Gen Math

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Available for a range of VCE subjects from your local bookseller OR purchase online at

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Fully worked solutions.

Official past examination questions.

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STANDARD GENERAL MATHEMATICS

VCE

Second edition

ESSENTIAL

PETER JONES

KAY LIPSON

DAVID MAIN

BARBARA TULLOCH

ESSENTIALStandard GeneralMathematics

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www.cambridge.edu.auwww.cambridge.org

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Revised edition enhanced with an interactive online textbook and TI-Nspire OS3 updates with colour screens

The Essential VCE Mathematics series has a reputation for mathematical excellence, with an approach developed over many years by a highly regarded author team of practising teachers and mathematicians. This approach encourages understanding through a wealth of examples and exercises, with an emphasis on VCE examination-style questions.

New in the Standard General Mathematics Enhanced TI-N/CP version: An additional chapter on bivariate data with an early introduction to regression analysis, a key topic in Further

Mathematics.

Updated worked examples and exercises, with revisions for CAS calculator use. Integrated CAS calculator explanations, examples and problems have been updated to reflect the TI-Nspire

OS3, and continue to feature the Casio ClassPad.

Page numbers in the printed text reflect the previous TI-Nspire and Casio ClassPad version, allowing for continuity and compatibility.

Digital versions of the student text are available in interactive HTML and PDF formats through Cambridge GO. The Interactive Textbook is an HTML version of the student text. It delivers interactive features designed to

enhance the teaching and learning experience. Features include formatting for on-screen reading, linked interactive spreadsheets and slide presentations, pop-up answers and multiple-choice quizzes.

The PDF Textbook, which is enabled for note-taking and bookmarking, is also available free to users of the student text.

Cambridge GO for students and teachersCambridge GO is the new home for the Essential VCE Mathematics companion website. It provides student and teacher resources, including digital textbook options for the enhanced versions and supplements for other models of calculators.

The PDF textbook may be activated using the access code printed in the front of the textbook. The Interactive Textbook may be accessed using the code in the Interactive Textbook sealed pocket, available

for purchase separately or with the student text.

www.cambridge.edu.au/GO

Also available for Essential Standard General Mathematics:

Essential Standard General Mathematics Solutions Supplement 978-0-521-61254-8 Worked solutions to the extended-response questions in the textbook

Essential Standard General Mathematics Teacher CD-ROM 978-0-521-61272-2 Valuable time-saving, planning and assessment resources for teachers

These titles are also supported by student resources on Cambridge GO, teacher resources and Solutions Supplements.

New edit

ion with C

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dates

Interact

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textbook

and PDF

TI-Nspire CAS OS3 and Casio ClassPad version TI-Nspire CAS OS3 and Casio ClassPad versionTI-Nspire CAS OS3 and Casio ClassPad

version

Pages changed in the Enhanced Version

12 Essential Standard General Mathematics

Constructing a histogram using a CAS calculatorIt is relatively quick to construct a histogram from a frequency table. However, if you only

have the data (as you mostly do), it is a very slow process because you have to construct the

frequency table rst. Fortunately, a CAS calculator will do this for us.

How to construct a histogram using the TI-Nspire CAS

Display the following set of 27 marks in the form of a histogram.

16 11 4 25 15 7 14 13 14 12 15 13 16 14

15 12 18 22 17 18 23 15 13 17 18 22 23

Steps1 Start a new document: Press c and select

New Document (or use / +N). If promptedto save an existing document, move cursor to

No and press .

2 Select Add Lists & SpreadsheetEnter the data into a list named marks.

a Move the cursor to the name space of

column A (or any other column) and type in

marks as the list name. Press .

b Move the cursor down to row 1, type in the

rst data value and press . Continue

until all the data has been entered. Press

after each entry.3 Statistical graphing is done through the Data &

Statistics application.Press / +I and select Add Data & Statistics(or press c, arrow to , and press ).

Note: A random display of dots will appear this is toindicate that data are available for plotting. It is not astatistical plot.

a Press e to show the list of variables that

are available. Select the variable marks.

Press to paste the variable marks to that

axis.

b A dot plot is displayed as the default plot. To

change the plot to a histogram, press

b>Plot Type>Histogram and then pressb or click (press a).

ISBN 978-1-107-66462-3 Photocopying is restricted under law and this material must not be transferred to another party.

Peter Jones, Kay Lipson, David Main, Barbara Tulloch 2012 Cambridge University Press


Chapter 1 Univariate data 13

Your screen should now look like that shown

opposite. This histogram has a column (or bin)

width of 2 and a starting point of 3.

4 Data analysis

a Move cursor onto any column. A will

appear and the column data will be

displayed as shown opposite.

b To view other column data values move the

cursor to another column.

Note: If you click on a column it will be selected.To deselect any previously selected columnsmove the cursor to the open area and press x.Hint: If you accidentally move a column or data point,press / + d to undo the move.

5 Change the histogram column (bin) width to 4 and the starting point to 2.a Press / +b to get the contextual menu as shown (below left).

Hint: Pressing / +b with the cursor on the histogram gives you access to a contextual menuthat enables you to do things that relate only to histograms.

b Select Bin Settingsc In the settings menu (below right) change the Width to 4 and the Starting Point

(Alignment) to 2 as shown. Press .

d A new histogram is displayed with column width of 4 and a starting point of 2 but

it no longer ts the viewing window (below left). To solve this problem press /

+b>Zoom>Zoom-Data and to obtain the histogram as shown below right.





6 To change the frequency axis to a percentage axis, press / +b>Scale>Percentand then press .

How to construct a histogram using the ClassPad

Display the following set of 27 marks in the form of a histogram.

16 11 4 25 15 7 14 13 14 12 15 13 16 14

15 12 18 22 17 18 23 15 13 17 18 22 23

Steps1 From the application menu screen,

locate the Statistics application. Tap

to open.

Note: Tapping from the icon panel(just below the touch screen) will displaythe application menu if it is not alreadyvisible.

2 Enter the data into a list named

marks.a Highlight the heading of the rst

list by tapping.

b Press k and tap .c Type marks and press E.d Then, starting in row 1, type in

each data value. Press E orto move down the list.

Your screen should be like the one

shown below (left).




P1: FXS/ABE P2: FXS0521672600Xc01.xml CUAU034-EVANS October 10, 2011 16:9


For example, the median of the following data set is 6, as there are ve observations on

either side of this value when the data are listed in order.

median = 6

2 3 4 5 5 6 7 7 8 8 11

When there is an even number of data values, the median is dened as the mid-point of the

two middle values. For example, the median of the following data set is 6.5, as there are six

observations on either side of this value when the data are listed in order.

median = 6.5

2 3 4 5 5 6 7 7 8 8 11 11

Example 13 Calculating the median

Find the median number of premierships in the AFL ladder using the data in Example 12.

Solution

1 As the data are already given in order, it only remains

to decide which is the middle observation.

16 16 15 12 11 10 10 8 4 4 3 2 1 1 1 0

2 Since there are 16 entries in the table there is no actual

middle observation, so the median is chosen as the value

half-way between the two middle observations, in this

case the eighth and ninth (8 and 4).

median = 12(8+ 4)

= 6

3 The interpretation here is, that of the teams in the AFL,

half (or 50%) have won the premiership 6 or more times

and half (or 50%) have won the premiership 6 or less times.

For larger data sets, the following rule for locating the median is helpful.

In general, to compute the median of a distribution:

Arrange all the observations in ascending order according to size.

If n, the number of observations, is odd, then the median is then + 1

2th observation

from the end of the list.

If n, the number of observations, is even, then the median is found by averaging the two

middle observations in the list. That is, to nd the median then

2th and the

(n2

+ 1)

th

observations are added together, and divided by 2.

The median value is easily determined from an ordered stem-and-leaf plot by counting to the

required observation or observations from either end.



How to calculate measures of centre and spread using the TI-Nspire CAS

The table shows the monthly rainfall gures for a year in Melbourne.

Month J F M A M J J A S O N D

Rainfall (mm) 48 57 52 57 58 49 49 50 59 67 60 59

Determine the mean and standard deviation, median and interquartile range, and the range

for this data set. Give your answers correct to 1 decimal point where necessary.

Steps1 Start a new document: Press c and select

New Document (or press / +N).2 Select Add Lists & Spreadsheet.

Enter the data into a list named rain as shown.

Statistical calculations can be done in the Lists& Spreadsheet application or the Calculatorapplication.

3 Press / +I and select Add Calculator (orpress c and arrow to and press )

a Press b>Statistics>Stat Calculations>One-Variable Statistics, then .

b Press the e key to highlight OK andpress

c Use the arrow and to paste in the list

name rain. Press to exit the popup

screen and generate statistical results screen

shown below.Notes:1 The sample standard deviation is sx.2 Use the arrows to scroll through the results

screen to see the full range of statistical valuescalculated.

4 Write the answers to the required

degree of accuracy (i.e. 1 decimal

place).

x = 55.4, S = 5.8M = 57, IQR = Q3 Q1 = 59 49.5 = 9.5R = max min = 67 48 = 19





Using a CAS calculator to construct a boxplot

How to construct a boxplot using the TI-Nspire CAS

Construct a boxplot to display the given monthly rainfall gures for Melbourne.

Month J F M A M J J A S O N D

Rainfall (mm) 48 57 52 57 58 49 49 50 59 67 60 59

Steps1 Start a new document: Press c and select New

Document (or use / +N).2 Select Add Lists & Spreadsheet

Enter the data into a list called rain as shown.

3 Statistical graphing is done through the Data &Statistics application.

Press / +I and select Add Data & Statistics.(or press c and arrow to and press )

Note: A random display of dots will appear this is toindicate list data is available for plotting. It is not astatistical plot.

a Press e to show the list of variables. Select

the variable rain. Press to paste the

variable to that axis. A dot plot is displayed by

default as shown

b To change the plot to a box plot press

b>Plot Type>Box Plot, then or click(press a). Your screen should now look like

that shown below.4 Data analysis

Move the cursor over the plot to display key

values. Alternatively, use b >Analyze>Graph Trace and the horizontal arrow keysto move the cursor directly to key points.

Starting at the far left of the plot, we see that the

minimum value is 48: MinX = 48 rst quartile is 49.5: Q1 = 49.5 median is 57: Median = 57 third quartile is 59: Q3 = 59 maximum value is 67: MaxX = 67





5 Find the largest value which is not an outlier.

The right hand whisker of the boxplot will

nish at this value.

6 The boxplot can now be constructed

as shown.

The largest value which is not an outlier is 226.

0 100 200 300

The graphics calculator will also construct a boxplot with outliers. Consider again the data

from Example 19.

How to construct a boxplot with outliers using the TI-Nspire CAS

Draw a box plot with outliers to show the number of hours spent on a project by

individual students in a particular school.

2 3 4 9 9 13 19 24 27 35 36

37 40 48 56 59 71 76 86 90 92 97

102 102 108 111 146 147 147 166 181 226 264

Steps1 Press c and select New Document

(or use / +N).2 Select Add Lists & Spreadsheet

Enter the data into a list called hours as shown.

3 Statistical graphing is done through the Data &Statistics application. Press / +I and selectAdd Data & Statistics (or press c, arrow to ,and press ).Note: A random display of dots will appear this is toindicate list data is available for plotting. It is not astatistical plot.

a Press e to show the list of variables. Select

the variable hours. Press to paste the

variable hours to that axis. A dot plot is

displayed as the default plot.





b To change the plot to a box plot press

b>Plot Type>Box Plot, then or click(press a). Outliers are indicated by a dot(s)

lying outside the main body of the plot.

4 Data Analysis

Move the cursor over the plot to display the

key values (or use b>Analyze>GraphTrace)Starting at the far left of the plot, we see that the minimum value is 2: minX = 2 rst quartile is 25.5: Q1 = 25.5. median is 71: Median = 71. third quartile is 109.5: Q3 = 109.5 maximum value is 264: maxX = 264. It is also an outlier.

How to construct a boxplot with outliers using the ClassPad

Draw a boxplot with outliers to show the number of hours spent on a project by individual

students in a particular school.

2 3 4 9 9 13 19 24 27 35 36

37 40 48 56 59 71 76 86 90 92 97

102 102 108 111 146 147 147 166 181 226 264

Steps1 Open the Statistics application and

enter the data into the column

labelled hours.

2 Open the Set StatGraphs dialogbox by tapping in the toolbar.

Complete the dialog box as given

below. For Draw: select On Type: select MedBox ( ) XList: select main \ hours ( ) Freq: leave as 1

Tap the Show Outliers box to adda tick ( ). Tap h to exit.





3 Tapy to plot the boxplot and then tap rto obtain a full-screen display.

Key values can be read from the boxplot by

tapping . Use the horizontal cursor

arrows ( and ) to move from point to

point on the boxplot.

Starting at the far left of the plot, we see that the minimum value is 2 (minX = 2) rst quartile is 25.5 (Q1 = 25.5) median is 71 (Median = 71) third quartile is 109.5 (Q3 = 109.5) maximum value is 264 (maxX = 264).

It is also an outlier.

Exercise 1H

1 A researcher is interested in the number of books people borrow from a library. She decided

to select a sample of 38 cards and record the number of books each person has borrowed in

the previous year. Here are her results:

7 28 0 2 38 18 0 0 4 0 0

2 13 1 1 14 1 8 27 0 52 4

0 12 28 15 10 1 0 2 0 1 11

5 11 0 13 0

a Determine the ve-number summary for this data set.

b Determine if there are any outliers.

c Draw a boxplot of the data, showing any outliers.

d Describe the number of books borrowed in terms of shape, centre, spread and outliers.

2 The winnings (in dollars) of the top 25 male tennis players in 2011 are given in the table

shown opposite.






Novak Djokovic 7 608 673

Rafael Nadal 5 250 169

Roger Federer 2 502 919

Andy Murray 2 495 054

David Ferrer 1 692 314

Robin Soderling 1 220 729

Jo-Wilfried Tsonga 1 041 323

Tomas Berdych 937 344

Nicolas Almagro 857 973

Bob Bryan 811 264

Mike Bryan 811 264

Richard Gasquet 748 460

Juan Martin del Potro 737 508

Alexandr Dolgopolov 720 622

Marty Fish 710 826

Stanislas Wawrinka 678 075

Viktor Troicki 667 013

Michael Llodra 659 438

Jurgen Mayer 659 362

Juan Ignacio Chela 641 287

Andy Roddick 621 392

Feliciano Lopez 620 824

Florian Mayer 594 956

Philipp Petzschner 575 585

Milos Raonic 574 005

a Draw a boxplot of the data, indicating any outliers.

b Describe the annual winnings of the top 25 male players in terms of shape, centre, spread

and outliers.

3 The time taken, in seconds, for a group of children to complete a puzzle is:

8 6 18 39 7 10 5 8 6 14 11

10 8 60 6 6 14 15 6 7 6 5

8 11 8 15 8 8 7 8 8 6 29

5 7

a Draw a boxplot of the data, indicating any outliers.

b Describe the time taken for the group to complete the puzzle in terms of shape, centre,

spread and outliers.



Rev

iew


3 In a small company, upper management

wants to know whether there is a difference

in the three types of methods used to train

its machine operators. One method uses a

hands-on approach. A second method uses

a combination of classroom instruction and

on-the-job training. The third method is based

completely on classroom training. Fifteen

trainees are assigned to each training

technique. The data shown is the result of

a test administered after completion of the

training for people trained using each method.

Method 1 Method 2 Method 3

98 79 70

100 62 74

89 61 60

90 89 72

81 69 65

85 99 49

97 87 71

95 62 75

87 65 55

70 88 65

69 98 70

75 79 59

91 73 77

92 96 67

93 83 80

a Draw boxplots of the data sets, on the

same axis.

b Write a paragraph comparing the three

training methods in terms of shape, centre,

spread and outliers.

c Which training method would you recommend?

4 It has been argued that there is a relationship between a childs level of independence

and the order in which they were born in the family. Suppose that the children in

13 three-child families are rated on a 50-point scale of independence. This is done

when all children are adults, thus eliminating age effects. The results are as follows.

Family 1 2 3 4 5 6 7 8 9 10 11 12 13

First-born 38 45 30 29 34 19 35 40 25 50 44 36 26

Second-born 9 40 24 16 16 21 34 29 22 29 20 19 18

Third-born 12 12 12 25 9 11 20 12 10 20 16 13 10

a Draw boxplots of the data sets on the same axis. Use a calculator.

b Write a paragraph comparing level of independence for the three birth orders in

terms of shape, centre, spread and outliers.

5 A study was conducted to determine the effect of choice on performance in student

essays. One group of students was allowed to choose their essay topics from a long

list of possibilities. Another group was given the same essay topics but without any

choice of topic. The marks, out of 50, obtained by the students are given below.

36 34 37 30 40 35 33 25Choice

35 37 48 36 50 45 44 30

44 36 36 35 26 42 21 25No choice

44 24 49 48 28 42 35 40

a Draw boxplots for each of the sets of marks on the same scale.

b Write a paragraph to discuss the effect of students choice on the distribution of

marks in terms of shape, centre, spread and outliers.




Example 1 Using a formula

The cost of hiring a windsurfer is given by the rule

C = 5t + 8where C is the cost in dollars and t is the time in hours.

Annie wants to sail for 2 hours. How much will it cost her?

Solution

1 Write the formula. C = 5t + 82 To determine the cost of hiring a

windsurfer for 2 hours, substitute t = 2into the formula.

C = 5(2)+ 8

Remember: 5(2) means 5 2.3 Evaluate. C = 184 Write your answer. It will cost Annie $18 to hire a

windsurfer for 2 hours.

Example 2 Using a formula

The perimeter of the shape shown can be given by the formula:

P = 2L + H(

1 + 2

)where L is the length of the rectangle and H is the height.

Find the perimeter correct to 1 decimal place, if L = 16.1 cmand H = 3.2 cm.

H

L

Solution

1 Write the formula. P = 2L + H(1+

2

)2 Substitute values for L and H

into the formula. P = 2 16.1+ 3.2(1+

2

)3 Evaluate. P = 40.4 (correct to 1 dec. place)4 Give your answer with correct units. The perimeter of the shape is 40.4 cm.

Exercise 2A

1 The cost of hiring a dance hall is given by the rule

C = 50t + 1200where C is the total cost in dollars and t is the number of hours for which the hall is hired.



Chapter 2 Linear relations and equations 65

Find the cost of hiring the hall for:

a 4 hours b 6 hours c 4.5 hours.

2 The distance, d km, travelled by a car in t hours at an average speed of v km/h is given by

the formula

d = v tFind the distance travelled by a car travelling at a speed of 95 km/hour for 4 hours.

3 Taxi fares are calculated using the formula

F = 1.3K + 4where K is the distance travelled in kilometres and F is the cost of the fare in dollars.

Find the costs of the following trips.

a 5 km b 8 km c 20 km

4 The circumference, C, of a circle with radius, r, is given by

C = 2rFind, correct to 2 decimal places, the circumferences of the circles with the following radii.

a r = 25 cm b r = 3 mm c r = 5.4 cm d r = 7.2 m

5 If P = 2(L + W ), nd the value of P if:a L = 3 and W = 4 b L = 15 and W = 8 c L = 2.5 and W = 9

6 If A = 12 h (x + y), nd A if:a h = 1, x = 3, y = 5 b h = 5, x = 2.5, y = 3.2 c h = 2.7, x = 4.1, y = 8.3

7 The formula used to convert temperature from degrees Fahrenheit to degrees Centigrade is

C = 59 (F 32)Use this formula to convert the following temperatures to degrees Centigrade.

Give your answers correct to 1 decimal place.

a 50F b 0F c 212F d 92F

8 The formula for calculating simple interest is

I = PRT100

where P is the principal (amount invested or borrowed), R is the interest rate per annum and

T is the time (in years). In the following questions, give your answers to the nearest cent

(correct to 2 decimal places).

a Frank borrows $5000 at 12% for 4 years. How much interest will he pay?

b Chris borrows $1500 at 6% for 2 years. How much interest will he pay?

c Jane invests $2500 at 5% for 3 years. How much interest will she earn?

d Henry invests $8500 for 3 years with an interest rate of 7.9%. How much interest will he

earn?



How to construct a table of values using the TI-Nspire CAS

The formula for converting degrees Celsius to degrees Fahrenheit is given by

F = 95 C + 32Use this formula to construct a table of values for F using values of C in intervals of 10

between C = 0 and C = 100.

Steps1 Start a new document: Press / +N2 Select Add Lists & Spreadsheet

Name the lists c (for Celsius) and f (for

Fahrenheit)

Enter the data 0 100 in intervals of 10

into the list named c, as shown.

3 Place cursor in the grey formula cell in

column B (i.e. list f ) and type in:

= 9 5 c + 32Hint: If you typed in c you will need to selectVariable Reference when prompted. This promptoccurs because c can also be a column name.Alternatively, press h key and select c fromthe variable list.

Press to display the values given. Use

the arrow to move down through the

table.

How to construct a table of values using the ClassPad

The formula for converting degrees Celsius to degrees Fahrenheit is given by

F = 95 C + 32Use this formula to construct a table of values for F using values of C in intervals of 10

between C = 0 and C = 100.






1 Write the equation.

2 Subtract 6 from both sides of the equation.

This is the opposite process to adding 6.

3 Check your answer by substituting the

found value for x into the original equation.

If each side gives the same value, the solution

is correct.

x + 6= 10x + 6 6= 10 6

x = 4LHS = x + 6

= 4+ 6= 10= RHS

Solution is correct.

Example 5 Solving a linear equation by hand

Solve the equation 3y = 18.

Solution


2 The opposite process of multiplying by 3 is

dividing by 3. Divide both sides of the

equation by 3.

3y = 183y3

= 183

y = 6

3 Check that the solution is correct by

substituting y = 6 into the original equation.LHS = 3y

= 3 6= 18= RHS


Example 6 Solving a linear equation by hand

Solve the equation 4(x 3) = 24.

Solution: Method1 Solution: Method 21 Write the equation.

2 Expand the brackets.

3 Add 12.

4 Divide by 4.

4(x 3) = 244x 12 = 24

4x 12+ 12 = 24+ 124x = 364x4

= 364

x = 9


2 Divide by 4.

3 Add 3.

4(x 3) = 244(x 3)

4= 24

4x 3 = 6

x 3+ 3 = 6+ 3 x = 9

Check that the solution is correct bysubstituting x = 9 into the original equation. LHS = 4(x 3)

= 4(9 3)= 4 6= 24= RHS

solution is correct.




Example 7 Solving a linear equation using CAS

Solve the equation 4 5b = 8.

Solution

1 Use the solve( command on your CAScalculator to solve for b as shown opposite.Note: Set the mode of your calculator to Approximate(T1-nspire) or Decimal (Classpad) before using solve(.

solve(45b = 8, b) b = 2.4

2 Check that the solution is correct by

substituting x = 2.4 into the originalequation.

LHS = 4 5b= 4 5 2.4= 4+ 12= 8= RHS


Exercise 2C

1 Solve the following linear equations.

a x + 6 = 15 b y + 11 = 26 c t + 5 = 10 d m 5 = 1e g 3 = 3 f f 7 = 12 g f + 5 = 2 h v + 7 = 2i x + 11 = 10 j g 3 = 2 k b 10 = 5 l m 5 = 7

m 2 + y = 8 n 6 + e = 9 o 7 + h = 2 p 3 + a = 1q 4 + t = 6 r 8 + s = 3 s 9 k = 2 t 5 n = 1

2 Solve the following linear equations.

a 5x = 15 b 3g = 27 c 9n = 36 d 2x = 16e 6 j = 24 f 4m = 28 g 2 f = 11 h 2x = 7i 3y = 15 j 3s = 9 k 5b = 25 l 4d = 18

mr

3= 4 n q

5= 6 o x

8= 6 p t2 = 6

qh

8 = 5 rm

3 = 7 s14

a= 7 t 24

f= 12

3 Solve the following linear equations using CAS. Give answers correct to 1 decimal place

where appropriate.

a 3a + 5 = 11 b 4b + 3 = 27 c 2w + 5 = 9 d 7c 2 = 12e 3y 5 = 16 f 4 f 1 = 7 g 3 + 2h = 13 h 2 + 3k = 6

i 4(g 4) = 18 j2(s 6)

7= 4 k 5(t + 1)

2= 8 l 4(y 5)

5= 2.4

m 2(x 3) + 4(x + 7) = 10 n 5(g + 4) 6(g 7) = 25 o 5(p + 4) = 25 + (7 p)




Exercise 2F

1 Balloons cost 50 cents each and streamers costs 20 cents each.

a Construct a formula for the cost, C, of x balloons and y streamers.

b Find the cost of 25 balloons and 20 streamers.

2 Tickets to a concert cost $40 for adults and $25 for children.

a Construct a formula for the total amount, C, paid by x adults and y children.

b How much money altogether was paid by 150 adults and 315 children?

3 At the football canteen, chocolate bars cost $1.60 and muesli bars cost $1.40.

a Construct a formula to show the total money, C, made by selling x chocolate bars and

y muesli bars.

b How much money would be made if 55 chocolate bars and 38 muesli bars were sold?

4 At the bread shop, custard tarts cost $1.75 and iced doughnuts $0.70 cents.

a Construct a formula to show the total cost, C, if x custard tarts and y iced doughnuts are

purchased.

b On Monday morning, Mary bought 25 custard tarts and 12 iced doughnuts. How much

did it cost her?

5 At the beach cafe, Marion takes orders for coffee and milkshakes. A cup of coffee costs

$2.50 and a milkshake costs $4.00.

a Using x (coffee) and y (milkshakes),write a formula showing the cost, C, of coffee and

milkshake orders taken.

b Marion took orders for 52 cups of coffee and 26 milkshakes. How much money did this

make?

6 Joe sells budgerigars for $30 and parrots for $60.

a Write a formula showing the money, C, made by selling x budgerigars and y parrots.

b Joe sold 60 budgerigars and 28 parrots. How much money did he make?

7 James has been saving 50c and 20c pieces.

a If James has x 50c pieces and y 20c pieces, write a formula to show the number, N, of

coins that James has.

b Write a formula to show the value, V dollars, of Jamess collection.

c When James counts his coins, he has forty-ve 50c pieces and seventy-seven 20c pieces.

How much money does he have in total?

8 A rectangular lawn is twice as long as it is wide. It has a path 1 metre wide all around it. The

length of the perimeter of the outside of the path is 48 metres. What is the width of the lawn?

Give your answer correct to the nearest centimetre.




2.7 Setting up and solving simplenon-linear equationsNot all equations that are solved in mathematics are linear equations. Some equations are

non-linear.

For example:

y = x2 + 2 is a non-linear equation with two unknowns, x and y.d2 = 25 is a non-linear equation with one unknown, d.6m3 = 48 is a non-linear equation with one unknown, m.

Example 17 Solving a non-linear equation

Solve the equation x2 = 81.

Solution 1 (by hand)1 Write the equation. x2 = 812 Take the square root of both sides of the equation.

(The opposite process of squaring a number is to

take the square root.)

x2 =

81

x = 9

Note: Both the positive and negative answers should begiven, as 9 9 = 81 and 9 9 = 81.

Solution 2 (using CAS)Use the solve( command as shownopposite.

solve(x 2 = 81, x) x = 9 or x = 9

Note: Set the mode of your calculator to Approximate(T1-Nspire) or Decimal (Classpad) before using solve(.


Solve the equation a3 = 512.

Solution 1 (by hand)1 Write the equation. a3 = 5122 Take the cube root of both sides of the

equation. (The opposite process of cubing

a number is to take the cube root.)

3a3 = 3512 a = 8

Note: 3512 = (512) 13

Note: (8) (8) (8) = 512but 8 8 8 = 512

Solution 2 (using CAS)Use the solve( command as shown opposite.

solve(a3 = 512, a ) a = 8





The surface area S of a sphere of radius r is given by the equation S = 4r2. Find the radiusof a sphere with a surface area of 600 cm2.

Solution (using CAS)1 Write the equation. S = 4r 22 Substitute S = 600 into the equation. 600= 4r 23 Use the solve( command as shown

opposite to solve the equation.solve(600 = 4r 2, r )

r = 6.90988 or r = 6.90988

4 Noting that the radius must have a

positive value, write down the value of

r correct to 2 decimal places.

Correct to 2 decimal places, theradius of the sphere is 6.91 cm.

Exercise 2G

1 Evaluate the following.

a 42 b (9)2 c 72d 33 e 23 f 63

g (5)3 h 44 i (10)4

2 Solve the following non-linear equations correct to 2 decimal places.

a a2 = 12 b b2 = 72 c c2 = 568d d3 = 76 e e3 = 300 f f 3 = 759

3 Solve the following non-linear equations correct to 2 decimal places.

a 3x2 = 24 b 5y2 = 25 c 2a2 = 11d 6 f 2 = 33 e 4h2 = 19 f 11c2 = 75g x3 = 81 h r3 = 18 i y3 = 96j 2r3 = 50 k 4m3 = 76 l 8b3 = 21

m 2p2 1 = 8 n 3q3 + 5 = 101 o 2(r2 + 8) = 64

4 The volume of a cylinder is given by

V = r2hwhere h is the height and r is the radius of the base.

Find, correct to 2 decimal places, the value of h

when V = 450 cm2 and r = 10 cm. r

h




5 Pythagoras Theorem states that, for any right-angled triangle, the hypotenuse, h, is given by

h2 = a2 + b2where a and b are the other two sides of the triangle.

Find a, correct to 1 decimal place, when h = 17.5 cmand b = 7.8 cm.

a

b

h

6 The volume, V, of a cone is given by

V = 13r2hwhere r is the radius and h is the height of the cone.

a Find, to the nearest cm, the radius if the height of the

cone is 15 cm and the volume is 392.7 cm3.r

h

b Find, to the nearest cm, the height if the radius of the cone

is 7.5 cm and the volume is 562.8 cm3.

2.8 Linear recursionA recursive relationship is one where the same

thing keeps happening over and over again.

A recursion is the process of using a repeated

procedure. Recursion can be used to solve problems

by repeating a sequence of operations.

For example, a slowly developing bacterial

population doubles every day. This situation can

be described by a recursive relationship.

Assuming that we start with two bacteria, ,the population of bacteria can be seen to develop

as in the following diagram.

Day

Term t1 = 2 t2 = 4 t3 = 8 t4 = 16 tn = 2tn1 tn+1 = 2tn

1 2 3 4 n n + 1n 1Numberof bacteria



On day 1, there are 2 bacteria.

On day 2, there are 2 2 = 4 bacteria.On day 3, there are 4 2 = 8 bacteria.On day 4, there are 8 2 = 16 bacteria.On day n, there are tn1 2 bacteriaOn the (n + 1)th day, there are tn 2 bacteria.Thus a rule for this recursive relationship is tn+1 = 2tn , with a starting value t1 = 2.

t1 = 2t2 = 2 t1t3 = 2 t2t4 = 2 t3tn = 2 tn1

tn+1 = 2 tn

We can use linear recursion on a graphics calculator to generate a sequence of terms.

How to generate a sequence of terms using linear recursion using the TI-Nspire CAS

A slowly developing bacterial population doubles every day. The rule for this recursive

relationship is

tn+1 = 2tnShow the terms of this relationship, if the starting value is 2.

Steps1 Press c or (or w then c on the

Clickpad), then A to open

Scratchpad:Calculate.See Appendix for more details on the

Scratchpad.Note: You can also use c>New DocumentAdd Calculator if preferred.

2 Type in 2, the value of the rst term. Press. The calculator stores the value 2 as

Answer (you cant see this yet).3 Now type in 2 (the screen will show

Ans 2) and press. The second term in thesequence is 4. This value is now stored as Ans.Note: To see the value stored as Ans at any time, press/v.

4 Press to generate the next term. Continue

pressing until the required number of

terms is generated.





Solution

1 On the calculation

screen, type in 32 and press

(or E)

2 Type in + 14 and press (or E)

3 Continue pressing

(or E) until ve terms havebeen generated.

4 Write your answer. The first ten terms are 32, 46, 60, 74, 88.

Example 21 Using linear recursion

A linear recursion relationship is given by

tn+1 = 3tn 2Write the rst six terms if the starting value is t1 = 6.

Solution

1 On the calculation

screen, type in 6 and press

(or E)

2 Type in 3 2 and press (or E)

3 Continue pressing

(or E) until six terms havebeen generated.

4 Write your answer. The first six terms are 6, 16, 46, 136, 406, 1216.

Example 22 Using linear recursion to solve practical problems

Maree has $3000 in her bank account. She adds $45 to it at the end of each month. How much

will she have after 8 months?





Solution

This is a linear recursion relationship for which the starting value is $3000.

1 On the calculation screen,

type in 3000 and press

(or E).2 Each month, $45 is added

to the account, so type in

+ 45 and press (orE). At the end of the rstmonth, Marie has $3045.

3 Continue pressing (orE) to generate all eightvalues.

4 Give your answer with the correct units. After 8 months, Maree will have $3360.

Example 23 Using linear recursion to solve problems

A person starts a job on an annual salary of $35 000 and receives annual increases of $3500.

What will be their salary at the beginning of the fth year?

Solution

This is a linear recursion relationship with a starting value of $35 000.



(or E)2 Each year, there is a salary

increase of $3500, so type

in + 3500 and press (or E).

3 Press (or E) threemore times.

At the start of the second year, the salary will be $38 500.

At the start of the third year, the salary will be $42 000.

At the start of the fourth year, the salary will be $45 500.

At the start of the fth year the salary will be $49 000.

4 Write your answer. At the start of the fifth year, the salary will be $49000.






A person inherits $25 000 and invests it at 10% per annum. A linear recursion relationship for

this is given by

tn+1 = 1.1tnShow how the amount of $25 000 increases over 4 years.

Note: An increase of 10% means that the amount is 110% of the original.

110% means110

100= 1.1

Each year is therefore multiplied by 1.1. The linear recursion relationship is thus dened by tn+1 = 1.1tn .Solution



(or E).2 Each year, there is a 10%

salary increase so type in

1.1 and press (orE).

3 Press (or E) threemore times.

4 Write your answer, showing how

the amount increases each year.

At the end of the first year, the amount is $27 500.At the end of the second year, the amount is $30 250.At the end of the third year, the amount is $33 275.At the end of the fourth year, the amount is $36 602.50.


Bruce invests $35 000 at 10% per annum and decides to spend $5000 each year. Show how the

balance changes over a 4-year period.





Solution

The linear recursion relationship is tn+1 = 1.1tn 5000.1 On the calculation screen,


(or E)2 Next type in 1.1 5000

and press (or E).3 Press (or E) three

more times.

4 Write your answer, showing how the

balance changes each year.

At end of the first year, the balance is $33 500.At end of the second year, the balance is $31 850.At end of the third year, the balance is $30 035.At end of the fourth year, the balance is $28 038.50.

Exercise 2H

1 A linear recursion relationship is given by tn+1 = tn 12. Write the rst ve terms if thestarting value is t1 = 200.

2 Sarah is saving up for a new car. She already has $1500 and she is able to save $400

a month. How much will she have:

a after 6 months? b after 12 months?

3 Peter owes $18 000 to his father. He decides to pay his father $800 every month.

a How much will he owe:

i after 10 months? ii after 1 year? iii after 18 months?

b How long will it take Peter to pay the money back to his father?

4 Erica is offered a job with a starting salary of $29 500 per year and annual increases of

$550.

a What her salary be would:

i at the start of her fth year on the job?

ii at the start of her eighth year on the job?

b At this rate, how many years would she have to be in the job to receive a salary of

$35 000?





A CAS calculator can also be used to nd the point of intersection.

How to nd the point of intersection of two linear graphs using the TI-Nspire CAS

Use a graphics calculator to nd the point of intersection of the simultaneous equations

y = 2x + 6 and y = 2x + 3.

Steps1 Start a new document (/ +N) and select

Add Graphs

2 Type in the rst equation as shown. Note that

f1(x) represents the y. Press , the edit line

will change to f2(x) and the rst graph will be

plotted. Type in the second equation and press

to plot the second graph.

Hint: If the entry line is not visible press e.Hint: To see all entered equations move the cursoronto the and press x.

Note: To change window settings, useb>Window/Zoom>Window Settings and change tosuit. Press when nished.

3 To nd the point of intersection, use

b>Points & Lines>Intersection Point(s)Move the cursor to one of the graphs until it

ashes, press x, then move to the other graph

and press x. The solution will appear.

4 Press to display the solution on the

screen. The co-ordinates of the point of

intersections are x = 0.75 and y = 4.5.

Note: you can also nd the intersection point usingb>Analyze Graph>Intersection






4 Find y by substituting x = 1 into eitherequation (1) or equation (2).

Substitute x = 1 into (1).y = 1+ 5

y = 65 Check by substituting x = 1 and y = 6 into

equation (2).

LHS= 6RHS= 3(1)+ 9= 3+ 9= 6

6 Write your answer as a pair of coordinates. Solution is (1, 6).

Method 2: EliminationWhen solving simultaneous equations by elimination, one of the unknown variables is

eliminated by the process of adding or subtracting multiples of the two equations.

Example 29 Solving simultaneous equations by elimination

Solve the pair of simultaneous equations x + y = 3 and 2x y = 9.

Solution

1 Number the two equations.

On inspection, it can be seen that if the

two equations are added, the variable y will

be eliminated.

x + y = 3 (1)2x y = 9 (2)

2 Add equations (1) and (2). (1)+ (2) : 3x = 12

3 Solve for x. Divide both sides of the equation by 3. 3x3

= 123

x = 44 Substitute x = 4 into equation (1) to nd the

corresponding y value.

Substitute x = 4 into (1).4+ y = 3

5 Solve for y. Subtract 4 from both sides of

the equation.

4+ y 4 = 3 4 y = 1

6 Check by substituting x = 4 and y = 1 intoequation (2).

LHS = 2(4) (1)= 8+ 1 = 9= RHS

7 Write your answer as a pair of coordinates. Solution is (4, 1).

Example 30 Solving simultaneous equations by elimination

Solve the pair of simultaneous equations 3x + 2y = 2.3 and 8x 3y = 2.8




Solution

1 Label the two equations (1) and (2). 3x + 2y = 2.3 (1)8x 3y = 2.8 (2)

2 Multiply equation (1) by 3 and equation (2) by 2 to give

6y in both equations.

(1) 3 9x + 6y = 6.9 (3)(2) 2 16x 6y = 5.6 (4)

Remember: Each term in equation (1) must be multiplied by 3and each term in equation (2) by 2.

3 Add equation (4) to equation (3) to

eliminate 6y.

(3)+ (4) 25x = 12.5

4 Solve for x. Divide both sides of the equation by 25. 25x25

= 12.525

or x = 0.5

5 To nd y, substitute x = 0.5 into equation (1). 3(0.5)+ 2y = 2.31.5+ 2y = 2.3

6 Solve for y. Subtract 1.5 from both sides of the equation. 1.5+ 2y 1.5 = 2.3 1.52y = 0.8

7 Divide both sides of the equation by 2.2y2

= 0.82or y = 0.4

8 Check by substituting x = 0.5 and y = 0.4 intoequation (1).

LHS = 3(0.5)+ 2(0.4)= 2.3= RHS

9 Write your answer as a pair of coordinates. Solution is (0.5, 0.4).

Exercise 2J

1 Solve the following pairs of simultaneous equations by any algebraic method (elimination or

substitution).

a y = x 13x + 2y = 8

b y = x + 36x + y = 17

c x + 3y = 15y x = 1

d x + y = 10x y = 8

e 2x + 3y = 124x 3y = 6

f 3x + 5y = 8x 2y = 1

g 2x + y = 113x y = 9

h 2x + 3y = 156x y = 11

i 3p + 5q = 174p + 5q = 16

j 4x + 3y = 76x 3y = 27

k 3x + 5y = 113x 2y = 8

l 4x 3y = 62x + 5y = 4



2 Solve the following pairs of simultaneous equations by any suitable method.

a y = 6 x2x + y = 8

b 2x + 3y = 5y = 7 2x

c 3x + y = 4y = 2 4x

d 3x + 5y = 9y = 3

e 3x + 2y = 03x y = 3

f 4x + 3y = 285x 6y = 35

2.11 Solving simultaneous linear equationsusing a CAS calculator

How to solve a pair of simultaneous equations using the TI-Nspire CAS

Solve the following pair of simultaneous equations:

24x + 12y = 3645x + 30y = 90

Steps1 Press c (or w, then c on

the Clickpad), then A to open.

Scratchpad:Calculate.Note: You can also use c>New Document>Add Calculator if preferred.

2 Press b>Algebra>Solve System ofEquations>Solve System of Equations andpress . Complete the pop-up screen as

shown (the default settings are for two

equations with variables x & y).

A simultaneous equation template is thenpasted to the screen.

3 Enter the equations as shown into the

template. Use the e key to move between

entry boxes

4 Press to display the solution, x = 0 andy = 3.

5 Check the solution x = 0 and y = 3 bysubstitution.

6 Write your answer as a pair of coordinates.

LHS = 24 (0)+ 12 (3) = 36 = RHSLHS = 45(0)+ 30(3) = 90 = RHSSol uti on i s (0,3).





How to solve a pair of simultaneous linear equations algebraically using the ClassPad

Solve the following pair of simultaneous equations:

24x + 12y = 3645x + 30y = 90

Steps1 Open the built-in Main

application.a Press k on the

front of the calculator

to display the built-in

keyboard.

b Tap the) tab and

locate simultaneous

equations icon:

c Enter the information{24x + 12y = 3645x + 30y = 90

x,y

2 Press E to display the solution, x = 0 andy = 3.

3 Check the solution x = 0 and y = 3 bysubstitution.

4 Write your answer as a pair of coordinates.

LHS = 24 (0)+ 12 (3) = 36 = RHSLHS = 45(0)+ 30(3) = 90 = RHSSol uti on i s (0,3).

Exercise 2K

Solve the following simultaneous equations:

a 2x + 5y = 3x + y = 3

b 3x + 2y = 5.52x y = 1

c 3x 8y = 132x 3y = 8

d 2h d = 38h 7d = 18

e 2p 5k = 115p + 3k = 12

f 5t + 4s = 162t + 5s = 12

g 2m n = 12n + m = 8

h 15x 4y = 62y + 9x = 5

i 2a 4b = 122b + 3a 2 = 2

j 3y = 2x 13x = 2y + 1

k 2.9x 0.6y = 4.84.8x + 3.1y = 5.6

2.12 Practical applications ofsimultaneous equationsSimultaneous equations can be used to solve problems in real situations. It is important to

dene the unknown quantities with appropriate variables before setting up the equations.






Example 31 Using simultaneous equations to solve a practical problem

The perimeter of a rectangle is 48 cm. If the length of the rectangle is three times the width,

determine its dimensions.

Solution

Strategy: Using the information given, set up a pair of simultaneous equations to solve.

1 Choose appropriate variables to represent the

dimensions of width and length.

Let W= widthL = length.

2 Write two equations from the information given

in the question. Label the equations as (1) and (2).

2W+ 2L = 48 (1)L = 3W (2)

Remember: The perimeter of a rectangle is the distancearound the outside and can be found using 2w + 2l.3 Solve the simultaneous equations by substituting

equation (2) in equation (1).

Substitute L= 3W into (1).2W+ 2(3W) = 48

4 Expand the brackets. 2W+ 6W= 485 Collect like terms. 8W= 486 Solve for w. Divide both sides by 8. 8W

8= 48

8 W= 6

7 Find the corresponding value for l by

substituting w = 6 into equation (2).Substitute W= 6 into (2).

L = 3(6) L = 18

8 Give your answer in the correct units. The dimensions of therectangle are width 6 cm andlength 18 cm.

Example 32 Using simultaneous equations to solve a practical problem

Mark buys 3 roses and 2 gardenias for $15.50.

Peter buys 5 roses and 3 gardenias for $24.50. How

much did each type of ower cost?



Solution

Strategy: Using the information given, set up a pair of simultaneous

equations to solve.

1 Choose appropriate variables to represent the

cost of roses and gardenias.

Let r be the cost of a rose andg be the cost of a gardenia.

2 Write two simultaneous equations using the

information given in the question. Label the

equations (1) and (2).

3r+ 2g = 15.5 (1)5r+ 3g = 24.5 (2)

3 Use your CAS calculator to solve the two

simultaneous equations.

4 Write down the solutions. r = 2.50 and g = 45 Check by substituting r = 2.5 and g = 4

into equation (2).LHS = 5(2.5)+ 3(4)

= 12.5+ 12 = 24.5 = RHS6 Write your answer with the correct units. Rosescost$2.50eachandgardenias

cost$4each.

Exercise 2L

1 Jessica bought 5 textas and 6 pencils for $12.75, and Tom bought 7 textas and 3 pencils for

$13.80.

a Using t for texta and p for pencil, nd a pair of simultaneous equations to solve.

b How much did one pencil and one texta cost?

2 Peter buys 50 L of petrol and 5 L of motor oil for $93. His brother Anthony buys 75 L of

petrol and 5 L of motor oil for $122. How much do a litre of petrol and a litre of motor oil

cost each?

3 Six oranges and ten bananas cost $7.10. Three oranges and eight bananas cost $4.60. Find

the cost each of oranges and bananas.






4 The weight of a box of nails and a box of screws is 2.5 kg. Four boxes of nails and a box of

screws weigh 7 kg. Determine the weight of each.

5 An enclosure at a wildlife sanctuary contains wombats

and emus. If the number of heads totals 28 and the

number of legs totals 88, determine the number of each

species present.

6 The perimeter of a rectangle is 36 cm. If the length of

the rectangle is twice its width, determine its dimensions.

7 Find a pair of numbers whose sum is 52 and whose

difference is 8.

8 Bruce is 4 years older than Michelle. If their combined age is 70, determine their individual

ages.

9 A chocolate thickshake costs $2 more than a fruit smoothie. Jack pays $27 for 3 chocolate

thickshakes and 4 fruit smoothies. How much do a chocolate thickshake and a fruit

smoothie cost each?

10 In 4 years time a mother will be three times as old as her son. Four years ago she was ve

times as old as her son. Find their present ages.

11 The fees for registering in a mathematics competition between two neighbouring schools

are $1.20 for students aged 812 and $2 for students 13 years and over. An amount of

$188.40 has been collected and 125 students have already registered. How many students

between the ages of 8 and 12 have registered?

12 A computer company produces 2 laptop models: standard and deluxe. The standard laptop

requires 3 hours to manufacture and 2 hours to assemble. The deluxe model requires 51/2 to

manufacture and 11/2 hours to assemble. The company allows 250 hours for manufacturing

and 80 hours for assembly over a limited period. How many of each model can be made in

the time available?

13 A chemical manufacturer wishes to obtain 700 litres of a 24% acid solution by mixing a

40% solution with a 15% solution. How many litres of each solution should be used?

14 In a hockey club there are 5% more boys than there are girls. If there is a total of 246

members in the club, what is the number of boys and the number of girls?

15 The owner of a service station sells unleaded petrol for $1.42 and diesel fuel for $1.54. In

ve days he sold a total of 10 000 litres and made $14 495. How many litres of each petrol

did he sell? Give your answer to the nearest litre.

16 James had $30 000 to invest. He chose to invest part of it at 5% and the other part at 8%.

Overall he earned $2100 in interest. How much did he invest at each rate?



Review


Extended-response questions

1 The cost, C, of hiring a boat is given by C = 8h + 25 where h represents hours.a What is the cost if the boat is hired for 4 hours?

b For how many hours was the boat hired if the cost was $81?

2 A phone bill is calculated using the formula C = 25 + 0.50n where n is the numberof calls made.

a Complete the table of values below for values of n from 60 to 160.

b What is the cost of making 160 phone calls?

n 60 70 80 90 100 110 120 130 140 150 160

C

3 An electrician charges $80 up front and $45 for each hour, h, that he works.

a Write a linear equation for the total charge, C, of any job.

b How much would a 3-hour job cost?

4 Two families went to the theatre. The rst family bought tickets for 3 adults and

5 children and paid $73.50. The second family bought tickets for 2 adults and

3 children and paid $46.50.

a Write down two simultaneous equations that could be used to solve the problem.

b What was the cost of an adults ticket?

c What was the cost of a childs ticket?

5 A bank account has $5000 in it. At the end of each month $300 is withdrawn.

a How much is in the account at the end of 6 months?

b How much is in the account after 1 year?

c How long until there is no more money left?

6 Mark invests $3000 at 3% per annum. The linear recursion relationship describing

this investment is given by tn+1 = 1.03tn . Show how $3000 increases over 5 years.7 The perimeter of a rectangle is 10 times the width. The length is 9 metres more than

the width. Find the width of the rectangle.

8 A secondary school offers three languages: French, Indonesian and Japanese. At the

Year 9 level, there are 105 students studying one of these languages. The Indonesian

class has two-thirds the number of students that the French class has and the

Japanese class has ve-sixths the number of students of the French class. How many

students study each language?



How to draw a straight-line graph and show a table of values using the TI-Nspire CAS

Use a graphics calculator to draw the graph y = 8 2x and show a table of values.Steps1 Start a new document (/ +N) and select Add

Graphs

2 Type in the equation as shown. Note that f1(x)

represents the y. Press to obtain the graph

below.Hint: If the function entry line is not visible, press e

3 Change the window setting to see the key features

of the graph. Use b>Window/Zoom>WindowSettings and edit as shown. Use the e key tomove between the entry lines. Press when

nished editing the settings. The re-scaled graph

is shown below.

4 To show values on the graph, use

b>Trace>Graph Trace and then use the arrows to move along the graph.Note: Press to exit the Trace tool.

5 To show a table of values, press / +T.Use the arrows to scroll through the values

in the table.






Solution

1 Write down the two-point

formula.

2 Write down the values of

x1, x2, y1 and y2.Note: It does not matter whether thepoint (2, 1) is called (x1, y1)or (x2, y2).

3 Substitute the values of

x1, y1, x2 and y2 into

the formula.

4 Simplify to make y the subject.

5 Write your answer. The equation of the line is y = 4.5x 8

y y1 = slope (x x1) where slope=y2 y1x2 x1

x1 = 2, y1 = 1; x2 = 4, y2 = 10

Slope = y2 y1x2 x1 =

10 14 2 = 4.5

y y1 = 4.5(x x1)y 1 = 4.5(x 2)y 1 = 4.5x 9

y = 4.5x 8

Exercise 3E

1 Find the equation of each of the lines

(A, B, C) shown on the graph below.

x

y

0 1 2

2

4

6

8

10

3 4 5

(1, 10) (3, 10)

(1, 0) (2, 1) (3, 1)

(5, 10)

A B

C

2 Find the equations of each of the lines

(A, B, C) shown on the graph below.

x

y

0 1 2

2

4

6

8

10

3 4 5

(1, 10) (2, 10)

(1, 0)

(0, 2) (5, 4)

(5, 8)

A

B

C

3.6 Finding the equation of a straight line from itsgraph: the CAS calculator methodWhile the interceptslope method of nding the equation of a line from its graph is relatively

quick and easy to apply, the two-point method can be tedious to apply. An alternative to using

either of these methods is to use the line-tting facility of your CAS calculator. If you go on to

study Further Mathematics, you will use this facility extensively. It is known as linear

regression.

The advantage of the CAS calculator method is that it works all the time, provided the

coordinates of the points are entered in the correct order. The disadvantage of using linear

regression is that it will give you the wrong results if you do not enter the coordinates of the

points in the correct order. So take care.


Chapter 3 Linear graphs and models 121

How to nd the equation of a line from two points using the TI-Nspire CAS

Find the equation of the line that passes through the two points (2, 1) and (4, 10).

Steps1 Write the coordinates of the two points.

Call one point A, the other B.

The line passes through the pointsA(2, 1) and B (4, 10).

2 Start a new document (/ +N) and selectAdd Lists & SpreadsheetEnter the coordinate values into lists

named x and y.

3 Plot the two points on a scatterplot.

Press / +I and select Add Data &Statistics.(or press and arrow to and press

)

Note: A random display of dots will appear thisis to indicate list data is available for plotting.It is not a statistical plot.

To construct a scatterplot

a Press e and select the variable x

from the list. Press to paste the

variable x to the x-axis.

b Press e again and select the variable

y from the list. Press to paste the

variable y to the y-axis axis to generate

the required scatter plot.





4 Use the Regression command to draw aline through the two points and

determine its equation.

Press b>Analyze>Regression>ShowLinear (a+bx) and to complete thetask.

Correct to one decimal place, the

equation of the line is:

y = 8.0 + 4.5x5 Write your answer. The equation of the line is Y = 8+ 4.5x.

How to nd the equation of a line from two points using the ClassPad

Find the equation of the line that passes through the two points (2, 4) and (4, 10).

Steps1 Open the Statistics application

and enter the coordinate values

into lists named x and y, as

shown.

2 Plot the two points on a

scatterplot.

a Tap from the toolbar to

open the Set StatGraphsdialog box.

b Complete the dialog box as

follows. For Type: select Scatter ( ) XList: select main \ x ( ) YList: select main \ y ( )

Leave Freq: as 1

Tap h to conrm your selections.





For example, in the scatterplot opposite, the

advertised prices of 12 second-hand cars are

plotted against the cars ages (in years).

In this relationship, the cars price is clearly

the dependent variable (DV) as it depends on

its age, so price is plotted on the vertical

axis. Age, the independent variable (IV), is

plotted on the horizontal axis. 0 2 4 6 8

Pri

ce (

$00

0)

Age (years)

16

14

12

10

8

Using a graphics calculator to construct a scatterplotWhile you need to understand the principles of constructing a scatterplot, and maybe to

construct one by hand for a few points, in practise you will use a graphics calculator to

complete this task.

How to construct a scatterplot using the TI-Nspire CAS

The data below give the marks that students obtained on an examination and the times

they spent studying for the examination.

Time (hours) 4 36 23 19 1 11 18 13 18 8

Mark (%) 41 87 67 62 23 52 61 43 65 52

Use a graphics calculator to construct a scatterplot. Treat time as the independent

(x) variable.

Steps1 Start a new document (/ +N) and select Add

Lists & SpreadsheetEnter the data into lists named time and mark.

2 Statistical graphing is done through the Data &Statistics application.Press / +I and select Add Data & Statistics.(or press c and arrow to and press )

Note: A random display of dots will appear this is toindicate list data is available for plotting. It is not astatistical plot.




Chapter 4 Bivariate data 1 143

3 To construct a scatterplot

a Press e and select the variable time from thelist. Press to paste the variable time to thex-axis.

b Press e again and select the variable markfrom the list. Press to paste the variable

mark to the y-axis axis to generate the requiredscatter plot. The plot is automatically scaled.

Note: To add colour (or change colour), move cursorover the plot and press / +b>Color>Fill Color.

How to construct a scatterplot using the ClassPad

The data below give the marks that students obtained on an examination and the

times they spent studying for the examination.

Time (hours) 4 36 23 19 1 11 18 13 18 8

Mark (%) 41 87 67 62 23 52 61 43 65 52

Use a graphics calculator to construct a scatterplot. Treat time as the independent

(x) variable.

Steps1 Open the Statistics application and

enter the coordinate values into

lists named time and mark, asshown.

2 Tap from the toolbar to open

the Set StatGraphs dialog box.





Rev

iew


11 For the scatterplot shown, the line drawn by eye

would have an equation closest to:

A velocity = 5 timeB velocity = 19 + 1 timeC velocity = 1 + 19 timeD velocity = 19 + 5 timeE velocity = 5 + 19 time

0

5

10

15

20

25

30

1 2 3 4 5Time (s)

Vel

ocit

y (m

/s)

12 For the scatterplot shown, the line drawn by

eye would have a slope closest to:

A 2000B 1000C 200D 2000

E 1000

0 2 4 6 8P

rice

($

000)

Age (years)

16

14

12

10

8

The following information relates to Questions 13 and 14The weekly income and weekly food costs for a group of 10 university students is given

in the following table.

Income ($) 150 250 300 300 380 450 600 850 950 1000

Food cost ($) 40 60 70 130 150 260 120 460 200 600

13 The equation of the two-mean line would be found by nding the equation of the

line passing through the points:

A (276, 90) and (770, 328) B (300, 70) and (850, 460)

C (90, 276) and (328, 770) D (150, 40) and (1000, 600)

E (276, 84) and (770, 334)

14 The equation of the two-mean line that would enable food cost to be predicted from

weekly income is closest to:

A food cost = 0.48 + 43 income B food cost = 0.48 43 incomeC food cost = 43 + 0.48 income D food cost = 240 + 1.4 incomeE food cost = 1.4 + 240 income

The following information relates to Questions 15 and 16For incomes between $600 and $1200 per week, the equation of a line that relates

weekly expenditure on entertainment (in dollars) to weekly income (in dollars) is given

by:

expenditure = 40 + 0.10 income



Rev

iew


4 The following table gives the gold-medal winning distance, in metres, for the mens

long jump for the Olympic games for the years 1896 to 1996. (Some years are

missing owing to the two world wars.)

Year 1896 1900 1904 1908 1912 1920 1924 1928 1932 1936 1948 1952 1956

Distance (m) 6.35 7.19 7.34 7.49 7.59 7.16 7.44 7.75 7.65 8.05 7.82 7.57 7.82

Year 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004

Distance (m) 8.13 8.08 8.92 8.26 8.36 8.53 8.53 8.72 8.67 8.50 8.55 8.59

a Which is the independent variable and which is the dependent variable?

b Construct a scatterplot of these data.

c Describe the association between the distance and year.

d Determine the value of the q-correlation coefcient for these data, and classify the

strength of the relationship.

e Determine the equation for the two-mean line and write down in terms of the

variables distance and year.

f Use your equation to predict the winning distance in the year 2008.

g How reliable is the prediction made in part f?

5 To test the effect of driving instruction on driving skill, 10 randomly selected learner

drivers were given a driving skills test. The number of hours of instruction for each

learner was also recorded. The results are displayed in the table below.

Hours 19 2 5 9 16 4 19 26 14 8

Test score 32 12 17 19 23 16 28 36 30 23

a Which is the independent variable and which is the dependent variable?

b Construct a scatterplot of these data.

c Describe the relationship between the number of hours of instruction and the

score.

d Determine the value of the q-correlation coefcient for these data and classify the

strength of the relationship.

e Determine the equation for a line by eye and write down, in terms of the variables,

the number of hours of instruction and score.

f Use the equation to predict the score after 10 hours of instruction.


P1: FXS/ABE P2: FXS0521672600Xc05a.xml CUAU034-EVANS October 9, 2011 14:12

Chapter 5 Shape and measurement 207

2 Calculate the missing dimensions, marked x and y, in these pairs of similar triangles.

a

10 cm

9 cm x cm

6 cm

18 cm y cm

b

48 m52 m

20 m 10 m

y mx m

3 A triangle with sides 5 cm, 4 cm and 8 cm is similar to a larger triangle with a longest side

of 56 cm.

a Find the lengths of the larger triangles other two sides.

b Find the perimeter of the larger triangle.

4 A tree and a 1 m vertical stick cast their shadows at a particular time in the day. The shadow

lengths are shown in the diagram below (not drawn to scale).

a Give reasons why the two triangles shown are similar.

b Find the scale factor for the side lengths of the triangles.

c Find the height of the tree.

Shadow of tree 30 30Shadow of stick

1 m

4 m


P1: FXS/ABE P2: FXS0521672600Xc05a.xml CUAU034-EVANS October 9, 2011 14:12


5 John and his younger sister, Sarah, are standing side by side. Sarah is 1.2 m tall and casts a

shadow 3 m long. How tall is John if his shadow is 4.5 m long?

Sarahs shadow 3 m

Johns shadow 4.5 m

1.2 m

6 The area of triangle A is 8 cm2.

Triangle B is similar to triangle A.

What is the area of triangle B?

Triangle ATriangle B

3 cm

9 cm

8cm2

5.12 Similar solidsTwo solids are similar if they have the same shape and the ratios of their corresponding linear

dimensions are equal.

Cuboids

3 cm

9 cm

6 cm

Cuboid A

2 cm3 cm

1 cm

Cuboid B

The two cuboids above are similar because:

they are the same shape (both are cuboids)

the ratios of the corresponding dimensions are the same.

3

1= 9

3= 6

2= 3

1Height of cuboid A

Height of cuboid B= width of cuboid A

width of cuboid B= length of cuboid A

length of cuboid B

Ratio of volumes = 9 6 33 2 1 =

162

6= 27

1= 3

3

1


P1: FXS/ABE P2: FXS0521672600Xc05b.xml CUAU034-EVANS October 11, 2011 7:5


The gures may be translated, rotated or reected to t into place.

Only three regular polygons will tessellate:

Equilateral triangles Squares Regular hexagons

These are called regular tessellations. The reason why these shapes tessellate is that, at each

point where the shapes meet, their angles sum to 360. A combination of these shapes couldtherefore also tessellate.

When two different types of regular polygon are used to tessellate, the pattern is called a

semi-regular tessellation. A pattern that includes one or more types of irregular polygons is

called an irregular tessellation.

Semi-regular tessellation Irregular tessellation

The 20th-century Dutch artist M. C. Escher is famous for his tessellations. Many of his works

can be seen in calendars, books and posters, and involve changing the original shape so that the

area remains the same. You can view much of his work online at http://www.mcescher.com.

Further information on tessellations can be found on the website

http://www.tessellations.org/.

Other tessellating shapes can be made by starting with a regular polygon, cutting out a shape

and placing that shape on the opposite side of the polygon.

Example 24 Making a tessellating shape

1 Start with a regular polygon that tessellates (e.g. a square).

2 Cut out a shape on one side:

3 Copy the shape onto the other side.

4 This shape will then tessellate.



Chapter 5 Shape and measurement 215

Exercise 5N

1 Which of the following shapes will tessellate on a at surface? Explain each of your

answers.

a b c d

2 The measure of one interior angle of a regular decagon (10-sided polygon) is 144. Explainwhether or not a regular decagon will tessellate.

3 Suggest three other regular polygons that will not tessellate. Give a reason for each.

4 Choose a polygon that will tessellate and use it to make an interesting pattern on square or

dot paper.

5 Does the following shape form a semi-regular tessellation? If so, copy and continue the

pattern.

6 Choose another combination of two or three polygons that will tessellate and use them to

make an interesting pattern.

7 Draw a regular polygon that tessellates and then use it to make other interesting patterns by

cutting a piece out.



Rev

iew


7 Find the circumferences of the following circles, correct to 2 decimal places.

a

5 cm

b

24 cm

8 Find the areas of the circles in Question 7, correct to 2 decimal places.

9 For the solid shown on the right, nd, correct to 2 decimal places:

a the area of rectangle BCDE

b the area of triangle ABE

c the length AE

d the area of rectangle AEGH

e the total surface area. 12 m

10 m

3 m3 m

A

B

C D

F

G

E

H

10 Find the volume of a rectangular prism with length 3.5 m, width 3.4 m and height

2.8 m.

Extended-response questions

1 A lawn has three circular ower beds in it, as shown

in the diagram. Each ower bed has a radius of 2 m.

A gardener has to mow the lawn and use a

whipper-snipper to trim all the edges. Calculate:

a the area to be mown

b the length of the edges to be trimmed.

Give your answer correct to 2 decimal places.8 m

8 m

8 m

16 m

2 Chris and Gayle decide to build a

swimming pool on their new housing

block. The pool will measure 12 m by 5 m

and it will be surrounded by timber decking

in a trapezium shape. A safety fence will

surround the decking. The design layout of

the pool and surrounding area is shown in the diagram.

a What length of fencing is required? Give your answer correct to 2 decimal places.

b What area of timber decking is required?

c The pool has a constant depth of 2 m. What is the volume of the pool?

d The interior of the pool is to be painted white. What surface area is to be painted?

16 m

4 m4 m

10 m12 m

5 m




If the money is invested for more or less than 1 year, the amount of interest payable is

proportional to the length of time for which it is invested.

Example 6 Calculating simple interest for periods other than one year

Calculate the amount of simple interest that will be paid on an investment of $5000 at 10%

simple interest per annum for 3 years and 6 months.

Solution

Apply the formula with P = $5000, r = 10%and t = 3.5 (since 3 years and 6 months isequal to 3.5 years).

I = Prt100

= 5000 10100

3.5= $1750

Example 7 Calculating the total amount borrowed or invested

Find the total amount owed on a loan of $16 000 at 8% per annum simple interest at the end of

2 years.

Solution

1 Apply the formula with P = $16 000,r = 8% and t = 2 to nd the interest.

I = Prt100

= 16000 8100

2= $2560

2 Find the total owed by adding the

interest to the principal.

A = P + I = 16000+ 2560= $18560

The graphics calculator enables us to investigate simple interest problems using both the tables

and graphing facilities of the calculator.

How to solve simple interest problems using the TI-Nspire CAS

How much interest is earned if $10 000 is invested at 8.25% simple interest for

10 years? Show that the graph of simple interest earned is linear.

Steps1 Substitute P = $10 000 and r = 8.25%

in the formula for simple interest.I = Prt

100= 10000 8.25 t

100= 825t


Chapter 6 Financial arithmetic 231

2 Start a new document (/ +N) and select AddLists & SpreadsheetName the lists time (to represent time in years)and interest.Enter the data 1 10 into the list named time asshown.

Note: you can also use the sequence command to do this.

3 Place the cursor in the grey formula cell in the list

named interest and type = 825 timeNote: you can also use the h key and paste time from thevariable list.

Press to display the values.

By scrolling down the table (use ) we can see

interest of $8250 will be earned after 10 years.

4 Press / +I and select Data & Statistics andplot the graph as shown.

a To connect the data points. Move the

cursor to the graphing area and press

/+b. Select Connect Data Pointsb To display a value:

Move the cursor over the data points

or use b>Analyze>Graph Trace andthe horizontal arrow keys to move

from point to point.

From the plot we can see that the

graph of the amount of simple interest

earned is linear. The slope of the

graph is equal to the interest paid

each year.

Note: you can also graph this example in the Graphs application.





How to investigate compound interest problems using the TI-Nspire CAS

a Determine, to the nearest dollar, the amount of money accumulated after 3 years if

$2000 is invested at an interest rate of 8% per annum, compounded annually.

b Determine the amount of interest earned.

c Show that the graph of the amount of money accumulated curves upwards.

Steps1 Substitute P = $2000 and r = 8 into the

formula for compound interest.A = 2000

(1+ 8

100

)t

2 Start a new document (/ +N) and select AddLists & SpreadsheetName the lists time (to represent time in years)and amount.Enter the data 1 10 into the list named time asshown.Note: you can also use the sequence command to dothis.

3 Place the cursor in the grey formula cell in the

list named amount and type in:= 2000 (1 + 8 100) timeNote: you can also use the h key and paste timefrom the variable list

Press to display the values as shown.

By scrolling down the table we can see

that

a the amount of money accumulated after

3 years is $2519.42

b interest earned = $2519.42 $2000 = $519.424 Press / +I and select Add Data & Statistics

and plot the graph as shown.Notes:1 To connect the data points. Move the cursor to

the graphing area and press / +b. SelectConnect Data Points

2 To display a value: Move the cursor over the datapoints or use b>Analyze>Graph Trace.

3 You can use / +b and select Zoom>WindowSettings and set the Ymin to 0 if you prefer.

From the plot we see that, for compound interest, the graph of amount of money

accumulated curves upwards with time.




Chapter 6 Financial arithmetic 249

How to determine at rate depreciation and book value using the TI-Nspire CAS

Michael purchases a new car for $24 000. If it decreases in value by 10% of the purchase

price each year:

a What is the amount of the annual de

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