tasmanian certificate of education mathematics applied 2007

7/31/2019 Tasmanian Certificate of Education Mathematics Applied 2007

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Pages: 12Questions: 2

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Tasmanian Qualifications Authority.

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Tasmanian Certificate of Education

MATHEMATICS - APPLIED

Senior Secondary 5C

Subject Code: MAP5C

External Assessment

2007

Part 1 Algebraic Modelling

Time: approximately 36 minutes

On the basis of your performance in this examination, the examinerswill provide results on the following criterion taken from thesyllabus statement:

Criterion 6 Select and apply algebraic or graphical models toanalyse and solve problems in linear and non-linear modelling situations.

TASMANIA

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Mathematics Applied Part 1

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CANDIDATE INSTRUCTIONS

You MUST ensure that you have addressed the externally assessed criterion on this examination paper.

The 2007 Mathematics Applied 5C Formulae Sheet can be used throughout the examination.

No other printed material is allowed into the examination.

1. ALL questions in this part should be attempted.

2. Answers must be written in the spaces provided on the examination paper.

3. In total it is recommended that you spend approximately 36 minutes answering the questions in thispart.

4. Graph paper is provided in the booklet when required.

5. Logical and mathematical presentation of answers and the statement of the arguments leading toyour answer will be considered when assessing this part.

6. You are expected to provide a graphics calculator approved by the Tasmanian QualificationsAuthority.

All written responses must be in English.

Spare graph paper has been provided in the back of the booklet for you to use if required.

If you use the spare space, you MUST indicate you have done so in your answer to that question.


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Question 1 (Approximately 22 minutes)

Scientists have been measuring the level of carbon dioxide CO2( ) in the atmosphere at a sitehigh on Mauna Loa Mountain in Hawaii since 1950. The table shows the levels recordedevery five years from 1950 to 2000, in parts per million.

(a) Determine the linear equation that best describes the relationship between C and T.

Give your numbers to two decimal places.

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(b) Determine the correlation coefficient, r, and the coefficient of determination, r2 , forthe linear relationship found in (a).

Correlation coefficient, r.

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Correlation coefficient, r2 .

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Question 1 continues opposite.

T

Years since 1950

C

Carbon dioxide level

(parts per million)

0 302

5 308

10 314

15 320

20 326

25 332

30 339

35 345

40 352

45 359

50 367

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Question 1 (continued)

(d) Determine the exponential equation (that is C= a" bT OR C= a" ebT) that bestdescribes the relationship between Cand T. Give a to two decimal places and b tofourdecimal places.

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(e) Determine the correlation coefficient, r, and the coefficient of determination, r2 , forthe exponential relationship found in (d).

Correlation coefficient, r.

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Correlation coefficient, r2 .

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(f) Use your calculator to find a residuals graph for the exponential relationship, setting theWindow values as shown in the graph below. Draw the residuals graph below.


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(g) State which, of the linear or exponential relationship, is the better model for the data.

Give reasons for your choice.

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(h) Using your exponential function from (d), determine the CO2 level in 2005 ( 55=T )

and in 2050 ( 100=T ). Comment on the reliability of the answers.

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A certain East Coast motel has 25 rooms available for guests. Guests pay $75 per room forone night.

The motel managers find that if 0 to 13 rooms are occupied, their fixed costs are $500 per

night (staff wages, plus advertising and other costs) and in addition each occupied room coststhem $28 per night (food, laundry and some electricity costs).

(a) Find the managers cost equation for 0 to 13 occupied rooms per night and theirrevenue equation.

Cost equation: ..............................................................................................................

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Revenue equation: ........................................................................................................

If 14 to 25 rooms are occupied, the managers need to employ extra staff, and this costs theman additional $300 per night, but their variable costs are reduced to $26 per room per night.

(b) Find the cost equationif 14 to 25 rooms are occupied per night.

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(c) Plot all three equations from (a) and (b) and indicate the two break-even points on thegraph paper. Add appropriate labels to the graph.


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(d) Algebraically, determine the two break-even points for the motel business.

0 to 13 occupied rooms per night.

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14 to 25 occupied rooms per night.

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How many rooms must be occupied per night for the motel managers to make a profit?

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Spare Graph for Question 1 (c)

Spare Graph for Question 1 (f)


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Spare Graph for Question 2 (c)


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Senior Secondary 5C

Subject Code: MAP5C

External Assessment

2007

Part 2 Calculus


On the basis of your performance in this examination, the examiners

will provide results on the following criterion taken from thesyllabus statement:

Criterion 7 Use calculus to analyse and solve problems.

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3. In total it is recommended that you spend approximately 36 minutes answering the questions in this

part.



6. You are expected to provide a graphics calculator approved by the Tasmanian Qualifications

Authority.


Spare space for a graph has been provided in the back of the booklet for you to use if required.

If you use this spare page, you MUST indicate you have done so in your answer to that question.


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Carol is taking her first parachute jump from a rising balloon. The distance (D metres) she hasfallen t seconds after jumping, is given by the equation:

D = 4.9t2"1.9t for 0 # t#10

(a) How far has Carol fallen after 4.5 seconds?

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The derivative ofD,dt

dDis the speed of Carols descent.

(b) Finddt

dD, and use this to find how fast Carol is descending after 4.5 seconds. Use

appropriate units in your answer.

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A group of Maths Applied students decide to collect

popcorn from a popcorn making machine in 10-second

intervals. The table shows the total number of pieces of

popcorn (N) collected after tseconds. After 100

seconds, no more popcorn came out of the popcornmaking machine.

The students find that the equation

1000for096.000064.023

!!+"= tttN

provides an excellent model for their data.

(a) The derivative

dt

dNcan be used to find the rate of change in the amount of popcorn

produced by the popcorn making machine.

(i) What are the units ofdt

dN?

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(ii) What is the rate of change when 15=t seconds?

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(b) Use calculus to show that 23 096.000064.0 ttN +!= has a maximum value of 320

when 100=t and a minimum value of 0 when 0=t .

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Time,

t seconds

Number of popcorn

pieces, N

0 0

10 9

20 3330 69

40 113

50 160

60 207

70 251

80 287

90 311

100 320

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(c) Using your calculator to assist, sketch a graph of the function N= "0.00064t3+ 0.096t

2

on the axes below. The scale shown indicates how the window of your calculator should

be set up.

Show on your graph the coordinates of the maximum and minimum points and zeros.

(d) Explain why the equation N= "0.00064t3 + 0.096t2 cannot sensibly be used to model

the amount of popcorn produced by the popcorn making machine if t< 0 or if t>100.

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l

w

25m

15m

Existing building


A rectangular enclosure is to be built

with length l and width w, and the

enclosure is to join an existing building

at two corners as shown in the diagram.

The existing building is 25 metres

long and 15 metres wide.

A total of 240 metres of fencing material

can be used to build the enclosure.

(a) Show that the area (A) of the enclosure (notincluding the existing building) is given by

the equation A = "l2+140l " 375.

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(b) Use calculus to find the length l which will result in the area of the enclosure being a

maximum.

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(c) What is the maximum area?

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Spare Space for Question 4 (c) graph


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Senior Secondary 5C

Subject Code: MAP5C

External Assessment

2007

Part 3 Applied Geometry



Criterion 8 Use geometrical models to solve problems.

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Spare diagrams have been provided in the back of the booklet for you to use if required.

If you use either of the spare diagrams, you MUST indicate you have done so in your answer to

that question.


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In May this year, Ken Gourlay arrived in Devonport after completing an around the worldvoyage. In doing so he became the fastest and oldest Australian to sail solo, unassisted andnon-stop around the world.

(a) Ken completed his journey in 179 days 21 hours and 38 minutes, and he travelled a totalof 24361 nautical miles. Calculate his average speed (in knots) for the journey.

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(b) At one point on his journey, Ken observed that the sun was at its highest point in the skywhen his chronometer showed that the time was 9:30 am GMT. What was his longitudeat this position?

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(c) At another point on his journey, Ken sailed close to Ilha da Trinidade, an island off thecoast of Brazil. The highest point of this island is 405 metres above sea level. How faraway from the island would Ken have been when he could just see the highest point ofthe island on the horizon? Assume that the radius of the Earth is 6371 km.

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(d) During the final part of his journey, Ken sailed from the position 41S 21E (near CapeTown, South Africa) due east to 41S 143E (near Tasmania).

Calculate the distance covered in this part of the journey, in nautical miles to the

nearest whole number. Working must be shown.

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Qantas flight QF93 leaves Melbourne (38S 145E) and flies non-stop to Los Angeles, USA(34N 118W).

(a) What is the standard time difference between Melbourne and Los Angeles?

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(b) Determine the shortest distance between Melbourne and Los Angeles. Give youranswer to the nearest kilometre and show working. Assume that the radius of the earthis 6371 km.

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.......................................................................................................................................Question 7 continues opposite.

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The flight leaves Melbourne at 11:20 am (local standard time) on 24 th November and arrivesin Los Angeles at 7:30 am (local standard time) on 24th November, taking the shortestdistance for the journey.

(c) Determine the average speed of the flight. Give your answer in kilometres per hour.

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(d) Due to prevailing winds, the return journey is longer, taking 15 hours and 15 minutes. Ifthe return journey departs from Los Angeles at 11:15 pm (local time) on 24 th November,what will be the local standard time and date when it arrives in Melbourne?

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There are two lighthouses near Wynyard and Burnie in north-west Tasmania. From theWynyard lighthouse (W), the Burnie lighthouse (B) is 23.5 km away on a bearing of S56 E.

When seen from a ship (S) at sea, the bearing of the Wynyard lighthouse (W) is S67W and the

bearing of the Burnie lighthouse (B) is S29E.

(a) How far is the ship from each lighthouse? Give both answers in kilometres correct toone decimal place.

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A flare is fired vertically upwards from the ship. At its highest point, the flare can be seenfrom the Wynyard lighthouse (W) on an angle of elevation of 345.

(b) Given that the Wynyard lighthouse is 176 metres above sea level, what is the heightabove sea level of the flare at its highest point? Give your answer to the nearest 10metres.

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Spare Diagram for Question 8 (a)

Spare Diagram for Question 8 (b)


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PLACE LABEL HERE



Senior Secondary 5C

Subject Code: MAP5C

External Assessment

2007

Part 4 Data Analysis


On the basis of your performance in this examination, the examinerswill provide results on the following criterion taken from the

syllabus statement:

Criterion 9 Analyse and describe distribution data associated

with populations and samples.

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3. In total it is recommended that you spend approximately 36 minutes answering the questions in this

part.



6. You are expected to provide a graphics calculator approved by the Tasmanian Qualifications

Authority.



If you use this spare graph, you MUST indicate you have done so in your answer to that question.


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The Stanford-Binet Intelligence Test generates IQ scores that are normally distributed with a

mean of 100 and a standard deviation of 15.

(a) The population of Tasmania is estimated to be 485300. How many of them could be

expected to have IQ scores above 150?

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(b) What IQ score must you exceed to be in the top 5% of IQ scores? An appropriate

diagram must be shown.

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Seventy car tyres of a particular brand were tested to see how far they could be driven on a

standard car before being worn out. The results of the test are summarised in the ogive below.

Note that the horizontal axis is scaled in thousands of kilometres.

(a) How many tyres travel less than 40 000 km before being worn out?

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(b) By drawing lines on the graph above to assist, estimate each of the following:

(i) the median distance travelled by the tyres;

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(ii) the lower quartile distance travelled by the tyres;

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(iii) the upper quartile distance travelled by the tyres.

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..............................................................................................................................Question 10 continues opposite.

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Distance travelled (thousands of km)

Number

of

tyres(cumulative)

Distance travelled (thousands of km)


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(c) The tyre manufacturer wishes to guarantee that the tyres will last more than a certain

number of kilometres. If at least 90% of tyres will be required to last longer than the

guaranteed number of kilometres, what should the guaranteed number be?

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The Australian Football League (AFL) has 16 teams.

At the end of 17 June 2007, some teams had played

12 games in the 2007 season and some had played

11 games. Two of the most famous teams in the AFL

are Carlton and Collingwood. By 17 June, Carltonhad played 12 games and Collingwood had played

11 games. The stem and leaf plot at right shows the

total points scored by each of these teams in these

games.

(a) On the graph below, draw side by side box and whisker diagrams of the scores of each

team.


Carlton

(12 games)

Collingwood

(11 games)

9 3

45

6 6

7 5

6 4 2 0 8 1 2 6

5 9 5

10 2 3

5 11 5 9

5 4 12 0

8 0 13

1 14

Key: 3 9 = 39 points

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(b) Using your box and whisker diagrams for evidence, discuss each of the following

questions.

(i) Which of these two teams has the more consistent scores?

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(ii) Which of these two teams tends to have the higher scores?

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Geelong and Hawthorn are another two teams in the Australian Football League (AFL). By 17June, both of these teams had played 12 games in the 2007 season.

The scores for each of these teams in their 12 games are shown in the table below.

Geelong 93 162 109 72 102 222 109 94 116 125 69 85

Hawthorn 44 116 91 76 93 131 80 72 99 109 66 180

Can we conclude from these results that Geelong is a higher-scoring team than Hawthorn?

Conduct a two-samplet-test to determine if this is the case or not.

(a) Null Hypothesis: ............................................................................................................

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Alternative Hypothesis: ..................................................................................................

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(b) t-test results

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(c) Conclusion

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Senior Secondary 5C

Subject Code: MAP5C

External Assessment

2007

Part 5 Finance



Criterion 10 Demonstrate a working knowledge of the standardfinancial models.

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Simon invests $4 000 at 4.8% p.a., compounded monthly over a 5 year period.

(Algebraic workings MUST be shown)

(a) Determine the balance of his account after 5 years.

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(b) How much interest does Simon get?

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Nicole wants to travel to London for the 2012 Olympic Games. She is able to invest $200monthly into an annuity which pays interest of 7.8% per annum compounded monthly.

Nicole makes her first payment in December 2007 and her final payment in January 2012, for

a total of 50 payments.

Algebraic working MUST be shown in both parts of this question.

(a) Determine the final balance of Nicoles annuity after 50 months.

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(b) If Nicole wishes to have $15 000 by the end of January 2012, how much extra will sheneed to invest each month, beginning in December 2007?

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Sarah and Andrew take out a housing loan for 25 years for $220 000 at a nominal interest rateof 7.2% per annum with monthly repayments.

(a) Show that they make monthly repayments of $1583.10.

(Algebraic workings MUST be shown).

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(b) Calculate the total amount they would repay over 25 years.

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You may use your calculators financial mode for the rest of this question.

(c) Determine how much Sarah and Andrew owe after 10 years (120 months) ofrepayments.

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After 10 years (120 months) of repayments, Sarah and Andrew inherit a sum of $50 000. Theyuse all of this amount to pay off part of their housing loan before making their 121 st payment.

(d) If Sarah and Andrew continue to pay the same monthly repayment of $1583.10,

determine the number of repayments required to pay off the loan, and determine howmuch they save when compared with your answer from (b).

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A company purchases a new item of machinery at a cost of $150 000. The company is able toclaim depreciation for tax purposes using either of the following two methods.

Method 1: Straight line depreciation method:depreciating it by12.5% of the original purchase price each year

Method 2: Reducing balance method:

depreciating it by22% p.a. of the reducing balance each year

(You may use your calculators financial mode in this problem).

(a) Determine which of the two methods will give the lower book value after six years.

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t C B


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(b) Determine the time to the nearest year when both methods will give the same bookvalue. Use a labelled graph and your calculator to assist, considering an eight-yearperiod.

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(c) Briefly, discuss under what circumstances the company would choose to depreciatetheir machinery on a reducing balance depreciation method and when it would be betterfor them to use the straight line method.

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t C B A

t C B A


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Spare Graph for Question 16 (b)


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tasmanian certificate of education mathematics applied 2007

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