10 - haesemathematics.com€¦ · an r value of 1 suggests that there is a perfect linear...

Contents:

10

Models of growth

A

B

C

D

E

F

G

H

Correlation

Two variable analysis

Laws of indices (Revision)

Graphs of exponentials

Growth and decay

Exponential equations

Some other applications of exponentials

Function fitting

Review set 10

SA_11APPmagentacyan yellow black

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Y:\HAESE\SA_11APP-2ed\SA11APP-2_10\497SA11APP-2_10.CDR Thursday, 6 September 2007 12:28:13 PM PETERDELL

When a quantity changes we say that it shows variation.

Variation could mean getting larger as in growth, getting faster, decreasing in price, or many

other things.

Some variation will be regular and predictable and some will not.

The variation in one quantity may depend on the variation in another.

For example:

² Does variation in a company’s profit depend on its sales, or vice versa?

² Does blood pressure depend on blood alcohol level, or vice versa?

In the examples above, a company’s profit is the dependent variable as it depends on the

company’s sales. Also blood pressure depends on blood alcohol level (amongst other things).

So, blood pressure is the dependent variable.

Often we want to know how two variables are associated or related. Does the increase in

one variable generally result in an increase or decrease in the other?

To find such a relationship, the first step is to construct and observe a scatterplot. On the

horizontal axis we put the independent variable and on the vertical axis the dependent

variable. A typical scatterplot could look like:

TWO VARIABLE ANALYSIS

The variables are level of histamine and the dose.

role: dependent independent

Medical researchers develop a new drug. They must find the appropriate dose forit to do its job, which is to reduce the level of histamines in the body.

Determine the variables involved in this scenario and their role.

Self TutorExample 1

The weight of a personis usually dependent on

their height.

or this

weight

height

dependent variable

independent variable

A variable which depends on another is called the and the othervariable is called the .

dependent variable

independent variable

498 MODELS OF GROWTH (Chapter 10) (T9)


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Y:\HAESE\SA_11APP-2ed\SA11APP-2_10\498SA11APP-2_10.CDR Monday, 17 September 2007 3:17:48 PM PETERDELL

OPENING PROBLEM

0

The height and weight of members of a cricket team is to be investigated.

Player Height Weight Player Height Weight

1 193 101 7 170 732 179 88 8 176 793 183 90 9 178 884 177 81 10 171 795 176 80 11 169 816 187 87 12 181 87

Things to think about:

² Are the variables categorical or quantitative?

² What is the dependent variable?

² What would the scatterplot look like? Are the points close to being linear?

² Does an increase in the independent variable generally cause an increase or a

decrease in the dependent variable?

² Is there a way of indicating the strength of a linear connection for the variables?

² How can we find the equation of ‘the line of best fit’ and how can we use it?

The scatterplot for the Opening Problem is

drawn alongside. Height is the independent

variable and is represented on the horizontal

axis.

In general, as the height increases, the

weight increases also.

Correlation is a measure of the relationship or association between two variables.

In looking at the correlation between two variables we should follow these steps:

Step 1: Look at the scatterplot for any pattern.

For a generally upward trend we say that the

correlation is positive.

An increase in the independent variable means

that the dependent variable generally increases.

70

80

90

100

110

165 170 175 180 185 190 195

height (cm)

weight (kg)

Weight versus Height

CORRELATIONA

The raw data is given below with heights in cm and weights in kg.

MODELS OF GROWTH (Chapter 10) (T9) 499


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For a generally downward trend we say that the

correlation is negative.

Step 2: Look at the spread of points to make a judgement about the strength of the corre-

lation. For positive relationships we would classify the following scatterplots as:

Similarly there are strength classifications for negative relationships:

Step 3: Look at the pattern of points to see whether or not it is linear.

Step 4: Look for and investigate any outliers.

These appear as isolated points away

from the main body of data.

Outliers should be investigated as they

are sometimes mistakes made in record-

ing or plotting the data.

Genuine extraordinary data should still

be included.

strong moderate weak

strong moderate weak

outlier

not anoutlier

These points appear to be roughly linear. These points do not appear to be linear.

An increase in the independent variablemeans that the dependent variable generally

.decreases

For points (with noupward or downward trend) there is usually

.

randomly scattered

no correlation



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Looking at the scatterplot of the Opening Problem data we can say that:

‘there appears to be a strong positive correlation between the cricketers’ heights and

weights. The relationship appears to be linear with no obvious outliers.’

The association between two variables can be measured using the concept known as

correlation.

The correlation coefficient (r) is a number calculated from the data using a known formula.

It indicates the strength and direction of the linear association between the variables.

The correlation coefficient always lies between ¡1 and 1.

An association between two variables is described as a positive correlation if an increase

in one variable results in an increase in the other in an approximately linear manner.

For positive correlation, r ranges between 0 and 1.

An r value of 0 suggests that there is no linear association present (or no correlation).

An r value of 1 suggests that there is a perfect linear association present (or perfect positive

correlation).

The correlation between the height and the weight of people is positive and lies between 0and +1. It is not an example of perfect positive correlation because, for example, not all

short people are of light weight. However, taller people are generally heavier than shorter

people.

Scatter diagrams for positive correlation:

The scales on each of the four graphs are the same.

An association between two variables is described as a negative correlation if an increase

in one variable results in a decrease in the other in an approximately linear manner.

For negative correlation, r ranges between 0 and ¡1.

An r value of ¡1 suggests that there is a perfect linear association present (or perfect negative

correlation).

THE CORRELATION COEFFICIENT ( )r

POSITIVE CORRELATION

NEGATIVE CORRELATION

y

x

��r =

y

x

��r =

y

x

��r =

y

x

��r =



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Scatter diagrams for negative correlation:

When analysing data, we must be aware of causation. A high degree of correlation between

two variables does not necessarily mean that a change in one variable causes the other to

change.

For example:

The heights and reading speeds of children were measured and a strong positive correlation

was found. Does this mean that increasing height makes you read faster or that increasing

your reading speed will cause you to grow? These suggestions are obviously not sensible.

The strong correlation results because both variables are closely associated with age. As age

increases, both the variables height and reading speed increase. It is age which causes height

and reading speed to increase.

If a change in one variable causes a change in the other variable then we say that a

causal relationship exists between them.

For example:

1 For each of the scatterplots below state:

i whether there is positive, negative or no association between the variables

ii whether the relationship between the variables appears to be linear or not

iii the strength of the association (zero, weak, moderate or strong).

a b c

y

x

��r =

y

x

��r =

y

x

��r =

y

x

�r =

CAUSATION

EXERCISE 10A

y

x

y

x

y

x

The age and height of a group of children is measured and there is a strongpositive correlation between these variables. This will be a

because an increase in age will cause an increase in height.causal

relationship



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y

x x

yy

x

d e f

2 Copy and complete the following:

a If the variables x and y are positively associated then as x increases, y .............

b If there is negative correlation between the variables m and n then as m increases,

n ..............

c If there is no association between two variables then the points on the scatterplot

appear to be ............. ...............

The statistician Pearson constructed a formula for calculating a correlation coefficient, which

he denoted r.

For a set of n data given as ordered pairs (x1, y1), (x2, y2), (x3, y3), ......, (xn, yn),

Pearson’s correlation coefficient r =

Pxy ¡ nxyp

(P

x2 ¡ nx2)(P

y2 ¡ ny2)

where x and y are the means of the x and y data respectively andP

means the

sum over all the data values.

To help describe the strength of asso-

ciation we calculate the coefficient of

determination (r2). This is simply the

square of the correlation coefficient (r)

and as such the direction of association

is eliminated.

Many texts vary on the advice they

give. We suggest using the table along-

side when describing the strength of

linear association.

value strength of association

r2 = 0 no correlation

0 < r2 < 0:25 very weak correlation

0:25 6 r2 < 0:50 weak correlation

0:50 6 r2 < 0:75 moderate correlation

0:75 6 r2 < 0:90 strong correlation

0:90 6 r2 < 1 very strong correlation

r2 = 1 perfect correlation

TWO VARIABLE ANALYSISB

PEARSON’S CORRELATION COEFFICIENT ( )r

THE COEFFICIENT OF DETERMINATION ( )r2

Pearson’s correlation coefficient, often called the , iscalculated directly from the data set.

linear correlation coefficient



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A chemical fertiliser company wishes

to determine the extent of correlation

between the ‘quantity of compound X

used’ and ‘lawn growth per day’.

Find and interpret the correlation

coefficient between the two variables.

Lawn Compound Lawn

X (gms) Growth (mm)

A 1 3B 2 3C 4 6D 5 8

x y x2 y2 xy

1 3 1 9 3

2 3 4 9 6

4 6 16 36 24

5 8 25 64 40

totals: 12 20 46 118 73

) x =

Px

n

=12

4

= 3

and y =

Py

n

=20

4

= 5

r =

Pxy ¡ nx yp

(P

x2 ¡ nx2)(P

y2 ¡ ny2)=

73 ¡ 4 £ 3 £ 5p(46 ¡ 4 £ 32)(118¡ 4 £ 52)

=13p

10 £ 18

+ 0:969 and so r2 + 0:939

There is a very strong positive correlation between the quantity of compound X

used and lawn growth.

This suggests that the more of compound X used, the greater the lawn growth

per day. However, the small amount of data may provide a misleading result.

1 Consider the three graphs given below:

A B C

Clearly: A shows perfect positive linear correlation

B shows perfect negative linear correlation

C shows no correlation.

a For each set of points, find r using r =

Pxy ¡ nxyp

(P

x2 ¡ nx2)(P

y2 ¡ ny2)

b Comment on the values of r for each graph.

Self TutorExample 2

EXERCISE 10B.1

x

y

( )��,

( )��,

( )��,

( )��,x

y

1

2

1 2 3

3

x

y

( )��,

( )��,

( )��,

( )��,



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2 Determine Pearson’s correlation coefficient for the variables in the following cases,

commenting on the result of each calculation.

a Length and Weight of mice:Length (cm) 5 6 7 8

Weight (g) 14 22 26 30

b Weedicide concentration and

Number of weeds surviving:Weedicide (mg/L) 5 10 15 20

Number of weeds surviving 43 25 10 2

We will now take the chemical fertiliser problem in Example 2 a little further.

We know that there is a very strong positive correlation between the two variables.

We should therefore be able to find a

linear function, often called a line of

best fit, to closely model the relation-

ship between length and mass. We

could fit a line ‘by eye’, as shown

alongside.

However, different people will use

different lines.

So, how do we find the line of best fit

mathematically?

Statisticians invented a method where the best line results. The process is minimisation of

the sum of the squares of the residuals.

The mathematics behind this method is generally

established in university mathematics courses,

but in brief what happens is that we find ver-

tical distances or residuals to the line of best fit:

d1, d2, d3, .... etc.

We then add the squares of the residuals,

i.e., d 21 + d 2

2 + d 23 + ::::::

The least squares regression line is the one

which makes this sum as small as possible.

Click on the icon and use trial and error to try to find the least

The least squares line has equation y = mx+ c where m =

Pxy ¡ 1

n(P

x)(P

y)

(P

x2) ¡ 1

n(P

x)2

and c = y ¡mx

LEAST SQUARES ‘LINE OF BEST FIT’

0

2

4

6

8

10

0 1 2 3 4 5 6 7 8 9 10

line of best fit (by eye)

Growth ( mm)y

Compound ( g)X x

d1

d2 d3

d4

d5

y

x

COMPUTER

DEMO


squares line of best fit for the data sets given.


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Y:\HAESE\SA_11APP-2ed\SA11APP-2_10\505SA11APP-2_10.CDR Friday, 7 September 2007 10:55:14 AM PETERDELL

Once the line of best fit is determined and you have its equation then you can use it to predict

other values that the experiment did not reveal.

For the chemical fertiliser data of Example 2, find the least squares ‘line of best fit’.

x y xy x2

1 3 3 12 3 6 44 6 24 165 8 40 25P12 20 73 46

x =

Px

n=

12

4= 3, y =

Py

n=

20

4= 5

m =

Pxy ¡ 1

n(P

x)(P

y)

(P

x2) ¡ 1

n(P

x)2

) m =73 ¡ 1

4£ 12 £ 20

46 ¡ 1

4£ 122

= 1:3

and c = y ¡mx = 5 ¡ 1:3 £ 3 = 1:1

) line of best fit is y = 1:3x + 1:1

With many points and much more complicated numbers, the task of finding r, r2, m and c

can be very tedious using the given formulae.

We use technology to ease the work load.

Self TutorExample 3

USING A COMPUTER PACKAGE

COMPUTER

PACKAGEClick on the icon to obtain the package. Enter thedata and observe how the correlation coefficient, coefficient of determination,and line of best fit are automatically calculated.

Two Variable Analysis



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Returning to the chemical fertiliser example we can use the line of best fit y = 1:3x + 1:1to predict the lawn growth per day for other quantities of compound x.

If we substitute a quantity in between the smallest and largest quantities in the experiment,

we say we are interpolating (in between the poles).

If we predict length values for quantities outside the smallest and largest quantities in the

experiment, we say we are extrapolating (outside the poles).

The accuracy of an interpolation depends on

how linear the original data was. This can be

gauged by determining the correlation coeffi-

cient and ensuring that the data is randomly

scattered around the line of best fit.

The accuracy of an extrapolation depends not

only on ‘how linear’ the original data was, but

also on the assumption that the linear trend

will continue past the poles.

The validity of this assumption depends

greatly on the situation under investigation.

In the chemical fertiliser example, there is

a point where the addition of more fertiliser

will no longer be beneficial, and may even be

harmful, to the lawn’s growth.

USING A GRAPHICS CALCULATOR

INTERPOLATION AND EXTRAPOLATION

lowerpole

upperpole

line ofbest fit

y

x

interpolationextrapolation extrapolation

Using a Casio fx-9860G

Select STAT from the main menu, then key

the x values into List 1 and the y values

into List 2.

To find the line of best fit, press (¤)

if the GRPH icon is not in the bottom left

corner of the screen, then press

(CALC) (REG) (X).

The line of best fit y = 1.3x + 1.1 is

given.

Using a Texas Instruments TI-83

Press 1 to enter the list editor, then

key the x values into L1 and the y values

into L2.

To find the line of best fit, press

4, which selects 4:LinReg(ax+b) from the

CALC menu.

Press 1 (L1) , 2 (L2)

ENTER to display the result. The line of

best fit is given as y = 1.3x + 1.1



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Y:\HAESE\SA_11APP-2ed\SA11APP-2_10\507SA11APP-2_10.CDR Friday, 14 September 2007 10:44:50 AM DAVID3

An example could be the world record for the long jump

prior to the Mexico City Olympic Games of 1968. A

steady regular increase in the World record over the pre-

vious 30 years had been recorded. However, due to the

high altitude and a perfect jump, the USA competitor

Bob Beamon shattered the record by a huge amount,

not in keeping with previous increases.

In general, interpolation between the poles is reliable, but extrapolation outside the poles can

be unreliable.

1 Tomatoes are sprayed with a pesticide-fertiliser mix. The figures below give the yield of

tomatoes per bush for various spray concentrations.

Spray concentration (x mL/L) 3 5 6 8 9 11

Yield of tomatoes per bush (y) 67 90 103 120 124 150

a Draw a scatterplot for this data.

b Determine the r and r2 values.

c Describe the association between yield and spray

concentration.

d Find the least squares line of best fit for the data.

e Predict the yield when the spray concentration is

7 mL/L. How reliable is this result?

f Predict the yield when the spray concentration is

15 mL/L. How reliable is this result?

2 The following data shows the annual cherry yield and number of frosts data for a cherry

growing farm over a 7 year period.

Number of frosts (x) 27 23 7 37 32 14 16

Cherry yield (y tonnes) 5:6 4:8 3:1 7:2 6:1 3:7 3:8



c Describe the association between the cherry yield

and the number of frosts.


e Predict the yield when the number of frosts is 20.

How reliable is this result?

f Predict the yield when the number of frosts is 45.

How reliable is this result?

CARE MUST BE TAKEN WHEN EXTRAPOLATING

EXERCISE 10B.2



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3 The following data shows the top speed of a Formula One racing car when it is filled

with various quantities of petrol.

Petrol quantity (xL) 50 37 18 85 27 102

Top speed (y km/h) 315 319 323 311 321 308



c Describe the association between the top speed and the petrol quantity.


e If the capacity of the petrol tank is 120 litres, predict the top speed of the car when

the tank is i full ii half full iii nearly empty.

f Which of the predictions in e is likely to be most accurate?

4 The following table gives the value of a television set at various times t years after it

was bought.

Time (t years) 2 5 7 10 12

Value (V dollars) 900 745 655 500 405



c Describe the association between value and time.

d Find the least squares line of best fit of the data.

e Estimate the value of the TV set 15 years after it was bought. How reliable is this

result?

f Is it reasonable to assume that the linear trend displayed by the data will continue

indefinitely? Explain.

Before examining exponential functions and their properties we will revise the laws of indices.

We are already familiar with index or exponential notation. For example, 8 = 2£2£2 = 23

In general, an = a£ a£ a£ a£ ::::£ a| {z }n factors

where n is a positive integer.

Recall that: ² 34 £ 32 = (3 £ 3 £ 3 £ 3) £ (3 £ 3) = 34+2 = 36

as we have six 3s multiplied together.

² 35

32=

3 £ 3 £ 3 £ 3 £ 3

3 £ 3= 35¡2 = 33

² (33)2 = 33 £ 33 = 33+3 = 36

LAWS OF INDICES (Revision)C

i.e., 8 = 23 index, power or exponent

base number



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So, to get from 34 £ 32 to 36, we add indices

35

32to 33, we subtract indices

(33)2 to 36 we multiply indices

² am £ an = am+n

² am

an= am¡n

² (am )n = amn

Notice also that 23 £ 20 = 23+0 = 23, so 20 must be 1.

In fact, a0 = 1 for all a 6= 0

The laws for expanding powers of products and quotients are:

(ab)n = anbn and

µa

b

¶n

=an

bn(b 6= 0)

Finally notice that a¡n £ an = a¡n+n = a0 = 1

Thus, a¡n and an are reciprocals, and a¡n =1

anfor all a 6= 0:

1 Simplify mentally:

a a2 £ a7 b m3 £m4 c d7 £ d d k8 £ k3

en4

n2f

m7

m3g

g6

g4h

s41

s17

i (a5)2 j (m3)4 k (n4)5 l (p7)3

2 Remove the brackets using an index law:

a (ab)5 b (mn)4 c (2xy)2 d (3ab2)3

e³an

´4f

³x2

´4g

µ3

y

¶4

h

µ2a

b

¶4

Examples like these helpus establish that:

EXERCISE 10C

Write as powers of 2: a 16 b 1

16c 1

a 16

= 2 £ 2 £ 2 £ 2

= 24

b 1

16

=1

24

= 2¡4

c 1

= 20

Self TutorExample 4



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3 Write as powers of 2:

a 4

g 2

4 Write as powers of 3:

a 9

f 1

32

l 1

128

f 1

3

g 81

b 1

4

h 1

2

b 1

9

h 1

81

c 8

i 64

c 27

i 1

d 1

8

j 1

64

d 1

27

j 243

e 32

k 128

e 3

k 1

243

5 Write as a single power of 2:

a 2 £ 2a

f2c

4


a 9 £ 3p

f3y

3

b 4 £ 2b

g2m

2¡m

b 27a

g3

3y

c 8 £ 2t

h4

21¡n

c 3 £ 9n

h9

27t

d (2x+1)2

i2x+1

2x

d 27 £ 3d

i9a

31¡a

e (21¡n)¡1

j4x

21¡x

e 9 £ 27t

j9n+1

32n¡1

Write without negative indices: a (3x)¡2 b³ a

2b

´¡3

a (3x)¡2

=

µ3x

1

¶¡2

=

µ1

3x

¶2

=1

9x2

b³ a

2b

´¡3

=

µ2b

a

¶3

=23b3

a3

=8b3

a3

Write as a single power of 2: a 4 £ 2n b2m

8

a 4 £ 2n

= 22 £ 2n

= 22+n

b2m

8

=2m

23

= 2m¡3

Self TutorExample 5

Self TutorExample 6 We can changea negative

index into itspositive by

reciprocatingthe contents ofthe brackets.



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7 Write without negative indices:

a a¡4 b (ab)¡2 c³ab

´¡1

d (2x)¡3

e³x

2

´¡2

f (4b)¡1 g

µ3m

n

¶¡1

h

µ2a

b

¶¡2

Since a12 £ a

12 = a

12+ 1

2 = a1 = a, ffor index laws to be obeyedgand

pa£p

a = a also, then

a1

2 =pa. fby direct comparisong

Likewise a13 £ a

13 £ a

13 = a1 = a,

compared with 3pa£ 3

pa£ 3

pa = a

suggests a1

3 = 3pa.

Thus in general, a1

n = npa. f n

pa reads “the nth root of a”g

Notice also that a23 £ a

23 £ a

23 = a2

)

³a

23

´3= a2 fusing (am)n = amn g

) a23 =

3pa2:

In general, amn = n

pam .

Write as a single power of 2: a3p

2 b1p2

c5p

4

a3p

2

= 213

b1p2

=1

212

= 2¡

12

c5p

4

= (22)15

= 22£ 1

5

= 225


a5p

2 b15p

2c 2

p2 d 4

p2 e

13p

2

f 2 £ 3p

2 g4p2

h (p

2)3 i1

3p

16j

1p8

RATIONAL INDICES

Self TutorExample 7



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a3p

3 b13p

3c

4p

3 d 3p

3 e1

9p

3

Use your calculator to evaluate a 275 b 5

p4

Calculator: Answer:

For 275 so press: 2:639 015

5p

4 = 415 so press: 4 1 5 1:319 507

10 Use your calculator to find:

a 334 b 2

78 c 2

¡13 d 4

¡35

ep

9 f4p

8 g5p

27 h13p

7

Split these fractions into separate terms with no fractions:

a2x2 + 1

xb

x3 + x2 ¡ 2x + 3

x2

a2x2 + 1

x

=2x2

x+

1

x

= 2x + x¡1

bx3 + x2 ¡ 2x + 3

x2

=x3

x2+

x2

x2¡ 2x

x2+

3

x2

= x + 1 ¡ 2x¡1 + 3x¡2

11 Split these fractions into separate terms with no fractions:

ax + 3

xb

x2 ¡ 2

xc

x2 + 3x ¡ 4

x

dx2 + 5x

x2e

x3 + 2x2 ¡ 6

x2f

x3 + 2x2 ¡ 3x + 5

x2

gx4 + x2 + 1

x3h

x5 + 3x2 ¡ 5

x3i

2x4 + 3x2 ¡ 6x + 5

x3

jx2 + x¡1 + 5

x¡1k

x + 2x2 ¡ 6

x¡2l

x2 + 3x ¡ 4x¡1

x¡3

Self TutorExample 9

Self TutorExample 8

2 7 5^ ( ÷ ) ENTER

^ ( ÷ ) ENTER


a

b


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Without using a calculator, write in simplest rational form: a 843 b 27

¡23

a 843 b 27

¡23

= (23)43 = (33)

¡23

= 23£ 4

3 f(am)n = amng = 33£¡

23

= 24 = 3¡2

= 16 = 1

9

12 Without using a calculator, write in simplest rational form:

a 432 b 8

53 c 16

34 d 25

32 e 32

25

f 4¡

12 g 9

¡32 h 8

¡43 i 27

¡43 j 125

¡23

The simplest exponential function has the form y = ax where a > 0, a 6= 1.

For example, y = 2x is an exponential function.

Table of values:x ¡3 ¡2 ¡1 0 1 2 3

y 1

8

1

4

1

21 2 4 8

We notice that for x = ¡10, y = 2¡10 + 0:001

and when x = ¡50, y = 2¡50 + 8:88 £ 10¡16

So, it appears that as x becomes large and negative, the

graph of y = 2x approaches the x-axis from above it.

We say that y = 2x is ‘asymptotic to the x-axis’, or

‘y = 0 is a horizontal asymptote’.

Self TutorExample 10

GRAPHS OF EXPONENTIALSD

INVESTIGATION 1 EXPONENTIAL GRAPHS

1 a On the same set of axes, use a graphing package or graphics calculator to graph

the following functions: y = 2x, y = 3x, y = 10x, y = (1:3)x.

b The functions in a are all members of the family y = bx:

i What effect does changing b values have on the shape of the graph?

ii What is the y-intercept of each graph?

iii What is the horizontal asymptote of each graph?

x

y

1

1

� � 2 3

2

4

6

8

y = 2x

What to do:

We will investigate families ofexponential functions.

TI

C

GRAPHING

PACKAGE



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2 a On the same set of axes, use technology to graph the following functions:

y = (0:5)x, y = (0:8)x, y = (0:96)x.

b The functions in a are all members of the family y = ax, 0 < a < 1.

i What effect does changing a have on the shape of the graph?

ii What is the y-intercept of each graph?

iii What value is y approaching as x is large and negative?

From your investigation you should have discovered that:

1 Given the graph of y = 2x we can find

approximate values of 2x for various

values of x.

For example:

² 21:8 + 3:5 (see point A)

² 22:3 + 5 (see point B)

Use the graph to determine approximate

values of:

a 212 (=

p2) b 20:8

c 21:5 d 2¡1:6

2

a y = 2x and y = 2x ¡ 2 b y = 2x and y = 2¡x

c y = 2x and y = 2x¡2 d y = 2x and y = 2 £ 2x

3 The graph of y = 3x is given alongside. Draw free-

hand sketches of the following pairs of graphs:

a y = 3x and y = 3¡x

b y = 3x and y = 3x + 1

c y = 3x and y = ¡3x

d y = 3x and y = 3x¡1

y

x

1

ƒ( )x ��x

GRAPHING

PACKAGE

for the general exponential function y = ax

² a controls how steeply the graph increases or decreases

² the graphs are all asymptotic to the x-axis (i.e., approach it)

² ²if thegraph isincreasingfor all

a>

x

� �1 ifthe graph isdecreasing forall

0 1� � � �<a<

x

EXERCISE 10D

Draw freehand sketches of the following pairs of graphsbased on your observations from the previous investigation.


11

1111�� 22

22

33

44

yy

xx

y = 2x

y = 2x

AA

1.81.8

3.53.5

BB55


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Y:\HAESE\SA_11APP-2ed\SA11APP-2_10\515SA11APP-2_10.CDR Friday, 14 September 2007 4:59:55 PM DAVID3

For the general exponential function

y = bkx + c, y = c is the horizontal asymptote.

We can actually obtain reasonably accurate sketch

graphs of exponential functions using

² the horizontal asymptote

² the y-intercept

² two other points, say when x = 2, x = ¡2

Sketch the graph of y = 2¡x ¡ 3:

For y = 2¡x ¡ 3the horizontal asymptote is y = ¡3

when x = 0, y = 20 ¡ 3

= 1 ¡ 3

= ¡2

) the y-intercept is ¡2

when x = 2, y = 2¡2 ¡ 3

= 1

4¡ 3

= ¡23

4

when x = ¡2, y = 22 ¡ 3

= 4 ¡ 3

= 1

4 Sketch the graphs of:

a y = 2x + 1 b y = 2 ¡ 2x c y = 2¡x + 3 d y = 3 ¡ 2¡x

In this section we will examine situations where quantities are increasing or decreasing

exponentially. We describe these situations as exponential growth and decay.

Populations of animals, people, bacteria, and so on, usually grow in an exponential way.

Radioactive substances and items that depreciate usually decay exponentially.

y

x

��

y��

y�� x

2

HORIZONTAL ASYMPTOTES

All exponentialgraphs aresimilar in

shape and havea horizontalasymptote.


GROWTH AND DECAYE



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So, if Pn is the population after n weeks, then:

P0 = 100 fthe original populationgP1 = P0 £ 1:2 = 100 £ 1:2

P2 = P1 £ 1:2 = 100 £ (1:2)2

P3 = P2 £ 1:2 = 100 £ (1:2)3

...

etc, and from this pattern we see that Pn = 100 £ (1:2)n:

An entomologist monitoring a grasshopper plague notices that the area affected

by the grasshoppers is given by An = 1000£ 20:2n hectares, where n is the

number of weeks after the initial observation.

a Find the original affected area.

b Find the affected area after i 5 weeks ii 10 weeks.

c Find the affected area after 12 weeks.

d Draw the graph of An against n.

a A0 = 1000 £ 20

= 1000 £ 1

= 1000

) original area was 1000 ha.

b i A5 = 1000 £ 21

= 2000

) affected area is 2000 ha.

ii A10 = 1000 £ 22

= 4000

) affected area is 4000 ha.

c A12 = 1000£ 20:2£12

= 1000£ 22:4 fPress: 1000 2 2:4 g+ 5278

) after 12 weeks, the area affected is about 5300 ha.

d

BIOLOGICAL GROWTH

Consider a population of mice which underfavourable conditions is increasing by each week.To increase a quantity by , we multiply it by

or .

10020%

20%120% 1 2:


×

b i

a

b ii

c

1412108642

6000

4000

2000

A (ha)

n (weeks)

^ ENTER

100

200

300

400

P

n (weeks)

n

1 2 3 4 5 6



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1 The weight of bacteria in a culture t hours after establishment

is given by Wt = 100 £ 20:1t grams. Find:

a the initial weight

b the weight after i 4 hours ii 10 hours iii 24 hours.

c Sketch the graph of Wt against t using only results from a and b.

d Use technology to graph Y1 = 100£ 20:1X and check your answers to a, b and c.

2 A breeding program to ensure the survival of

pygmy possums was established with an ini-

tial population of 50 (25 pairs). From a previ-

ous program, the expected population Pn in n

years’ time is given by Pn = P0 £ 20:3n .

a What is the value of P0?

b What is the expected population after:

i 2 years ii 5 years iii 10 years?

c Sketch the graph of Pn against n using

only a and b.

d Use technology to graph Y1 = 50 £ 20:3X and use it to check your answers in b.

3 The speed Vt of a chemical reaction is given by Vt = V0 £ 20:05t where t is the

temperature in oC. Find:

a the speed at 0oC

b the speed at 20oC

c the percentage increase in speed at 20oC compared with the speed at 0oC.

Now consider a radioactive substance of original

weight 20 grams which decays (reduces) by 5% each

year. The multiplier is now 95% or 0:95.

So, if Wn is the weight after n years, then:

W0 = 20 grams

W1 = W0 £ 0:95 = 20 £ 0:95 grams

W2 = W1 £ 0:95 = 20 £ (0:95)2 grams

W3 = W2 £ 0:95 = 20 £ (0:95)3 grams... etc.

and from this we see that Wn = 20 £ (0:95)n.

If the original weight W0 is unknown then JM

Wn = W0 £ (0:95)n.

EXERCISE 10E

GRAPHING

PACKAGE

DECAY

10 20

25

20

15

10

5

Wn (grams)

n (years)



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Y:\HAESE\SA_11APP-2ed\SA11APP-2_10\518SA11APP-2_10.CDR Friday, 7 September 2007 10:56:06 AM PETERDELL

The weight of radioactive material remaining after t years is given by

Wt = 11:7 £ 2¡0:0067t grams.

a Find the original weight.

b Find the weight after i 10 years ii 100 years iii 1000 years.

c Graph Wt against t using a and b only.

Wt = 11:7 £ 2¡0:0067t

a When t = 0, W0 = 11:7 £ 20 = 11:7 grams

b i W10

= 11:7 £ 2¡0:067

+ 11:17 g

c

ii W100

= 11:7 £ 2¡0:67

+ 7:35 g

iii W1000

= 11:7 £ 2¡6:7

+ 0:11 g

4 The temperature Tt of a liquid which has been placed in a refrigerator is given by

Tt = 100 £ 2¡0:02t degrees Celsius, where t is the time in minutes. Find:

a the initial temperature

b the temperature after i 15 min ii 20 min iii 78 min.

c Sketch the graph of Tt against t using a and b only.

5 The weight Wt of radioactive substance remaining after t years is given by

Wt = 1000 £ 2¡0:03t grams. Find:

a the initial weight

b the weight after i 10 years ii 100 years iii 1000 years.

c Graph Wt against t using a and b only.

6 The weight Wt of radioactive uranium remaining after t years is given by the formula

Wt = 25 £ 2¡0:0002t grams, t > 0. Find:

a the original weight b the percentage weight loss after 2000 years.

7 The current It flowing in a transistor radio t seconds after it is switched off is given by

It = 35 £ 2¡0:02t amps. Find:

a the initial current

b the current after 1 second

c the percentage change in current after 1 second

d I50 and I100 and hence sketch the graph of It against t.


12001000800600400200

12

10

8

6

4

2

(10' 11"17)

(100' 7"35)

(1000' 0"11)

Wt (grams)

t (years)

11.7



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Consider graphs of y = 2x and y = (12)x = 2¡x:

This function is increasing for all x. This function is decreasing for all x.

In each case, for each value of y there is one and only one value of x.

Consequently, if 2m = 2n then m = n.

In general, if am = an then m = n.

This is referred to as ‘equating indices’ and is used when solving exponential equations.

Solve for x: a 4x = 8 b 32x¡1 = 1

9

a 4x = 8

) (22)x = 23

) 22x = 23

) 2x = 3 fequating indicesg) x = 3

2

b 32x¡1 = 1

9

) 32x¡1 = 3¡2

) 2x¡ 1 = ¡2

) 2x = ¡1

) x = ¡1

2

1 Solve for x:

a 2x = 8 b 2x = 32 c 2x = 1

4d 4x = 2

e 4x = 8 f 4x = 1

2g 4x = 1

8h 8x = 4

i 8x = 32 j 8x = 1

4k 4x = 1 l 4x = ¡4

m 23x¡1 = 4 n 21¡2x = 1 o 43x = 8 p 9x = 27

q 9¡x = 3 r 3x = 1 s 5x = 0 t 3x+1 = 1

9

u 3¡x = 1

9v 32¡x = 1

27w 49x = 7 x 252¡x = 0:2

EXPONENTIAL EQUATIONSF

y

x

1

� ��( )x x

y

x

1

� �� ( ) (Qw_ )x x x��


EXERCISE 10F



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Solve for x: a 5 £ 2x = 40 b 18 £ 3x = 2

a 5 £ 2x = 40

) 2x = 8 f¥ both sides by 5g) 2x = 23

) x = 3 fequating indicesg

b 18 £ 3x = 2

) 3x = 1

9f¥ by 18g

) 3x = 3¡2

) x = ¡2

2 Solve for x:

a 3 £ 2x = 24 b 7 £ 2x = 56 c 3 £ 2x+1 = 24

d 12 £ 3¡x = 4

3e 4 £ (1

3)x = 36 f 5 £ (1

2)x = 20

Often it is not possible to select a common base and then equate indices.

For example, in 2x = 5. We use technology to solve these equations.



Select EQUA from the main menu, then

press (Solver).

Enter the equation by pressing 2 ^. (=) 5 .

Make sure the cursor is on the guess for X,

then press (SOLV) to solve the

equation.

The solution x + 2:32 is given.


Press 0 to select 0:Solver… Press

to delete the previous equation.

We must rearrange the equation so that one

side is equal to zero, so the equation

becomes

Enter this by pressing 2 ^ 5

ENTER .

Make sure the cursor is on the guess for X,

then press ENTER (SOLVE) to

solve the equation.

2x ¡ 5 = 0.

The solution x + 2:32 is given.



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3 Solve for x, correct to 3 s.f.:

a 2x = 100 b 3x = 0:067 c 5x = 2578

d (1:2)x = 14:89 e (0:97)x = 0:032 f 2¡x = 0:0075

g 3¡x = 0:000 842 h 2x+2 = 23:96 i 100 £ 2x = 602:8

j 200 £ 3x = 10:8 k 50:8 £ 5x = 1 l 0:0873 £ 2x = 14

4 The weight Wt of bacteria in a culture t hours after establishment is given by

Wt = 20 £ 20:15t grams. Find the time for the weight of the culture to reach:

a 30 grams b 100 grams.

5 The temperature T of a liquid which has been placed in a refrigerator is given by

T = 100 £ 2¡0:03t degrees Celsius, where t is the time in minutes. Find the time

required for the temperature to reach:

a 25oC b 1oC.

6 The weight Wt of radioactive substance remaining after t years is given by

Wt = 1000 £ 2¡0:04t grams. Find the time taken for the weight to:

a halve b reach 20 grams c reach 1% of its original value.

The weight of radioactive material remaining after t years is given by

W = 30 £ 2¡0:001t grams. Find:

a the original weight

b how long it would take for the material to decay to 10 grams

c how long it would take for the material to decay to the ‘safe level’ of 1%of its original value.

a When t = 0, W = 30 £ 20 = 30

So, 30 g is the original weight.

b For W = 10

30 £ 2¡0:001t = 10

) 2¡0:001t = 1

3

) t + 1584 fusing technologyg) it would take about 1580 years.

c For W = 1% of the original weight

30 £ 2¡0:001t = 1

100£ 30

) 2¡0:001t = 0:01

) t + 6644 fusing technologyg) it would take about 6640 years.




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7 The weight W of radioactive uranium remaining after t years is given by the formula

W = 1200 £ 2¡0:0002t grams, t > 0. Find the time taken for the original weight to

fall to:

a 25% of its original value b 0:1% of its original value.

8 A man jumps from the basket of a stationary hot

air balloon. His speed of descent is given by

V = 50(1 ¡ 2¡0:2t) m/s where t is the time in

seconds.

Find the time taken for his speed to reach 40 m/s.

9 Hot cooking oil is placed in a freezer and

its temperature after m minutes is given by

Tm = (225 £ 2¡0:17m ¡ 6) oC. Find the time

taken for the temperature to fall to 0oC.

There is a very large range of sound intensity that is audible to the human ear.

The intensity of a sound is measured by the formula I = I0 £ 100:1S

where I0 = the threshold intensity for hearing (minimum audible)

and S = the perceived loudness (in decibels).

How many times louder is:

a 30 decibels than the threshold intensity

b sound of 28 decibels than sound of 15 decibels?

a If S = 30,

then I = I0 £ 100:1£30

) I = I0 £ 103

) I = 1000 £ I0

) it is 1000 times louder.

b If S = 15, I1 = I0 £ 101:5

If S = 28, I2 = I0 £ 102:8

)I2

I1=

I0 £ 102:8

I0 £ 101:5= 101:3 + 20

) I2 + 20I1

) it is about 20 times louder.

SOME OTHER APPLICATIONS

OF EXPONENTIALSG

SOUND INTENSITY MEASUREMENT




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The size (or magnitude) of an earthquake’s intensity depends on the amplitude of the ground

movements during the quake.

These variables are related by the formula: A = 10M

where M = quake magnitude on the Richter Scale

and A = amplitude of ground movements.

Note: ² A quake measuring M > 6 usually causes great damage.

² A quake measuring M < 2 usually cannot be noticed by humans.

The energy released by an earthquake of magnitude M is given by

E + 101:5M+4:8 joules.

a How much more intense is a quake of magnitude 6:1 compared with one of

magnitude 4:7?

b Find the energy released by a quake of magnitude 6:1.

a When M1 = 4:7, A1 = 104:7 and when M2 = 6:1, A2 = 106:1

So,A2

A1

=106:1

104:7= 101:4 + 25:1 and ) A2 + 25A1

) the ground movement is just more than 25 times greater.

b When M = 6:1, E + 101:5£6:1+4:8 + 1013:95 + 8:9 £ 1013 joules

1 How many times louder is:

a an aeroplane at 125 db compared with the airport lounge at 50 db

b a conversation at 55 db compared with ‘library silence’ at 15 db

c a motor car at 70 db compared with a busy street at 53 db?

2 Calculate the increase in ground movement that occurs between two earthquakes with

Richter Scale measurements:

a 2 and 4 b 3:3 and 6:3 c 4 and 8:5

3 Calculate the total energy released by a quake of magnitude:

a 3 b 6:5 c 9:2

4 How many times greater is the energy released by an earthquake of magnitude 7 compared

with one measuring 5?

EARTHQUAKE INTENSITY MEASUREMENT


EXERCISE 10G



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Suppse we have just conducted a scientific experiment, and have obtained a set of data points

as results. What do we need to do to fit an appropriate function to them?

For example, given the points:

² Linear ² Quadratic

f(x) = ax + b f(x) = ax2 + bx + c

² Exponential ² Power

f(x) = abx f(x) = axb

Graph must pass through O(0, 0).

There are many other function families, but in this course we will consider only those given

above. Given only a few points, the problem we face is in identifying which is the most

appropriate function type to fit our data. We will use technology to assist us in this section.

“Which function should we choose to model the data?” is an interesting question.

There are thousands of possible function types which we could try to fit a given data set.

Some would fit very well and others would fit very poorly.

Web sites exist with software which helps us with identification of possible models.

One such site is

FUNCTION FITTINGH

we may attempt to fit aquadratic or exponentialfunction to them, ormaybe some otherfunction?

y

x

FUNCTION IDENTIFICATION (REVIEW)

http://home.comcast.net/~curveexpert/



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The function fitting process is:

Step 1: Use technology to obtain a scatterplot of the data.

Step 2: Eliminate those types where the model cannot apply.

For example: is clearly not linear. It is also not

a simple power function as it does

not pass through the origin.

Step 3: Suppose you suspect the model may be one of several types. For each of them:

² calculate the coefficient of determination r2, and

² obtain a residual plot of the data.

These two aids should be used wisely either to help find an appropriate model or

to reject a model.

The model where r2 is closest to 1 fits the given data best, but it is still possible

for this model to be inappropriate.

The residual plot is often used to eliminate models.

If the points are randomly distributed about the horizontal axis the model may be

appropriate, but if there is a pattern we would likely reject the model.

Step 4: Write down the equation of the ‘function of best fit’ and plot this function over the

scatterplot of the experimental points.

Note: ² There are real problems in relying on r2 values only.

Many data sets which are clearly not linear have r2 values very close to 1.

If a data set has r2 = 0:98 for model A and r2 = 0:95 for model B it may

be unwise to reject B in favour of A on this evidence alone.

² There are also problems in relying on a residual plot only.

For example,

For example, its residual plot may show a distinct pattern whereas B’s plotmay be random. Using model A may lead to incorrect predictions, especiallywhen extrapolating outside the range of the data.

This residual plot was obtainedfrom data from a Hooke’s Lawexperiment where the elastic limitof the spring was not exceeded.

A linear model was expected.

The random scattering indicates thatthe model may be appropriate.

The pattern indicates that we shouldmost likely reject the model.



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It is possible, due to human or experimental error, that a

scatterplot could appear to have a regular pattern, and on

this basis we could wrongly reject the model.

This residual plot is not random about the horizontal axis.

So, should we reject the model in this case?

Actually we have an almost perfect fit, and it would be

wrong to reject the model.

² Adding just one extra data point can completely change our views on the possible model.

So, if you can obtain more data, do so.

We know that the higher we get above the earth,

the further we can see. We call the distance we can

see the distance to the horizon, d km. If h metres

is the distance we are above the earth’s surface,

the following table shows values of d for various

h values.

h (m) 2 10 20 50 100 200

d (km) 5 11:3 19:6 25:3 35:8 50:6

a Obtain a scatterplot of the data and indicate the most likely model which would

fit the data.

b Obtain the model of best fit, giving r2 or residual plot evidence.

c Use the model to find the distance to the horizon from the top of a 500 m high

hill.

d Use your results from c and the rule of Pythagoras to find the radius of the earth.

a d + 3:72 £ 0:498

Evidence:

² r2 + 0:990 which is very close to 1

² Residual plot is random about the

horizontal axis.

² None of the other three models are

appropriate

c For h = 500,

d + 3:72 £ 5000:498

d + 82:2

) the distance is about 82:2 km.

d

As (R + 0:5)2 = R2 + 82:22

then R2 + R + 0:25 = R2 + 82:22

) R = 82:22 ¡ 0:25

) R + 6760

) the radius is about 6760 km.


horizon

h d

0

10

20

30

40

50

60

0 50 100 150 200 250

h ( )m

d�( )km

R

R

500 m

b h



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Press 1 to enter the list editor, then key the h values

into L1 and the d values into L2.

To obtain a scatterplot of the data, press Y= (STAT

PLOT) 1 to select Plot 1, then ensure the screen is set up as

shown alongside.

Press 9 to select 9:ZoomStat and view the scatterplot.

We now obtain the residual plot.

Press Y= (STAT PLOT) 1, scroll down to Ylist and

select the list of residuals by pressing (LIST) 1

which selects 1:RESID.

Press 9 to select 9:ZoomStat and view the residual

plot.

The points are randomly distributed about the horizontal axis,

indicating the model is appropriate.

To find the power model of best fit, press , scroll

down to A:PwrReg and press ENTER .

Press 1 (L1)

The power model d + 3:72 £ h0:498 is given with

r2 + 0:990 .

Following are graphics calculator instructions for obtaining models from data:


Select STAT from the main menu, then key the h values into

List 1 and the d values into List 2.


2 (L2) ENTER to display the

result.

,


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(Using only linear, quadratic, exponential or power models.)

1 Strawberry plants are sprayed with a pesticide-fertiliser

mix. The data below give the yield of strawberries per

plant for various spray concentrations per litre of water:

Spray concentration (mL), x 0 2 4 5 8

Yield of strawberries per plant, y 8 14 19 22 30

a Generate a scatterplot of the data.

b Which of the possible models can be eliminated?

c For each of the remaining models, fit a curve, state

the value of the coefficient of determination r2, and find the residual plot.

d State the equation of the function you think best fits the data and plot this function

over the scatterplot of the experimental points.

e Predict the number of strawberries per plant if the spray concentration per litre of

water is i 3 mL ii 10 mL.

EXERCISE 10H

Press (SET UP), scroll down to Resid List,

and select List 3 by pressing (LIST) 3 .

This stores the residual values into List 3 when we fit our

function. Press when you are done.

To obtain the scatterplot, press (¤) if the GRPH icon is

not in the bottom left corner of the screen, then press

(GRPH) (SET) and ensure the screen is setup as

shown.

Press (GPH1) to draw the scatterplot. From the

scatterplot, a power model appears the most appropriate

model.

To find the power model of best fit, press (CALC)

(¤) (Pwr).

Press to view the residual values stored in List 3.

To obtain the residual plot, press (SET), and change the

YList to List 3, which holds the residual values.

Press (GPH1) to draw the residual plot.

The points are randomly distributed about the horizontal axis,

indicating the model is appropriate.

The power model d + 3:72 £ h0:498 is given with

r2 + 0:990 .


F4


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2 Pinkie’s Pizza Parlour prices pepperoni pizzas as

follows:

Size (in inches), x 9 12 15 18

Cost ($), y 5:70 8:70 12:70 17:70


b Can any of the possible models be eliminated?

c For each of the remaining models, fit a curve,

state the value of the coefficient of determina-

tion r2, and find the residual plot.

d State the equation of the function you think best

fits the data and plot this function over the scatterplot of the data points.

e If Pinkie plans to make 23 inch pizzas, predict the possible price.

3

Years since 1980, x 5 10 15 20 25

Population, y 730 995 1365 1870 2560



c For the possible models remaining, fit a curve, state the value of the coefficient of

determination r2, and find the residual plot.

d State the equation of the ‘function of best fit’ and plot this function over the scat-

terplot of the data points.

e Predict the population of the villiage in: i 2015 ii 1965:

f Suggest reasons why your prediction in e ii may be inaccurate.

4 The maximum speed of a Chinese ‘dragonboat’ with increasing numbers of paddlers is

recorded in the table:

Number of paddlers, x 4 6 10 18 30

Maximum speed (kmph), y 8:7 10:2 12:6 15:9 19:5

a Draw a scatterplot of the data.

b Which of the possible models can be

eliminated?

c For the possible models remaining, fit

a curve, state the value of the coeffi-

cient of determination r2, and find the

residual plot.

d State the equation of the function that

you think best fits the data and plot

this function over the scatterplot of the experimental points.

e Predict the maximum speed of a dragonboat if there are:

i 24 paddlers ii 40 paddlers.

A remote village of hill tribes people was discovered in and its populationrecorded every years after . The results were:

19805 1980



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5 The agreed resale value of a car to a dealer if all

service conditions are met is set out in the following

schedule:

Age (years), x 1 2:5 5 7

Resale value ($), y 39 000 26 500 14 000 8500



c Fit a curve to the possible models, and in each case state the value of the coefficient

of determination r2 and find the residual plot.

d State the equation of the function that you think best fits the data and plot this

function over the scatterplot of the experimental points.

e Predict the i original value of the car ii value after 10 years.

f Are either of the values found in e reliable?

6 A steel ball is dropped from various heights and the time to reach the ground is measured.

The results were:

Time (t sec) 1:4 2:1 2:9 3:3 3:6 4:3 4:9

Distance fallen (d m) 10:0 21:3 40:6 52:1 63:4 89:2 117:7


b Does a power or exponential law apply? Give evidence.

c Determine the law connecting d and t.

d Predict the distance fallen after i 4 seconds ii 8 seconds

e Which of the results in d is more likely to be accurate?

7 Mansar Nominees have their business car valued every June 30th. Their records show:

n years after purchase 1 2 3 4

Value $V (nearest $100) 51 100 38 200 28 700 21 400


b Hence find the approximate law connecting V and n.

c Use the result of b to estimate:i the car’s original cost ii the approximate value after 8 years.

d Which of the results in c is likely to be accurate?

8 To determine the maximum speed of a small boat with different powered engines, the

following experimental data was obtained:

Horse power p (hp) 2 5 10 20 25 30 50

Max. speed S (kmph) 8:8 12:0 15:1 19:0 20:5 21:8 25:8


b Determine the law connecting S and p.

c Hence: i estimate the horse power of a motor which is capable of producing

a maximum speed of 30 kmph

ii estimate the maximum speed of the boat for a 40 h.p. motor.

d Which of the results in c is more likely to be accurate?



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PROJECT THE PENDULUM

9 The table below shows the concentration of chemical X in the blood at various times

after an injection was administered.

Time (t minutes) 10 20 30 40 50 60 70

Concentration of X

(C micrograms/cm3)104:6 36:5 12:7 4:43 1:55 0:539 0:188


b Hence determine the approximate exponential law connecting C and t.

c If the injection is effective while the concentration of X is greater than

2 £ 10¡4 micrograms/cm3, how long will it take before the injection ‘wears off’?

10 The table below shows the temperature of a cup of coffee as it cools. Note that the

temperature T oC is the temperature above room temperature.

Time of cooling (t minutes) 5 10 15 20 25 30 35

Temperature (T oC aboveroom temperature)

31:5 13:3 5:6 2:3 1:0 0:41 0:17


b

c If room temperature was constant at 20oC throughout the cooling, estimate the

temperature of the coffee when the experiment started.

d Estimate the temperature of the coffee after it has been cooling for 2 minutes.

1 Which of the following ideas have merit when finding the period of the pendulum for

a particular length?

² Several students should time one period using their stopwatches.

² Timing 8 complete swings and averaging is better than timing one complete swing.

² If several students do the timing, the highest and lowest scores should be removed

and the remaining scores should be averaged.

2 List possible factors which could lead to inaccurate results.

A can be made by tying an object such as a mobile phone to ashoe lace and allowing the object to swing back and forth when supportedby the string.

pendulum

We will find the , which is the time taken forone complete oscillation, back and forth, for variouslengths of the pendulum.

The maximum angle of the pendulum should be .The length of the pendulum is the distance from thepoint of support to the object’s centre of mass.

period

µ 15o

What to do:

The object of this investigation is to determine any rule which mayconnect the period of the pendulum ( secs) with its length ( cm).T l

l cm

point ofsupport

��

object

Determine the law connecting T and t.



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REVIEW SET 10

3 After deciding on a method for determining the

period, measure the period for pendulum lengths

of 20 cm, 30 cm, 40 cm, ...., 100 cm and record

your results in a table like the one alongside.

length (l cm) period (T s)

203040...4 Use technology to determine the law connecting

T and l.

1 A population of seals t years after a colony was noticed is given by Pt = 40£ 20:3t,t > 0. Find:

a the initial size of the population

b the population after

i 5 years ii 10 years iii 20 years.

c Sketch the graph of Pt against t using only your results from a and b.

d Use technology to graph Y1 = 40£ 20:3t and check your answers to a, b and c.

2 The weight of a radioactive substance after t years is given by the formula

Wt = 5 £ 3¡0:003t grams. Find:

a the initial weight of the radioactive substance

b the percentage remaining after

i 100 years ii 500 years iii 1000 years.

c Graph Wt against t.

3 a What is the total energy released by an earthquake of magnitude 7:2 on the

Richter scale?

b What increase in ground movement occurs between two earthquakes with mag-

nitudes 3:4 and 5:1?

4 How many times louder is a sound of 82 decibels than a sound of 42 decibels?

5 The figures below show the number of surviving beetles in a square metre of lawn

two days after it was sprayed with a new chemical.

Amount of chemical (x grams) 2 3 5 6 9

Number of lawn beetles (y) 11 8 6 4 3


b Determine the r and r2 values for the linear model.

c Describe the association between the number of lawn beetles and the amount of

chemical sprayed.

d Use your technology to determine the equation of the line of best fit.

e Give an interpretation for the slope and vertical intercept of this line.

f Use your equation to predict the amount of chemical required to remove all lawn

beetles.



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The following table shows the profit figures for McKenzie Mattresses over a yearperiod:

76

a Use technology to draw a scatterplot for this data.

b Determine the r and r2 values for the linear model.

c Find the equation of the line of best fit and comment on its reliability to predict

future profits.

7 Write as a power of 3:

a 81 b 1 c 1

27d 1

243

8 Write as a power of 2:

a 1

4b 32 c

1p2

dp

8

9 Write as a single power of 2 or 3: a 4a £ 8 b4x+2

21¡xc

27

9a

10 Write without negative indices:

a mn¡2 b (mn)¡3 c³mn

´¡2

d (4m¡1n)2

11 Without using a calculator, write in simplest rational form:

a 843 b 27

43 c 32¡

25 d 25¡

32

12 On the same set of axes draw the graphs of:

a y = 2x and y = 2x + 2 b y = 2x and y = 2x+2

13 Sketch the graph of y = 2¡x ¡ 5:

14 Sketch the graph of y = 4 ¡ 2x.

15 Without using a calculator, solve for x: a 27x = 3 b 91¡x = 1

3

16 Solve for x, giving your answer correct to 3 significant figures:

a 5x = 7 b 20 £ 20:5x = 500 c 53¡x = 32x+1

17 The weight W grams, of radioactive substance remaining after t years is given by

W = 500 £ 2¡0:05t grams. Find the time taken for the weight to:

a halve b reach 50 grams

18

a Draw a scatterplot of the data

b Hence determine the law connecting P and t.

c Estimate the population number after i 18 months ii 8 years

d Which of the results in c is reliable?

The table below shows the population numbers of a colony of albatross years afterthe colony was established.

t

Year 2000 2001 2002 2003 2004 2005 2006

profit ($ £ 1000) 17 24 20 27 23 38 31

Time (t years) 0 1 2 3 4

Population (P ) 208 282 383 521 707



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19

Distance (d m) 10 20 30 40

Time (t sec) 1:43 2:02 2:47 2:86


b Which of the possible function models can be eliminated?

c For each of the remaining models, fit a curve, calculate the value of the coefficient

of determination r2, and find the residual plot.

d State the equation of the ‘function of best fit’ and plot this function over the

scatterplot of experimental points.

e Predict the time taken for the stone to fall 25 m.

20 The concentration of chemical Y in the blood of an accident victim at various times

after an injection was administered is recorded in the table below:


b Which of the possible function models can be

eliminated?

c For the possible function models, fit a curve,

calculate the value of the coefficient of deter-

mination r2, and find the residual plot.

d State the equation of the ‘function of best fit’

and plot this function over the scatterplot of

the experimental points.

e Predict the concentration of chemical Y after 40 minutes.

21 The value of a rare stamp t years after it was purchased is given by

V (t) = 100 £ 20:2t dollars.

a Sketch the graph of V (t) for t > 0.

b Find the value of the stamp (to the nearest dollar) 7 years after it was purchased.

c When will the value of the stamp reach $1000?

22 Alex sells cartons of milk in his store. In the past, he has altered the price of his

milk, and the table below shows the weekly profit received when the milk was sold

at various prices:


b Which of the possible function models is the best fit for the data? Explain.

c Find the equation of the ‘function of best fit’, and state the value of r2.

d Predict Alex’s weekly profit if he sells his milk for: i $1:85 ii $2:35.

The time taken ( secs) for a stone to fall various distances ( m) was observed andrecorded in the given table.

t d

Price ($p) 1:75 2:10 1:90 1:60 2:25 2:00

Profit ($P / week) 129 135 140 105 113 138

Time (t minutes) 5 10 15 20 25 30 35

Concentration of Y

(C micrograms/cm3)29 18 12 7:3 4:7 2:9 1:8



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e Which of the predictions in d is likely to be more accurate?

f What price should Alex set for his milk to maximise his profit?

23 The pulse rate of a runner t minutes after he finishes

training is given by P (t) = 90£ 3¡0:25t + 50 beats

per minute.

a Sketch the graph of P (t) for t > 0.

b Find the runner’s pulse rate immediately after he

finished training.

c How long did it take for his pulse rate to drop to:

i 70 beats per minute ii 55 beats per minute?

24 The table below shows the amount of television watched per week by a group of

students, as well as their score (out of 100) in a recent examination.


b Explain why a linear model appears most appropriate.

c Find the least squares ‘line of best fit’ for the data, and state the values of r and

r2.

d Describe the association between examination score and television watched.

e Predict the examination score for a student who watches television for

i 14 hours per week ii 25 hours per week?

25 Seven students, who live varying distances from their school, were asked how long it

takes them to get to school each morning. The results were:


b Kevin thinks the data looks linear, but Caitlin thinks it looks like a power model.

Find the ‘function of best fit’, state the value of r2 and draw a residual plot for:

i a linear model ii a power model.

c Which model appears to fit the data better? Explain.

d Kevin lives 15 km from school. Using the model chosen in c, predict how long

it takes Kevin to get to school each morning.

Television watched (t hours/week) 5 20 8 2 16

Examination score (S) 87 62 81 91 70

Distance from school (d km) 13 2 17 8 28 3 10

Time (T min) 23 6 28 16 40 8 19



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10 - haesemathematics.com€¦ · an r value of 1 suggests that there is a perfect linear...

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