / April / May / June / July / Aug. / Sept. / Oct. / Nov. / Dec.-23.6 / -24.3 / -19.7 / -10.9 / -0.4 / 9.2 / 15.1 / 15.9 / 11.2 / 2.5 / -8.0 / -17.6
a) Using January as Month 1, represent the data using a sinusoidal equation.
b) According to your equation, what is the maximum temperature for this location?
23. The centre of a bicycle wheel is 33.0 cm above the ground. A reflector mounted on the spokes is located 17.5 cm from the centre of the wheel that is turning at a constant rate of one revolution per second. An equation that represents the situation is (expressed in radians):
y =17.5sin (6.28x - 4.71) + 33
where y represents the height in cm and x represents time in seconds.
a) Sketch and label a sinusoidal graph of this equation for two revolutions.
b) At what time does the reflector first pass through a height of 25 cm above the ground?
c) How would the equation change if the reflector was moved 2 cm further from the centre of the wheel?
24. A rookie player is presented with two different contracts for his first year of playing hockey as follows:
Contract A: He will receive $20 000.00 in each month for the duration of the season.
Contract B: He will receive $16 000.00 in Month 1. Each month, this amount will increase by 8%.
a) If the rookie player accepts Contract B, what will he earn in Month 3?
b) The regular hockey season lasts 6 months. After 6 months, only the top teams will have an additional 2 months of play-offs.
Calculate the player's total earnings under each contract for the entire season including play-offs.
c) Which contract would you advise him to accept? Support your answer. State any assumptions you have made.
25. The original drawing of an 81 cm by 81 cm square is shown below. Successive generations are formed by dividing each side of the figure into three equal segments. New squares are thus formed on the exterior (outside) of the face as shown below.
a) Draw Generation 2. The original is provided below:
Original Generation 1 Generation 2
b) What is the total area of the figure in Generation 2?
26. A veterinarian prescribes a 35 mg pill to be given to a cat each morning. Thirty percent of the medication is eliminated from the cat's body over the next 24 hours.
How many milligrams of the medication remain in the cat's body immediately after the fourth pill is given? Show how you arrived at your answer.