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Copland CollegeMathematics Faculty Student’s Name: ...............................................

Unit: MA4 Optimisation and Modelling

Assessment Item: Investigative Task

Dates: Given: Week 5 Due: Friday, September 7, 2007

Weighting: This assignment contributes 16% of your final score for this unit.

CRITERIA FOR ASSESSMENT

In this project/assignment, students will be assessed on the degree to which they demonstrate:knowledge of a range of mathematical facts, techniques and relationshipsability to select and apply a range of mathematical skills in the solution of problemsability to interpret and communicate mathematical ideas in various formsability to develop logical arguments to support solutions, inferences and generalisationsappropriate use of technology.

SUBMISSION AND EXTENSION POLICY

i) SUBMISSION OF WORKAll work must be submitted no later than 3:00 pm on the due date whether it is

complete or incomplete and each submission must be accompanied by this Cover Sheet with the declaration overleaf signed.

If the student is absent on the due date, arrangements must be made by the student to comply with the above guidelines.

A Receipt will be issued to the student by the teacher receiving the work.

ii) GRANTING OF AN EXTENSIONStudents seeking an extension beyond the due date must apply to their teacher in writing

documenting reasons. After consideration of the reasons the teacher may give an extended date due which is applicable only to that particular student.

iii) LATENESS PENALTIESA penalty of 5% of the total possible mark per calendar day up to a maximum of 7 days

will apply to late assignments. (The penalty will not apply if those days are covered by a doctor’s certificate.)

Assignments that have not been submitted by 1 week after the due date (7 calendar days) will be deemed not to have been done for assessment purposes (as most projects/assignments will by then be returned to students).

APPLICATION FOR EXTENSION

GRANTED: Yes No NEW DATE: ...... / ...... / 2007

PROJECT SUBMITTED (Student to complete)

DATE: ...... / ...... / 2007TIME: .........................

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Copland College Maths Apps Assignment – September 2007

Copland College

YEAR 12 MATHS APPLICATIONSMA 4: Investigative Task Due: Friday, 7 September 2007

NAME:GUIDANCE AND AUTHENTICITY IN ASSESSMENT DETAILS.Students may have easy access to information contained on the Internet, CD-ROMs and other such sources. Although it may be very tempting to use a "cut and paste" technique, this practice is unacceptable, except when the author is acknowledged and details of the source are provided. Students may also wish to consult experts outside the school. Generally speaking, talking through an assignment and discussing related mathematical concepts is acceptable. However, it is not acceptable for students to receive written information on specific areas of an assignment or to have any part of the assignment completed for them.

Student declaration:Where the work submitted is not my own, I have acknowledged the contributions of others.

Signed: _________________________________

CRITERIA for ASSESSMENT: Your grade is based on the degree to which you demonstrate:

knowledge of mathematical facts, techniques and relationships (C1) appropriate selection and application of mathematical skills in mathematical modelling

and problem solving (C2) interpretation and communication of mathematical ideas in a form appropriate for a

given use or audience (C3) development of logical arguments to support solutions (C4) appropriate use of technology (C5)

SCORE for ASSESSMENT: Your score is based on marks achieved for each question:

Q1 (Criteria assessed C1, C2, C5) /11Q2 (Criteria assessed C1, C2, C5) /10Q3 (Criteria assessed C1, C2, C5) /11Q4 (Criteria assessed C1, C2, C5) /8Q5 (Criteria assessed C1, C2, C5) /6Q6 (Criteria assessed C1, C2, C5) /11Q7 (Criteria assessed C1, C2, C3, C4, C5) /20TOTAL /77

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Copland College

YEAR 12 MATHS APPLICATIONSInvestigative Task – Term 3/2007

To assist you with this assignment you should refer to the following internet site.

http://members.ozemail.com.au/~rdunlop/CoplandMain/MathsApps/MathsAppsMatrices_07.htm

If you do not have access to a graphics calculator for matrix operations the link above has a matrix multiplier applet that can be used.

Q1) a) The diagram below shows a spreadsheet used to store the sales of a small shop.

i) What was the number of sales of pins on Friday of week 1?ii) Which item sold least on Thursday of week 2?iii) Which item sold most on Monday of Week 1?iv) On what day were there no sales of plugs?

b) The sale price of each of the items is shown in the following table.

Nuts Bolts Pegs Pins Clips Taps Plugs$0.30 $0.40 $0.35 $0.20 $0.15 $3.37 $1.50

i) From the sales table from Weeks 1 and 2 construct two matrices to represent this information.ii) The value of each item on sale can also be expressed in matrix form. Construct this matrix.iii) Clearly demonstrate how these matrices can be used to determine the total value of sales on

(a) each day in week 1and (b) on each day of week 2. iv) Use the resulting matrices from part (iii) to show how the total value of sales can be calculated

for the two week period.

NOTE: For question (1) you must clearly demonstrate that the problem can be approached using matrix techniques.

Q2) For their holidays Tommy and Angelica are planning to spend time at a popular tourist resort. They will need accommodation at one of the local motels and they are not certain how long they will stay. Their initial planning is for is for three nights and includes three breakfasts and two dinners. They have gathered prices from three separate motels.

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The Bay View Motel: Rooms are $125 per night. A full breakfast costs $22 per person and therefore $44 for both of them. An evening meal for two usually costs $75 including drinks.

The Terrace Motel: Rooms are $150 per night, breakfast $40 per double and dinner costs on average $80.

The Staunton Star Motel: Rooms are $140 per night. Breakfast for two is $40 whilst an evening meal for two costs $65.

i) Write down a numbers matrix as a 1 x 3 row matrix representing the days stayed, breakfasts required and dinners required.

ii) Write down a 3 x 3 prices matrix summarising the costs of each of the items at the three motels.

iii) Clearly demonstrate how matrix multiplication can be used to determine total prices for each venue.

iv) Instead of Tommy and Angelica staying three nights, the alternative is to stay two nights. In that event they decide on having breakfast just once and one evening meal before moving on. Construct a new matrix to show this alternative. Recalculate the total cost for each venue.

v) Reconstruct the numbers matrix into a 2 x 3 matrix to show both the three night and two night scenario. Generate a new costs matrix that clearly shows the costs of staying at each venue.

Q3) Phil and Lil opened a new business in 1997. Their annual profit was $160000 in 2000, $198000 in 2001 and $240000 in 2002. Based on the information from these three years they believe that their annual profit could be predicted using the following model.

(Note: P is in dollars and P(t) represents the profit in the given year)

where t is the number of years after the year 2000, i.e., t = 0 gives 2000 profit

i) By appropriate substitution write three equations showing the profit for each of the years 2000, 2001 and 2002. Note each of these equations will contain a, b and c as unknowns.

ii) Write a suitable matrix equation in the form of AX = B iii) Solve this matrix equation to determine values for a, b and c.iv) If the profit was $130000 in 1999, does this profit fit the model as shown by the equation

above given the values of a, b and c that you have just determined.v) Phil and Lil believe that their profit will continue to grow according to this model. Predict

their profit in 2003 and 2005.

Q4) The following represents a transition matrix.

i) What is the probability of going from state P to state Q.

ii) What is the probability of going from state Q to state R in exactly two steps.

iii) What is the probability of going from state R to state P in exactly three steps.

iv) To what matrix do these transition matrices appear to converge after a large number of steps? Your answer needs to be correct to two decimal places.

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Q5) If Chuckie makes it to school on time he will make it on time the following day 85% of the time. However, if he doesn’t make it on time, he will be late the next day 60% of the time. i) Write down the transition matrix showing Chuckie’s chances of getting or not getting to

school on time. ii) Chuckie’s was on time on Monday. What are the chances that he will be,

a) on time on Wednesday?b) be late on Thursday?

Q6) Narnia is blessed with many wonderful attributes; however good weather is generally not considered to be one of these. It appears that they never have two nice days in a row. If they have a nice day, they are just as likely to have snow as rain the next day. If they have snow or rain, they have an even chance of having the same the next day. If there is change from snow or rain, only half of the time is this a change to a nice day.i) From this information write a transition matrix to summarise this information.ii) If Wednesday is raining find the probability that it is,

1. Fine on Friday2. Fine on Saturday3. Fine on both Friday and Saturday

iii) From this transition matrix determine the long term probability that it will be either raining, snowing or fine.

Q7)

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Note: To answer the questions in this section you must clearly show the matrix operations required. The arithmetic associated with such manipulations need not be shown, rather the results ONLY as determined by a graphics calculator or similar need be recorded.

1) Use the decoder matrix to check that the original message can be obtained. Remember you need only show how you operate on the matrices not the details of the calculations.

2) Use the code given to decode the message

21 12 1 9 22 15 18 25 20 22 2 21

21 1 2 25 10 12 0 20 23 1 21 20

8 1 21 10 15 2 5 23 3 6 12 4

3) Breaking code involving matrix multiplication is much more difficult. A chosen encoder matrix is required.

Suppose it is

The word SEND is encoded as x =

which is converted to

a) What is the coded form of SEND MONEY PLEASE ?

b) What decoder matrix needs to be supplied to the receiver so that the message can be read?

c) Check this decoder message by decoding the message.

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d) What is the special feature of this decoder matrix? Try another decoder matrix with the same feature. Encode and decode the word SEND

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Copland College

YEAR 12 MATHS APPLICATIONSInvestigative Task – Term 3/2007

Mark Sheet – Due Date: 7 September 2007 - NameItem Mark Your Mark

Assignment coversheet attached with signed authenticity declaration

1

Q1 A i) What was the number of sales of pins on Friday of week 1Q1 A ii) Which item sold least on Thursday of week 2? 1Q1 A iii) Which item sold most on Monday of Week 1? 1Q1 A iv) On what day were there no sales of plugs 1Q1 B i) From the sales table from Weeks 1 and 2 construct two matrices to represent this information.

2

Q1 B ii) The value of each item on sale can also be expressed in matrix form. Construct this matrix.

1

Q1 B iii) Clearly demonstrate how these matrices can be used to determine the total value of sales on (a) each day in week 1and (b) on each day of week 2

2

Q1 B iv) Use the resulting matrices from part (iii) to show how the total value of sales can be calculated for the two week period.

2

Total Question 1 11Q2 i) Write down a numbers matrix as a 1 x 3 row matrix representing the days stayed, breakfasts required and dinners required.

2

Q2 ii) Write down a 3 x 3 prices matrix summarising the costs of each of the items at the three motels.

2

Q2 iii) Clearly demonstrate how matrix multiplication can be used to determine total prices for each venue

2

Q2 iv) Instead of Tommy and Angelica staying three nights, the alternative is to stay two nights. In that event they decide on having breakfast just once and one evening meal before moving on. Construct a new matrix to show this alternative. Recalculate the total cost for each venue

2

Q2 v) Reconstruct the numbers matrix into a 2 x 3 matrix to show both the three night and two night scenario. Generate a new costs matrix that clearly shows the costs of staying at each venue.

2

Total Question 2 10Q3 i) By appropriate substitution write three equations showing the profit for each of the years 2000, 2001 and 2002. Note each of these equations will contain a, b and c as unknowns.

3

Q3 ii) Write a suitable matrix equation in the form of AX = B 2Q3 iii) Solve this matrix equation to determine values for a, b and c.

2

Q3 iv) If the profit was $130000 in 1999, does this profit fit the model as shown by the equation above given the values of a, b and c that you have just determined

2

Q3 v) Phil and Lil believe that their profit will continue to grow according to this model. Predict their profit in 2003 and 2005.

2

Total Question 3 11Q4 i) What is the probability of going from state P to state Q 2Q4 ii) What is the probability of going from state Q to state R in exactly two steps.

2

Q4 iii)What is the probability of going from state R to state P in 2____________________________________________________________________________________________


exactly three stepsQ4 iv) To what matrix do these transition matrices appear to converge after a large number of steps? Your answer needs to be correct to two decimal places.

2

Total Question 4 8Q5 i) Write down the transition matrix showing Chuckie’s chances of getting or not getting to school on time

2

Q5 ii A) Chuckie’s was on time on Monday. What are the chances that he will be, a) on time on Wednesday?

2

Q5 ii B) Chuckie’s was on time on Monday. What are the chances that he will be, b) be late on Thursday?

2

Total Question 5 6Q6 i) From this information write a transition matrix to summarise this information

3

Q6 ii 1) If Wednesday is raining find the probability that it is,1. Fine on Friday

2

Q6 ii 1) If Wednesday is raining find the probability that it is, 2. Fine on Saturday

2

Q6 ii 2) If Wednesday is raining find the probability that it is, 3 . Fine on both Friday and Saturday

2Q6 iii) From this transition matrix determine the long term probability that it will be either raining, snowing or fine

2Total Question 6 11

Q7. 1) Use the decoder matrix to check that the original message can be obtained. Remember you need only show how you operate on the matrices not the details of the calculations.

4

Q7. 2) Use the code given to decode the mess 4Q7 .3 a) What is the coded form of SEND MONEY PLEASE ? 4Q7.3 b) What decoder matrix needs to be supplied to the receiver so that the message can be read.

4Q7.3.c) Check this decoder message by decoding the message 2Q7.3 d) What is the special feature of this decoder matrix? Try another decoder matrix with the same feature. Encode and decode the word SEND

2

Total Question 7 20TOTAL 77

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