equations and functions century year 12/01 equations and...equations and functions 1 1 equations and...

EQUATIONS AND FUNCTIONS

1

11

Equations and functions

ALGEBRAIC MODELLING

Jenni is comparing two Internet access plans. Optnet has a monthly access fee of $8 and charges 60 cents per hour of Internet use. OzExpress charges $1 per hour of Internet use but has no monthly access fee. The two plans are represented by the formulas

Optnet:

C

=

0.6

t

+

8OzExpress:

C

=

t

where

C

is the cost in dollars and

t

is the number of hours of Internet use.

Can you advise Jenni? What will you tell her?

This problem involves analysing and comparing functions and formulas, which are the main themes of this chapter. Such a problem may be solved:

�

algebraically, by solving equations

�

graphically, on graph paper or using a graphics calculator or graphing software, or

�

by a ‘guess, check and refine’ method, using a table of values, spreadsheet or calculator.

In this chapter you will learn how to:

�

add, subtract, multiply and divide algebraic terms

�

calculate with numbers in scientific notation

�

solve different types of linear equations

�

solve equations involving powers, including the use of ‘guess, check and refine’

�

substitute values into formulas and solve equations for a variety of practical problems

�

change the subject of a formula

�

examine real life situations that can be modelled by linear functions

�

draw a line of best fit to a set of empirical data and find its equation

�

interpret the point of intersection of the graphs of two linear functions drawn frompractical contexts, including ‘break-even’ points.

!NNC Yr12 maths ch 01 Tuesday, November 21, 2000 4:45 PM

2

NEW CENTURY MATHS GENERAL: HSC

ALGEBRAIC EXPRESSIONS

An algebraic expression is a general statement involving

pronumerals

or

variables

. It is made up of

terms

. For example, 2

p

2

+

7

p

−

4 has three terms: 2

p

2

, 7

p

and

−

4.

Types of algebraic expressions

When writing algebraic expressions, terms with higher powers are usually written first. For example, we write 8

t

3

−

5

t

2

−

1, not

−

5

t

2

−

1

+

8

t

3

. Because of this, the term with the highest power is called the

leading term

. The leading term of 8

t

3

−

5

t

2

−

1 is 8

t

3

.

An algebraic expression is classified by the power of its leading term:

�

8

t

3

−

5

t

2

−

1 is called a

cubic expression

because its leading term has a power of 3 (the variable is ‘cubed’).

�

4

y

2

−

y

+

5 is called a

quadratic expression

because its leading term has a power of 2 (the variable is ‘squared’).

Quad

means ‘square’ in Latin.

�

2

n

−

1 is called a

linear expression

because its leading term has a power of 1. It is ‘linear’ because, when graphed as a function on the number plane, the graph is a straight line.

Simplifying algebraic expressions

Example 1

Simplify these expressions.(a) 3

d

+

8

d

2

−

3

−

d

(b) 4

xz

+

4

x

2

+

x

2

−

2

z

(c)

−

5

k

×

4

k

2

t

(d) 18

mp

2

�

45

m

2

p

Solution

(a) 3

d

+

8

d

2

−

3

−

d

=

3

d

−

d

+

8

d

2

− 3 Collecting like terms 3d and −d= 2d + 8d2 − 3= 8d2 + 2d − 3

(b) 4xz + 4x2 + x2 − 2z = 4x2 + x2 + 4xz − 2z Collecting like terms 4x2 and x2

= 5x2 + 4xz − 2z

(c) −5k × 4k2t = −5 × 4 × k × k2 × t= −20k3t

(d) 18mp2 � 45m2p =

=

Expanding algebraic expressionsExample 2Expand and simplify these expressions.(a) 3(2m + 7) − 2(m2 + 4) (b) r(4r − 10) − (4r − 10)

Solution(a) 3(2m + 7) − 2(m2 + 4) = 6m + 21 − 2m2 − 8

= −2m2 + 6m + 13(b) r(4r − 10) − (4r − 10) = 4r2 − 10r − 4r + 10

= 4r2 − 14r + 10

18m p2

45m2 p-----------------

2 p5m-------


EQUATIONS AND FUNCTIONS 3

1. Write these algebraic expressions in the correct order, then classify them as being linear (L), quadratic (Q) or cubic (C).(a) 3 − 4x + 6x3 (b) 10 − 2x (c) 5 + x2 − x

(d) 8 + x (e) 7x − x3 + x2 (f) 1 + 20x

(g) 7 + 5x2 (h) 2 − x (i) x − x2 + 1

(j) x + 7x2 − x3 (k) −5x2 + x3 + 14 (l) 4 − x

2. Simplify these expressions.(a) x3 + 2x2 + 5x2 − x3 (b) 4m + 7 + 18m − 5(c) 3bd + 4d + d2 − 5d (d) 6c − 6 − c + 20(e) 2r + r + 2r2 − r (f) 7tu + t2 − 8t2 + tu(g) −3e + 9 + e + e2 (h) 5p + 4p − 9p2 + 7p2

(i) 2xy + 3yz + 4xy − 8yz (j) 7k + 11y − 20 + 3k − 6(k) 16j − 3jk − 5j − 4jk (l) 8u2 + 6 − 10 − 5u2

3. Simplify these expressions.

(a) 3m2 × 2mn (b) −5p × −6p (c) (−2rt2)2

(d) 16w5y2 � 8wy (e) 10u2 × (f)

(g) 4p2 � 10p (h) 6r × −3r2 (i) (3x2)3

(j) 5u × u (k) 5u � u (l)

4. Expand and simplify these expressions.(a) 3(k + 4) + 4(k + 4) (b) 2(d + 1) − 2(d − 1)(c) 5(r + 3) + 4(r + 10) (d) 8(m + 5) − 3(m − 5)(e) 2(5a + 6) + (a − 1) (f) 3(2t + 1) − (4t + 8) (g) x(x + 7) − 5(x − 3) (h) 5u(2 − u) + u(6u + 3)(i) 2πr(r + h) − πr2 (j) 2x2(7 − x) + x(x − 1)(k) 4(4m + 1) − 3(3m − 2) (l) 8(5 − 2r) + r(3r + 1) (m) 6(2d − 3) + (2d − 3) (n) f(5f + 5) + 5(f + 5)(o) 10x(x − 1) + x(5x + 4) (p) 2c(2c + 7) + 2c(4 − c)


(a) −3b2 × (b) (−4ab)2 (c) 5x + 4xz − 4x + 6

(d) 3w − 12y + 2y2 − 4w (e) 8kp2 � 12k (f) 4de × 5d2e

(g) (h) 6jr + 2r − 4jr − 5r (i) 6wz ×

(j) 8tu + 4u2 − u2 − 8t (k) 18tu � 2tu2 (l) (5f3)2

(m) 8a2y × y2 (n) 4ay + y − ay + y (o) 2mn + n − n + m2

(p) −6hk × −2k2 (q) (r)

Exercise 1-01: Algebraic expressions

12---

34--- 1

3---

3u2

------ 18k3y3ky2---------------

6de2

14d2e2-----------------

4bd6

----------

10ut–4t2

-------------- 5w2

2z----------

12---

15dy2

20y2--------------- 24– a2b2

16a– b2---------------------


4 NEW CENTURY MATHS GENERAL: HSC

SCIENTIFIC NOTATIONScientific notation (also called standard form) is a shorthand for writing very large numbers or very small numbers using powers of 10.

Large numbers are written with positive powers of 10. Small numbers are written with negative powers of 10.

Example 3Express these numbers in scientific notation.(a) 13 700 000 000 (b) 20 480 (c) 0.000 000 61

Solution(a) 13 700 000 000 = 1.37 × 1010 10 places after the first significant figure, 1(b) 20 480 = 2.048 × 104 4 places after the first significant figure, 2(c) 0.000 000 61 = 6.1 × 10−7 7 places up to and including the first significant figure, 6

Example 4Express these numbers in normal decimal form.(a) 8.5 × 104 (b) 8.5 × 10−4 (c) 9.31 × 10−6

Solution(a) 8.5 × 104 = 85 000 4 places after the first significant figure, 8(b) 8.5 × 10−4 = 0.000 85 4 places up to and including the 8(c) 9.31 × 10−6 = 0.000 009 31 6 places up to and including the 9

Example 5Evaluate the following.

(a) (7.4 × 105) − (8.3 × 104) (b) (c)

SolutionCalculator keys

(a) (7.4 × 105) − (8.3 × 104) = 657 000 7.4 5 8.3 4

(b) = 9 × 109 or 9 000 000 000 2.7 6 3 4

(c) = 650 000 4.225 11

Scientific notation has the formm × 10n

where m is a number between 1 and 10 and n is an integer.

An integer is a positive or negative whole number, or zero.

2.7 106×3 10 4–×----------------------- 4.225 1011×

EXP − EXP =

2.7 106×3 10 4–×----------------------- EXP ÷ EXP +/− =

4.225 1011× EXP =



1. Express these numbers in scientific notation.(a) 230 000 (b) 8 950 000 (c) 0.000 17(d) 0.006 (e) 20 000 000 (f) 0.0308(g) 17 300 (h) 902 (i) 0.24(j) 0.000 036 (k) 4 800 000 (l) 700 000 000

2. Express these numbers in normal decimal form.(a) 7.2 × 105 (b) 7.2 × 10−3 (c) 6 × 104

(d) 8.19 × 107 (e) 2 × 10−6 (f) 3.2 × 10−5

(g) 9.6 × 108 (h) 4 × 106 (i) 1.75 × 10−2

(j) 2.8 × 103 (k) 3.087 × 10−4 (l) 5.129 × 107

3. Evaluate:(a) (3.6 × 105) + (1.1 × 103) (b) (2.7 × 10−4)2

(c) (5.46 × 107) + (8.2 × 104) (d)

(e) (f)

(g) (7.7 × 104)3 (h) (6.2 × 10−4) − (4.11 × 10−5)

4. Evaluate, giving your answer in scientific notation correct to 2 significant figures:(a) (8.4 × 105)2 (b) (3.905 × 107) + (1.1 × 105)

(c) (d)

(e) (8.6 × 108) × (9.4 × 10−3) (f) (3.98 × 103)3

(g) (h)

FORMULASExample 6Heron’s formula for calculating the area of a triangle with side lengths a, b, c is

A = where s = (a + b + c)

Use the formula to find (correct to 1 decimal place) the area of a triangle with sides of length 5 cm, 8 cm and 10 cm.

SolutionLet a = 5, b = 8, c = 10.

s = (5 + 8 + 10) = 11.5

A =

=

= = 19.8100 …Area ≈ 19.8 cm2

Exercise 1-02: Scientific notation

4.5 10 4–×6 10 6–×

-------------------------

1.089 10 9–× 18 106×------------------

5.8 108× 2.7 108×4.1 105×-----------------------

15.4 10 9–×------------------------- 6 1012×3

a

b

c

s s a–( ) s b–( ) s c–( ) 12---

12---

11.5 11.5 5–( ) 11.5 8–( ) 11.5 10–( )

11.5 6.5 3.5 1.5×××

392.4375



Example 7The time (in seconds) it takes a swing to go back and forth once is

T = 2π

where l is the length of the swing (in metres) and g is the gravitational acceleration. Find T correct to 1 decimal place if l = 2.8 m and g = 9.8 m/s2.

SolutionT = 2π

= 3.3585 … ≈ 3.4 s

1. The formula for the surface area of a closed cylinder isS = 2πr(r + h)

where r is the radius of its base and h is its height. Calculate (correct to 2 significant figures) the surface area of a cylinder with radius 0.8 m and height 2.3 m.

2. The kinetic energy K (in joules, J) of an object of mass m kg travelling at speed v m/s is

K = mv2

Calculate the kinetic energy of an object of mass 2.5 kg travelling at 4 m/s.

3. The value of an item depreciating over time isS = V(1 − r)n

where V is its original value, r is the annual rate of depreciation expressed as a fraction or decimal, and n is the number of years. Calculate the value of a $4200 computer after 3 years if it is depreciating at 27% p.a. Answer to the nearest dollar.

4. The body mass index (BMI) of an adult is

B =

where m is the mass in kilograms and h is the height in metres. (a) Glen is 1.83 m tall and weighs 88 kg. Calculate his body mass index correct to

1 decimal place.(b) If a BMI between 20 and 25 is an indication of good health, can Glen improve his

health? If so, how?

5. The formula for converting Australian dollars ($A) to New Zealand dollars ($NZ) is NZ = 1.2831A. Convert the following $A amounts to $NZ, correct to the nearest cent.(a) $A40 (b) $A84.95 (c) $A12.20

6. The number of days fresh milk will keep if stored at temperature T°C is

D =

How many days will fresh milk keep if it is stored at:(a) 3°C? (b) 5°C? (c) freezing point (0°C)?

7. The size of each angle (in degrees) in a regular polygon with n sides is

a = 180 −

Calculate the size of each angle in:(a) a regular nonagon (9 sides) (b) a regular decagon (10 sides)

lg---

2.89.8-------

Exercise 1-03: Formulas

12---

mh2-----

6T 1+-------------

360n

---------



8. Given that E = mc2, calculate the value of E when m = 0.002 and c = 3 × 108.

9. The number of times per minute a cricket chirps on a hot summer’s night isn = 8T − 24

where T is the air temperature in °C. (a) How many times per minute will a cricket chirp if the temperature is 22°C?(b) This formula is an example of an algebraic model. According to this model, will the

frequency of chirping increase or decrease as the night gets hotter?

10. The area of a trapezium is given by the formula

A = (a + b)h

where a and b are the lengths of the parallel sides and h is the distance between them. Calculate the area of the trapezium illustrated.

11. The velocity V m/s required for a rocket to escape the Earth’s gravity is

V = where g = 9.8 m/s2 (the gravitational acceleration) and r = 6.38 × 106 m (the radius of the Earth). Calculate the escape velocity of the rocket correct to 3 significant figures.

12. When prescribing medicine for infants, the dosage is given by the formula

D = (Fried’s rule) or D = (Young’s rule)

where A is the adult dosage, m is the infant’s age in months and y is the infant’s age in years. Jessie, aged 24 months, needs to take cough medicine with a recommended adult dosage of 12 mL. Calculate Jessie’s dosage, correct to 2 decimal places, using:(a) Fried’s rule (b) Young’s rule

13. The blood alcohol content (BAC) of a person who consumes alcoholic drinks isBAC = 0.0012na

where n is the number of drinks taken and a is the amount of alcohol in each drink measured in millilitres. Sally drank three stubbies (250 mL each) of light beer (2.7% alcohol). Calculate:(a) the amount of alcohol in each drink (b) Sally’s BAC level

14. The air temperature, T °C, outside an aeroplane is given by the formulaT = 15 − 0.006h

where h is the height of the plane above sea level. Calculate correct to 1 decimal place the temperature outside the plane at height:(a) 800 m (b) 975 m

15. The area of this irregular figure can beapproximated using the formula

A ≈ (a + 4b + c) (Simpson’s rule)

Calculate the approximate area of the figure if h = 18 m, a = 21 m, b = 25 m and c = 28 m.

6 cm

10 cm7 cm

12---

2gr

mA150--------- yA

y 12+---------------

h h

a b c

h3---



16. A country’s population density (in persons per square kilometre) is given by the formula

D =

where P is the population and A is the area in square kilometres. Calculate the population density of Australia, correct to 3 significant figures, if its population is 1.92 × 107 and its area is 7.69 × 106 km2.

17. The area of this triangle is given by the formula

A = ab sin �

Find A to 2 decimal places if a = 7 cm, b = 8 cm and θ = 51°.

18. The distance (in kilometres) that an observer can see to the horizon from the top of a structure of height h m is

d = 8

What distance can be seen from the top of the Eiffel Tower in Paris, a height of 320 m?

19. The volume of a sphere is given by the formula

V = πr3

Calculate the volume of the illustrated sphere correct to 2 significant figures.

20. Elena earns a taxable income of $57 821. Calculate her income tax using the formula T = 11 380 + 0.42(I − 50 000)

where T is the tax payable and I is the taxable income.

21. The area of an annulus (doughnut shape) is A = π(R2 − r2)

where R is the longer radius and r is the shorter radius. Calculate the area of the illustrated annulus correct to 2 decimal places.

22. The price of an item n years ago before inflation was

P =

where A is its current price and r was the annual rate of inflation over this period written as a decimal. Use the formula to calculate (to the nearest $100) the price of a $27 000 car 8 years ago if the inflation rate during this time was 1.2%.

23. The volume (in cubic centimetres) of wood in a tree isV = 0.4724d2h + 9.86

where d is the diameter of the trunk and h is the vertical distance between the trunk and the lowest branch. All dimensions are in centimetres. (a) Calculate the volume V if d = 55 cm and h = 210 cm.(b) Express this volume in cubic metres correct to 1 decimal place.

PA---

θb

a12---

h5---

4.1 cm

43---

8 cm

3 cm

A1 r+( )n

--------------------



SOLVING EQUATIONSExample 8Solve these equations.

(a) = (b) − = 4

Solution

(a) =

5(8x) = 7(2x − 1)

40x = 14x − 7 Expanding26x = −7 Subtracting 14x from both sides

x = − Dividing both sides by 26

Just for the record

BODY MASS INDEX

The body mass index, calculated by the formula

B =

where m is mass in kg and h is height in m, was adopted by the World Health Organisation (WHO) in 1997 as an international standard of health and fitness. It is a convenient measure that represents the health of an individual by a single value, applicable to both male and female adults, and does not require reference to height–weight charts. The table interprets a range of BMI scores.

According to this measure, about 40% of Australians and 59% of Americans are classified as being overweight or obese. People who fall into this category have a greater risk of heart disease, stroke, diabetes, higher blood pressure and higher cholesterol.

Note, however, that the BMI does not take into account individual differences in frame size, muscle mass or distribution of body fat. The BMI is an example of an algebraic model and, like all mathematical models, it has some limitations. For example, the BMI cannot be used to measure the fitness levels of the following types of people:� body builders� pregnant women� growing children below the age of 19� the frail and sedentary elderly.

Give a reason why the BMI does not apply to each of these types.

mh2-----

Health status BMI

Severe starvationMild starvationGood healthIdeal healthOverweightObesitySevere obesity

16–1718–1920–2521–2326–2930–39

40 and above

8x7

------ 2x 1–5

--------------- 4t3----- t 2+

10-----------

8x7

------ 2x 1–5

---------------

Cross-multiplying:

(equivalent to multipying both sides by 35)

8x7

------ 2x 1–5

---------------

726------

P



(b) − = 4

− = 30(4) Multiplying each term by 30

10(4t) − 3(t + 2) = 12040t − 3t − 6 = 120 Expanding

37t − 6 = 120 Collecting like terms37t = 126 Adding 6 to both sides

t = (or 3 ) Dividing both sides by 37

Solve these equations.

1. 3(p − 4) = −27 2. 4x + 4 = x − 11

3. 12d = 8 + 8d 4. = 29

5. = 6. = 4

7. −4(2k + 1) = 0 8. 3c + 8 = 2(c − 10)

9. 2 + = −8 10. 6 − 2k = 10

11. = −4 12. 6(2 − 3w) = −24

13. = 14. =

15. + 7 = − 4 16. 10k − 11 = 4(2k − 10)

17. 8c + 16 = c − 12 18. =

19. = 3 20. 8(4 − 2c) = −16

21. = −2 22. =

23. = −5 24. 10y − 8 = 46 − 2y

25. = 26. 9r + 24 = 3r

27. 5(d + 5) = 2(2d + 18) 28. = −3

29. 12g − 11 = 4(5g + 7) 30. − 1 = + 6

4t3----- t 2+

10-----------

3010 4t

3-----

1

303 t 2+

10-----------

1

12637--------- 15

37------

Exercise 1-04: Solving equations

5h 2–2

----------------

2c3

------ 29--- 3r

7----- r

5---–

t3---

6b 7–2

----------------

h3--- h

4---+

34--- 3y

5------ y 1+

2------------

a5--- 2a

3------

4q 3–4

---------------- q 8+10

------------

m3---- 3m

10-------–

3d 1–5

---------------- 4d 7+10

----------------+35--- x 1+

12------------

6z 10–3

------------------

e4--- 1

5---–

35---

2n7

------ n2---+

4b5

------ b3---



EQUATIONS INVOLVING POWERS AND ROOTSSquares and cubes, square and cube rootsExample 9Solve these equations.

(a) = 2 (b) w3 + 6 = 5 (c) = 7 (d) = 4

Solution

(a) = 2

x2 − 13 = 36 Multiplying both sides by 18x2 = 49 Adding 13 to both sides

x = ± Taking the square root of both sides= ±7

Note: Equations involving a pronumeral ‘squared’ (x2) usually have two answers.

(b) w3 + 6 = 5w3 = −1 Subtracting 6 from both sides

w = Taking the cube root of both sides= −1

Note: Equations involving a pronumeral ‘cubed’ (w3) only have one answer.

(c) = 7

= 72 Squaring both sides4d − 9 = 49

4d = 58 Adding 9 to both sides

d = Dividing both sides by 4

= (or 14 )

(d) = 4

= 43 Cubing both sides3p = 64

p = (or 21 ) Dividing both sides by 3

Power equations and the ‘guess, check and refine’ method

Example 10In the following equations, the unknown pronumeral is a power. Solve them using the ‘guess, check and refine’ method, giving answers correct to 1 decimal place where appropriate.

(a) 4x = 4096 (b) (2.2)k = 5 (c) 3r = 1.5

x2 13–18

------------------ 4d 9– 3 p3

x2 13–18

------------------

49

1–3

4d 9–

4d 9–( )2

584------

292------ 1

2---

3 p3

3 p3( )3

643------ 1

3---



SolutionThe ‘guess, check and refine’ method involves guessing the answer, checking it, then refining (improving) the guess until the answer (or a good approximation) is found.

(a) 4x = 4096

The solution to 4x = 4096 is x = 6.

(b) (2.2)k = 5

As 2.22 = 4.84 and 2.22.05 ≈ 5.0346, k must be between 2 and 2.05, so k ≈ 2.0(correct to 1 decimal place).Note: The solution to this equation is only an approximation.

(c) 3r = 1.5

r must be between 0.35 and 0.4, so r ≈ 0.4 (correct to 1 decimal place).

Equations of the form ax = b can also be solved using , the logarithmic function key on your calculator, using the formula

x =

For example, to solve the equation 1.2x = 5, evaluate x = by pressing the following keys on a calculator:

Display

5 1.2

(or 5 1.2 on older calculators)

The solution is x ≈ 8.8 (correct to 1 decimal place). Checking: 1.28.8 ≈ 4.975 02 ≈ 5.

Guess Check Result

x = 8 48 = 65 536 Too high

x = 4 44 = 256 Too low

x = 6 46 = 4096 Correct

Guess Check Result

k = 2 2.22 = 4.84 Too low

k = 2.5 2.22.5 = 7.1788 … Too high

k = 2.2 2.22.2 = 5.6666 … Too high

k = 2.1 2.22.1 = 5.2370 … Too high

k = 2.05 2.22.05 = 5.0346 … Too high

Guess Check Result

r = 1 31 = 3 Too high

r = 0.5 30.5 = 1.7320 … Too high

r = 0.4 30.4 = 1.5518 … Too high

r = 0.3 30.3 = 1.3903 … Too low

r = 0.35 30.35 = 1.4689 … Too low

Calculator keys

4 8 xy =

Technology: The key on your calculatorlog

log

log blog a------------

log 5log 1.2----------------

log � log = 8.8274 …

log � log =



Another way of finding an unknown power (or a close guess) in a power equation is to use the repeated (or constant) multiplication feature of your calculator. For example, to solve 4x = 4096, we need to multiply 4 by itself repeatedly until the display shows 4096.

To do this, first press: Display

4 4 (‘count 1’ for 41)

or 4 on older calculators.

Then press repeatedly to continue multiplying by 4:

(‘count 2’ for 42)



Count the number of times you press the key, including the very first time. In this

example, has been pressed 4 times, so 44 = 256. Press a further 2 times and you

will discover that 46 = 4096. So the solution to the equation 4x = 4096 is x = 6.

1. Solve these equations.

(a) = 7 (b) 3m2 + 5 = 80 (c) 5u3 = −40

(d) = 3 (e) −3 + = 9 (f) = 18

(g) = −6 (h) 4k3 + 11 = 267 (i) =

2. Solve these equations, expressing answers correct to 2 decimal places.

(a) d2 = 96 (b) 5y2 − 10 = 14 (c) 3c3 = 20

(d) = 7 (e) p3 + 12 = 6 (f) −7w2 = −10

(g) 8h3 − 12 = 18 (h) = 9 (i) p3 = 6

3. Solve these equations, correct to 1 decimal place where appropriate.

(a) 3r = 531 441 (b) 6x = 279 936

(c) 10p = 10 000 (d) 2d = 32 768

(e) 5y = 1 953 125 (f) 1.5a = 7.593 75

(g) 2t = 2000 (h) 5c = 63 470

(i) 3w = 945 (j) 1.7n = 5

(k) 2.5h = 88 (l) 1.06h = 4

4. Use the key on your calculator and the formula x = to solve these equations, correct to 1 decimal place where appropriate.

(a) 2h = 65 536 (b) 5k = 48 828 125 (c) 4u = 16 384

(d) 7n = 16 807 (e) 3p = 128 (f) 2r = 89

(g) 1.07x = 2 (h) 1.5t = 3 (i) 1.1a = 2.2

Technology: Using the repeated multiplication feature of your calculator

= ANS × 4

× ×

=

= 16

= 64

= 256

=

= =

Exercise 1-05: Equations involving powers and roots

5y 4+

2a 1+3 yd2

8-----

2e3 x5

------- 410------

n2

4-----

r2 4+5

--------------- 14---

loglog blog a------------



The spreadsheet (or List feature of a graphics calculator) is a useful tool for implementing the ‘guess, check and refine’ method of solving equations. Create a spreadsheet that allows you to solve equations of the form ax = b by calculating values of ax for different values of x. The value of a can be entered, followed by a starting guess for x and a step size for increasing the values of x. See the example below for solving the equation 1.2x = 5.

1st guess: values of x from 2 to 12 2nd guess: values of x from 8 to 9

From the first-guess spreadsheet, x lies between 8 and 9 (closer to 9). From the second-guess spreadsheet, x lies between 8.8 and 8.9 (closer to 8.8). So the value of x in 1.2x = 5 is 8.8 (correct to 1 decimal place).

CHANGING THE SUBJECT OF A FORMULAIf a moving object has (initial) speed u m/s and acceleration a m/s2, its final speed v m/s after time t seconds is given by the formula

v = u + atThis formula has four variables: v, u, a and t. Because the formula is written for v, with v on the left hand side of the ‘=’ sign, we say that v is the subject of the formula. However, a formula can be rearranged so that one of its other variables becomes the subject.

Example 11For the formula v = u + at, make:(a) u the subject (b) t the subject

A B

1 a 1.2

2 Starting guess, x 2

3 Step size 1

4

5 x ax

6 2 1.44

7 3 1.728

8 4 2.0736

9 5 2.48832

10 6 2.985984

11 7 3.5831808

12 8 4.29981696

13 9 5.159780352

14 10 6.1917364224

15 11 7.43008370688

16 12 8.916100448256

Spreadsheet activity: Guess, check and refine

A B

1 a 1.2

2 Starting guess, x 8

3 Step size 0.1

4

5 x ax

6 8 4.29981696

7 8.1 4.378930909610

8 8.2 4.459500506538

9 8.3 4.541552533785

10 8.4 4.625114267146

11 8.5 4.710213484270

12 8.6 4.796878473900

13 8.7 4.885138045273

14 8.8 4.975021537698

15 8.9 5.066558830310

16 9 5.159780352



SolutionChanging the subject of a formula is the same as solving an equation except that the solution is not a number but an algebraic expression (another formula).(a) v = u + at

u + at = v Swapping sides so that u appears on the left hand sideu = v − at Subtracting at from both sides to make u the subject

(b) v = u + atu + at = v Swapping sides so that t appears on the left hand side

at = v − u Subtracting u from both sides

t = Dividing both sides by a to make t the subject

Example 12The volume of a cone is given by the formula V = πr2h, where r is the radius of its circular base and h is its height. (a) Make h the subject of the formula.(b) Make r the subject of the formula.

Solution(a) V = πr2h

πr2h = V Swapping sides so that h appears on the left hand side

πr2h = 3V Multiplying both sides by 3

h = Dividing both sides by πr2 to make h the subject

(b) V = πr2h

πr2h = V Swapping sides so that r appears on the left hand side

πr2h = 3V Multiplying both sides by 3

r2 = Dividing both sides by πh

r = ± Taking (positive and negative) square roots

But since r, the radius, must be a positive value, we can omit the negative square root.

r =

1. Make y the subject of each of these formulas.

(a) 2p + y = 10 (b) w = 3 + y (c) x = y − 1

(d) 2m = 5 − y (e) u = 2n + 2y (f) x =

(g) z2 = x2 + y2 (h) = (i) =

(j) r = 14xy2 (k) d = 8 (l) b = c − 2ay

v u–a

------------

13---

13---

13---

3Vπr2--------

13---

13---

3Vπh-------

3Vπh-------

3Vπh-------

Exercise 1-06: Changing the subject of a formula

14---

6y 3+

a5--- y

2r----- y 1+

10------------ 3k

2------

y5---



2. Change the subject of each formula to the variable shown in brackets.

(a) I = Prn [P] (b) y = mx + b [m]

(c) S = [t] (d) S = V − Dn [D]

(e) d = [v] (f) V = x2h [x]

(g) v2 = u2 + 2as [s] (h) v2 = u2 + 2as [u]

(i) A = bh [h] (j) E = mc2 [m]

(k) E = mc2 [c] (l) tan θ = [x]

(m) d = [A] (n) s = ut + at2 [a]

(o) K = mv2 [v] (p) v = u + at [a]

(q) z = [x] (r) T = [n]

(s) A = πr2 [r] (t) B = [m]

(u) m = [k] (v) T = 2π [l]

(w) S = (a + b + c) [a] (x) v = [r]

(y) C = 60 + 2.7k [k] (z) P = 110 + [y]

3. Change the subject of each formula to the variable shown in brackets.

(a) C = (F − 32) [F] (b) V = πr3 [r]

(c) A = π(R2 − r2) [R] (d) S = 180(n − 2) [n]

(e) S = 2πr2 + 2πrh [h] (f) d = gt2 [t]

(g) B = [h] (h) D = [T]

(i) C = 12 + 8(h − 1) [h]

Listed below are 26 formulas that have been used so far in the General Mathematics course. Match each formula to its correct description below. (Some descriptions are used twice.)

1. I = Prn 2. c2 = a2 + b2 3. S =

4. = 5. P( ) = 1 − P(E) 6. A = πr2

7. C = 2πr 8. y = kx 9. y = mx + b

10. A = (a + b)h 11. V = Ah 12. A = P(1 + r)n

13. I = A − P 14. V = πr2h 15. A = xy

16. A = s2 17. C = πd 18. V = πr3

dt---

v2

g----- 1

3---

12---

hx---

mA150--------- 1

2---

12---

x M–s

-------------- n 24+8

----------------

mh2-----

5k8

------ lg---

12--- 2gr

y2---

59--- 4

3---

12---

mh2----- 6

T 1+-------------

Group activity: Getting the right formula

dt---

xΣxn

------~E

12---

13--- 1

2---

43---



19. A = lb 20. = 21. V = Ah

22. V = πr2h 23. A = bh

24. P(E) =

25. Percentage error = × 100%

26. m = =

DescriptionA. area of a trapezium B. area of a circleC. probability of a complementary event D. mean of a data set E. volume of a prism F. volume of a cylinderG. Pythagoras’ theorem H. circumference of a circleI. probability of an event J. volume of a coneK. linear variation L. area of a squareM. area of a rectangle N. speedO. area of a triangle P. gradient of a lineQ. simple interest earned R. compound interest earnedS. volume of a sphere T. area of a rhombusU. volume of a pyramid V. equation of a lineW. error of a measurement expressed as a percentageX. final amount of an investment after compound interest

xΣfxΣf-------- 1

3---

12---

number of favourable outcomestotal number of outcomes

----------------------------------------------------------------------------

absolute errormeasurement---------------------------------

riserun-------- vertical change in position

horizontal change in position---------------------------------------------------------------------

Study tips

YOUR STUDY ROUTINE

Start a weekly routine to develop good habits and cover all of your study commitments. For example, do more Maths on a Monday. However, don’t overplan and become obsessed with the number of hours worked each night. The quality of study is more important than the quantity. Be task-oriented rather than time-oriented.

Considerations for your study routine:� How much study will I do each week?� When’s my best/worst time of day for studying?� Will I alternate ‘heavy’ study days with ‘light’ ones?� Do I want one night or day that is comparatively study-free?� Will I study more or less on weekends?

Alternate between easy and hard tasks. Start studying straight away. Avoid time wasters such as cleaning your room, decorating title pages or even devising elaborate study timetables. Aim to develop a deep understanding of the subject rather than just memorising facts.

Take a short break after completing each task. Take a longer break after working for 45 minutes or more. Reward yourself when you have completed a significant amount of work. What’s significant will depend on you.



EQUATIONS AND FORMULASSometimes when solving a problem involving a formula, the answer is not immediately found after substituting a value. Instead an equation results, which must then be solved.

Example 13The mean of three numbers x, y and z is

M =

If three numbers have a mean of 22, and two of the numbers are 25 and 26, find the third number.

SolutionSubstitute M = 22, x = 25, y = 26 into the formula to find the third number z.

22 =

66 = 25 + 26 + z66 = 51 + z15 = z

z = 15The third number is 15.

Example 14The compound interest formula

A = P(1 + r)n

calculates the amount ($A) to which a principal ($P) will grow if invested at a rate of r per annum over n years, where r must be written as a fraction or decimal. If a principal of $5000 is invested at 12.5% p.a., how many years will it take to grow to $10 000? Answer correct to 1 decimal place.

SolutionA = 10 000, P = 5000, r = 0.125.

10 000 = 5000(1 + 0.125)n

10 000 = 5000(1.125)n

= (1.125)n

(1.125)n = 2By guess, check and refine: n ≈ 5.9 years.

1. The formula for converting temperatures from the Fahrenheit scale to the Celsius scale is

C = (F − 32)

Use the formula to convert a temperature of 38°C to °F.

2. The number of days fresh milk will keep if stored at temperature T°C is

D =

If a carton of milk lasted 4 days, at what temperature was it stored?

x y z+ +3

---------------------

25 26 z+ +3

----------------------------

10 0005000

----------------

Exercise 1-07: Equations and formulas

59---

6T 1+-------------



3. The volume of a cylinder is V = πr2h

where r is the radius of its base and h is its height. If a cylinder has volume 904.78 cm3 and height 8 cm, find the radius of its base to the nearest centimetre.

4. The compound interest formulaA = P(1 + r)n

shows the amount ($A) to which a principal ($P) will grow if invested at a rate of r per annum over n years, where r must be written as a decimal.(a) What principal needs to be invested at 9% p.a. for it to grow to $4000 in 5 years?

Express your answer to the nearest cent.(b) For how many years must a principal of $4000 be invested at 9% p.a. to grow to

$10 000? Answer to the nearest year.

5. The angle sum S of a polygon with n sides is S = 180(n − 2)°

How many sides has the polygon with an angle sum of:(a) 360°? (b) 1440°?

6. The formula for converting speeds from kilometres per hour to metres per second is

M =

Convert 12 m/s to kilometres per hour.

7. The distance in kilometres that an observer can see to the horizon from a height of h m is

d = 8

How high (to the nearest metre) would you need to be to see a distance of 100 km?

8. The simple interest earned when a principal $P is invested at an interest rate of r per annum for n years is

I = Prnwhere r must be written as a decimal. For how long must a principal of $2300 be invested at 6.5% p.a. to earn simple interest of $1196?

9. The average speed of a moving object in kilometres per hour is

S =

where d is the distance travelled in kilometres and t is the time taken in hours.(a) Find the distance travelled by a car in 3 hours if its average speed is 90.2 km/h.(b) Find the time taken (to the nearest minute) for a cyclist to travel 250 km if his

average speed is 17.5 km/h.

10. The volume of a sphere of radius r is V = πr3

Find the radius (correct to 1 decimal place) of the sphere with a volume of 2854.54 cm3.

11. Young’s rule for calculating the medicine dosage for infants is

D =

where y is the age of the infant in years and A is the adult dosage. How old is Libby if her medicine dosage is 0.6 mL and the adult dosage is 15 mL?

12. The surface area of a rectangular prism of length l, breadth b and height h is given byS = 2lb + 2lh + 2bh

If a rectangular prism with surface area 222 cm2 has breadth 3.5 cm and height 5 cm, find its length.

5k18------

h5---

dt---

43---

yAy 12+---------------



13. Washing removes 20% of a deep stain at each wash. After further washes, the percentage of the original stain left is given by the formula

A = 100(0.8)w

where w is the number of washes.(a) What percentage of the stain remains after 2 washes?(b) Show that 3 washes remove about half of the original stain.(c) At least how many washes are needed to reduce the stain to less than 5% of its

original content?

14. The formula for converting Australian dollars ($A) to New Zealand dollars ($NZ) isNZ = 1.2831A

Convert the following $NZ amounts to $A, correct to the nearest cent.(a) $NZ20 (b) $NZ38 (c) $NZ100

15. The number of times per minute a cricket chirps on a hot summer’s night is n = 8T − 24

where T is the temperature in °C. What is the temperature if the cricket chirps 144 times per minute?

16. The area of an annulus isA = π(R2 − r2)

where R is the longer radius and r is the shorter radius. Calculate the shorter radius of this annulus (to the nearest centimetre) if its area is 794.82 cm2.

17. The maximum distance d m that a ball covers if thrown with velocity v m/s is

d =

where g = 9.8 m/s2 is the gravitational acceleration. At what velocity was a ball thrown if it covered a distance of 25.8 m? Answer correct to 1 decimal place.

18. The body mass index (BMI) of an adult is

B =

where m is the mass in kilograms and h is the height in metres.(a) Kate is 1.76 m tall and has a BMI of 24.2. Calculate her weight to the nearest kg.(b) Stefan weighs 81 kg and has a BMI of 24.8. Calculate his height to the nearest cm.

19. Given that E = mc2, find c if m = 0.05 and E = 4.5 × 1015.

20. The time (in seconds) it takes a swing to move back and forth once is

T = 2π

where l is the length of the swing (in metres) and g = 9.8 m/s2 is the gravitational acceleration. Find the length of the swing (correct to 2 decimal places) if it takes 3.17 seconds to move back and forth once.

21. Use the compound interest formulaA = P(1 + r)n

to determine the number of years it will take a principal to double in value if invested at 5% p.a. Answer to the nearest 0.1 year. Hint: Let A = 2P.

17 cm

r

v2

g-----

mh2-----

lg---



22. The surface area of a cone isS = πr(r + s)

where r is the radius of the base and s is the slant height of the cone. Calculate the slant height of the cone (correct to 2 decimal places) that has a base radius of 3.5 cm and surface area of 101.16 cm2.

23. The blood alcohol content (BAC) of a person who consumes alcoholic drinks isBAC = 0.0012na

where n is the number of drinks and a is the amount of alcohol in each drink. If Mark drank 4 glasses of wine (each 150 mL) and registered a BAC level of 0.0792:(a) how many millilitres of alcohol were present in each glass?(b) what percentage of the wine was alcohol?

24. The population density of a country (in persons per square kilometre) is given by the formula

D =

where P is the population and A is the area in square kilometres. Calculate the population of China correct to 3 significant figures if its area is 9.57 × 106 km2 and its population density is 248 persons/km2.

25. The formula for calculating Samantha’s income tax T isT = 11 380 + 0.42(I − 50 000)

where I is her taxable income. Calculate Samantha’s taxable income if her income tax was $13 102.

There are numerous models for prescribing medicine to infants. Two of them are Fried’s rule and Young’s rule.

� Fried’s rule is D = , where m is the age of

the infant in months and A is the adult dosage.

� Young’s rule is D = , where y is the age

of the infant in years and A is the adult dosage.

Create a spreadsheet like the one on the next page that investigates and compares the two models. Enter the adult dosage and the infant’s age in months, then use the spreadsheet to calculate:� the age of the infant in years� the infant dosage using both rules� the difference between the two values.

s

r

PA---

Spreadsheet modelling activity: Two models for infant medicine

mA150---------

yAy 12+---------------



1. Is there an age value for which the infant dosages given by both rules are identical?

2. When do the two rules differ the most?

LINEAR FUNCTIONSA function like y = −2x + 10 is an algebraic relationship between two variables. It can be thought of as a ‘number machine’ that converts an input value, x, into an output value, y.

6 FUNCTION −2

x y = −2x + 10 y

Independent variable Dependent variable

Because the value of y depends on the value of x, y is called the dependent variable and x is called the independent variable. y = −2x + 10 is called a linear function because it involves a linear expression (−2x + 10) and its graph on the number plane is a straight line.

A B C D E

1 INFANT DOSAGE CALCULATOR

2

3 Adult dosage (mL) 15

4 Age of infant (months) 30

5

6 AGE OF INFANT DOSAGE (mL)

7 Months Years Fried’s rule Young’s rule Difference

8 30 2.50 3.00 2.59 0.4138

9 31 2.58 3.10 2.66 0.4429

10 32 2.67 3.20 2.73 0.4727

11 33 2.75 3.30 2.80 0.5034

12 34 2.83 3.40 2.87 0.5348

13 35 2.92 3.50 2.93 0.5670

14 36 3.00 3.60 3.00 0.6000

15 37 3.08 3.70 3.07 0.6337

16 38 3.17 3.80 3.13 0.6681

17 39 3.25 3.90 3.20 0.7033

18 40 3.33 4.00 3.26 0.7391

Input Output

� A linear function has the form y = mx + b.� The gradient, m, is the rate of change of y relative to x.� The y-intercept or vertical intercept, b, represents the value of y when x is zero.� The graph of y = mx + b is a straight line, demonstrating that y changes at a steady

rate.m = =

m = =

change in ychange in x----------------------------- change in dependent variable

change in independent variable--------------------------------------------------------------------------------

riserun--------- vertical change in position

horizontal change in position--------------------------------------------------------------------------



Functions involving higher powers of x such as x2 and x3 are called non-linear functions because their graphs are not lines but curves. Examples include y = 3x2 − x + 1, a quadraticfunction (whose graph is a parabola), and y = −2x3 + x + 8, a cubic function. These non-linear functions will be examined in Chapter 9.

Linear modelsWhen scientists and researchers observe number patterns occurring in nature and society, they try to find or fit a mathematical formula to represent the relationship. This is called algebraic modelling. A model approximates real life phenomena algebraically and can be described using a formula, a table of values or a number plane graph.

If the observed number pattern suggests a linear relationship, we use the linear functiony = mx + b to model the situation. This is called a linear model.

Example 15A criminologist studying crime in a major city found a linear relationship between P, the number of police patrolling the city, and C, the number of crimes committed per week. Some of her results are illustrated in the table.

(a) What is the independent variable?(b) Find the linear function of the form C = mP + b.(c) What is the gradient and what does it represent?(d) What is the vertical intercept and what does it represent?(e) Use the function to predict the number of crimes per week when 400 police are on patrol.

Solution(a) P, the number of police on patrol, is the independent variable.

(b) m =

= Choosing (50, 3100) and (150, 2800)

= = −3

∴ C = −3P + bTo find b, substitute another point, say (300, 2350).

2350 = −3(300) + b2350 = −900 + b

b = 3250∴ C = −3P + 3250

(c) The gradient −3 represents the reduction in the number of crimes for each new police officer added. As P increases by 1, C decreases by 3. According to this linear model, for every new police officer added, the number of crimes decreases by 3.

(d) The vertical intercept 3250 represents the number of crimes committed per week if P = 0 (i.e. no police on patrol).

(e) When P = 400C = −3(400) + 3250

= 2050 crimes per week

P (no. of police) 50 150 250 300

C (crimes per week) 3100 2800 2500 2350

12---

change in Cchange in P-----------------------------

2800 3100–150 50–

------------------------------

300–100

------------



Line of best fitReal observed data, also called empirical or experimental data, usually do not fit the linear relationship y = mx + b so strictly or reliably. There are two reasons for this:� There are always errors and inaccuracies associated with the measurement of data.� Often, the patterns and relationships in real life phenomena are actually more complicated

than a simple linear formula.

In these cases, we use a linear model to approximate the pattern because it is simpler. We introduce the linear model by drawing a line of best fit through the graphed empirical data.

Example 16This table shows the progressive score (total runs) of a cricket team playing in a one-day 50-over match. (An over is a bowler’s round of six balls bowled.)

(a) Graph this data and construct a line of best fit.(b) Find the equation of the line.(c) Use the equation to estimate the final score after 50 overs.(d) Use the line to estimate the over in which the team’s score reached 100.(e) A cricket team’s run rate is the average number of runs scored per over. According to

the linear model, what was this team’s run rate?(f) What are the limitations of this linear model?

Solution(a)

n (overs) 7 17 20 25 33 45

S (runs) 60 87 105 123 144 172

P

5 10 15 20 25 30 35 40 45 50

180

160

140

120

100

80

60

40

20

0

Run

s, S

Overs, n

Progressive score of a cricket team in a 50-over match

Use a transparent ruler when constructing a line of best fit.

Your line may differ slightly from this one.

(20, 105)

(30, 132)



(b) The equation has the form S = mn + b.Choosing two points on the line, (20, 105), (30, 132),

m =

= =

= 2.7∴ S = 2.7n + b

Substitute (20, 105) to find b.105 = 2.7(20) + b

= 54 + bb = 51

∴ S = 2.7n + 51(c) When n = 50 S = 2.7(50) + 51

= 186 runs(d) The score reached 100 in the 19th over.(e) The team’s run rate is the gradient of the line. Run rate = 2.7 runs/over.(f) The vertical intercept of 51 implies that the team already had a score of 51 when the

game started (0 overs). This is not possible. Also, a straight line graph suggests that the team’s score increases at a steady rate, but in reality the players all have different batting abilities and would not score at the same rate.

1. Of the functions listed below, which ones are:(a) linear? (b) quadratic? (c) cubic? (d) none of theseA. y = –3x2 B. y = –4x – 1C. y = x2 – 2x + 1 D. y = –x3 + x + 4E. y = 12 – x F. y = x3 – x2

G. y = 2x2 + 9 H. y = 2x

I. y = – J. y = x

K. y = x3 + 6x2 + 7x – 10 L. y =

2. Find the linear function for each of these tables of values.

(a) x 4 7 12 20 (b) k 6 10 13 15

y 10 31 66 122 C 152 120 96 80

change in Schange in n----------------------------

132 105–30 20–

------------------------(20, 105) is actually one of the observed data points.

2710------

From the graph, a vertical intercept of 51 looks reasonable.

From the graph, a final score of 186 looks reasonable.

A line of best fit:� represents most or all of the points as closely as possible� goes through as many points as possible� has roughly half of the points above it and roughly half of the points below it� is drawn so that the distance between each point and the line is kept at a

minimum.

Exercise 1-08: Linear functions

1x--- 1

4---

12---

x 7–2

------------



3. The table below shows the cost of water usage for different volumes of water.

(a) Find a linear formula for C in terms of v.(b) What does the gradient represent?(c) What does the vertical intercept represent?(d) What is the independent variable and what does it represent?

4. A road safety analyst studied the relationship between the populations of towns and the road accident rate (per day) in those towns.

(a) Graph this data and construct a line of best fit.(b) Find the equation of the line.(c) What is the daily road accident rate per 1000 population?(d) What is the daily road accident rate when the population is 24 000?(e) Estimate the population for which the daily road accident rate is 20.

5. This conversion graph shows the linear relationship between the Celsius and Fahrenheit scales for measuring temperature.

Volume, v (kL) 8 15 20 26 38 50

Cost, C ($) 98.8 104.75 109 114.1 124.3 134.5

Population, P (thousands) 12 17 20 32 45 50 61 67

Accidents per day, A 2 3 4 7 10 11 13 15

5 10 15 20 25 30 35 40 45 50

90

80

70

60

50

40

30

0

Fah

renh

eit

scal

e, F

(°F

)

Celsius scale, C (°C)

100

110

Celsius–Fahrenheit temperature conversion graph

120

(20, 68)

(35, 95)



(a) Find the formula for F in terms of C.(b) What is the independent variable?(c) What is the vertical intercept and what does it represent?(d) Use the formula to convert:

(i) 12°C to °F (ii) 86°F to °C(e) Use the graph to convert:

(i) 12°C to °F (ii) 86°F to °C

6. Lizzie recorded the value of her computer every 6 months. She graphed this data and drew a line of best fit as shown below.

(a) Find the equation of Lizzie’s line of best fit.(b) Use the equation to calculate the original value of her computer.(c) What is the gradient of the line and what does it represent?(d) Calculate the value of her computer after:

(i) 20 months (ii) 3 years

(e) When will the value of the computer be zero? Answer in years and months.

Months after purchase, m 6 12 18 24 30 36

Value of computer, $V 2900 2700 2510 1850 1200 750

12---

6 12 18 24 30 36 42 48

3600

3200

2800

2400

2000

1600

1200

800

400

0

Val

ue o

f co

mpu

ter,

V (

$)

Months after purchase, m

4000

Value of computer over time



7. The length of a shoe (in inches) has a linear relationship with its shoe size.

(a) What is the independent variable?(b) Find the linear function for L in terms of S.(c) What does the gradient represent?(d) What does the vertical intercept represent?(e) What is the length of a shoe of:

(i) size 6? (ii) size 7 ?(f) What size is a shoe of length:

(i) 8 inches? (ii) 13 inches?

8. In a factory, the weekly cost $C of manufacturing palm-sized computers is shown in the table, where p is the number of units produced.

(a) What is the dependent variable?(b) Find the linear function.(c) What does the gradient represent?(d) What does the vertical intercept represent?(e) What is the cost of producing 200 palm-sized computers?(f) How many units can be produced for $60 000?

9. Erin observed the following relationship between the volume of petrol V (in litres) in her car and the distance d (in kilometres) travelled on this volume.

(a) Graph this data and construct a line of best fit.(b) Find the equation of this line.(c) Use the equation to predict:

(i) the distance travelled on 45 L of petrol(ii) the number of litres of petrol needed to travel 350 km

(d) What is the gradient, m, and what does it represent?

(e) Use the formula F = to calculate the fuel consumption rate of Erin’s car in

litres per 100 km correct to 1 decimal place.

S (size) 2 5 7 8 12

L (inches) 9 10 10 11 12

p (units) 50 85 120 136 157

C ($) 21 900 31 350 40 800 45 120 50 790

V (L) 18 22 28 38 40 56

d (km) 179 225 301 387 412 580

23--- 1

3---

12---

23---

100m

---------



Programmable, statistical and graphics calculators usually have a feature that allows you to input the (x, y) coordinates of a set of points, then fits a straight line of the form y = mx + b through them, such that the distances between the points (observed data) and the line (linear model) are minimised. This is called a linear regression or line of best fit function.

Use the linear regression function of your calculator to redo Example 16 or question 9 in the previous exercise. Enter the coordinates of the set of observed data and see how the calculator generates a line of given gradient and vertical intercept. Compare the calculator’s linear model to your own.

Equipment: Tape measure or height chart, graph paper, graphing software or graphics calculator (if available).

Is it possible to predict a person’s shoe size from their height?

1. Select a sample of about 20 people. Measure their heights, h, to the nearest centimetre and record their shoe sizes, s.

2. Plot the data on a graph with s on the vertical axis (dependent variable).

3. Construct a line of best fit and find its equation in the form s = mh + b.

4. Looking at your data, do you think a linear model is appropriate or accurate?

5. Test your linear model by measuring the heights of some more people and predicting their shoe sizes.

6. If you have graphing software or a graphics calculator, enter the sample data from step 1 and compare the linear model generated to the one you created.

Technology: Line of best fit on the calculator

Modelling activity: Predicting shoe size

Study tips

READING AHEAD

If you can afford the time, read the textbook ahead of the lesson. Pre-reading provides your mind with a structure for categorising the new concepts and skills taught by your teacher. You can also ask questions in class straight away when you are unsure of something. Reading ahead is the reverse of revising your course notes and another way of maximising your learning. Many students learn better when they read ahead and know what to expect in the lesson before entering the classroom. You could also pre-read a study guide, your school’s teaching program (ask your teacher for one) or an ex-student’s course/summary notes.



INTERSECTION OF LINESExample 17Ally is organising an end-of-year outdoor lunch for her office colleagues. The cost of hiring a marquee (tent) is $116 and the food catering is $6.20 per guest. To cover these costs, Ally is charging each guest $12. The cost and revenue functions are represented by the formulas

Cost: C = 6.2n + 116Revenue: R = 12n

where C and R are amounts expressed in dollars and n represents the number of guests.(a) Graph both the C and R functions on the same axes for values of n from 0 to 50.(b) Use the graph to find the cost (to the nearest $10) of running the lunch for:

(i) 12 guests (ii) 30 guests(c) Why would it be unwise to run a lunch for 12 guests?(d) For what value of n does the cost of the lunch equal the revenue?(e) Why is this value called the ‘break-even’ point? What happens for values of n above this

break-even point?

Solution(a)

5 10 15 20 25 30 35 45

450

400

350

300

250

200

150

100

50

0

Cos

t/re

venu

e ($

)

No. of guests, n

500

Cost and revenue graphs for a lunch

550

5040

Cost, C = 6.2n + 116

Reven

ue, R

= 12

n

(20, 240)



(b) (i) $190 (ii) $300(c) The revenue from the lunch would not cover the costs. Ally would lose money.(d) n = 20 (C = R = $240)(e) Because when n = 20, the revenue covers the cost of the lunch exactly. For values of n

above 20, the revenue is more than the cost so Ally would make a profit.

Finding the point of intersection (20, 240) of the two lines above is an example of solving simultaneous equations: that is, finding a pair of values that solve both equations at the same time.

Linear functions may be graphed using graphing software or a graphics calculator, by entering either the equation of the line or simply the coordinates of two points on the line. All of the questions in Exercise 1-09 below can be solved using graphing software or a graphics calculator. Learn how to:

� graph a line by entering its equation (graph function);

� graph a line by entering the coordinates of two points on the line (plot function);

� use the trace and zoom functions to find the coordinates of points on a line, including the x- and y-intercepts;

solve a pair of simultaneous linear equations graphically by using the trace and zoom functions to locate the point of intersection.

1. For each question, find the equations of both lines and write the coordinates of their point of intersection.

(a) (b)

2. Solve this pair of simultaneous equations graphically:y = 2x – 1 and y = –x + 5

Technology: Graphing software and graphics calculators

Exercise 1-09: Intersection of lines

2 4

4

2

−2

1 3

−3

−4

1

y

x

3

−1−2−3−4 4

4

2

−2

1 3−1−2

1

y

x

3

5 62

−3

−4



3. At CopyCat Express, the charge for photocopying is 13 cents per sheet. Klaas is investigating the option of purchasing his own photocopier for $1200, which brings the cost down to 5 cents per sheet. The two cost functions have been graphed on the following page, represented by the formulas

CopyCat Express: C = 0.13xOwn photocopier: C = 0.05x + 1200

where x represents the number of copies made.(a) From the graph, which option is cheaper for making 5000 copies?(b) Calculate the cost of making 5000 copies under each option.(c) Why do you think one of the options is much more expensive at first?(d) At what value of x would you find the break-even point?(e) What happens when x exceeds this break-even value?

4. The monthly cost of purchasing new movies for a video store is given by the formulaC = 16n + 576

while the monthly income earned from each new movie bought isI = 48n

where C and I are in dollars and n represents the number of new videos bought in a month.

2 4 6 8 12 14 16 18

2250

2000

1750

1500

1250

1000

750

500

250

0

Pho

toco

pyin

g co

st, C

($)

No. of copies, x (thousands)

2500

Cost graphs for two photocopiers

10 20

2750

Own photocopier, C = 0.05x + 1200

CopyC

at Exp

ress

, C = 0.

13x



(a) Graph both functions for values of n from 0 to 40 and find the break-even point.(b) What would happen if the video store bought fewer videos than the value of n given

by the break-even point?(c) Calculate the monthly cost and income from purchasing 35 videos.(d) According to the formula, how much monthly income (on average) can be gained

from one new video?(e) The formula C = 16n + 576 suggests that the cost of purchasing new videos is $576

plus $16 per video. Why do you think there are two types of costs?

5. Dave claims that a quick ‘rule-of-thumb’ for converting Celsius temperatures to Fahrenheit is ‘double and add 30’.(a) Graph Dave’s rule and the actual conversion formula for values of C from 0 to 100:

Dave’s rule: F = 2C + 30

Actual conversion formula: F = C + 32

where C represents the temperature in °C and F represents the temperature in °F.(b) For what value of C is Dave’s rule exactly equal to the actual conversion formula?(c) For what values of C does Dave’s rule give answers that are too high?(d) For what values of C is the difference between Dave’s rule and the actual conversion

formula not more than 5° in Fahrenheit?(e) Is Dave’s rule-of-thumb a good one? Justify your answer.

6. Jenni is comparing two Internet access plans. Optnet has a monthly access fee of $8 and charges 60 cents per hour of Internet use. OzExpress charges $1 per hour of Internet use but has no monthly access fee. The two plans are represented by the formulas:

Optnet: C = 0.6t + 8OzExpress: C = t

where C is the cost in dollars and t is the number of hours of Internet use.(a) Graph both formulas for values of t from 0 to 50, and identify the break-even point.(b) If Jenni uses the Internet for an average of 24 hours per month, which plan is better

for her?(c) Calculate the cost of 24 hours’ Internet use under each plan.(d) Write a short explanation outlining the advantages of each plan, taking the break-

even point into account.(e) Which plan would you choose for yourself? Why?

7. The trip charges for two taxi companies are shown below as formulas:Burntrubber Taxis: C = 0.9k + 4.6 ($4.60 plus 90 c/km)Whiteknuckle Cabs: C = 1.2k + 2.5 ($2.50 plus $1.20/km)

where C represents the charge in dollars and k represents kilometres travelled.(a) Graph both formulas and find their point of intersection.(b) What does the point of intersection represent?(c) Describe the advantages of using each taxi company.(d) John travels to work each day by taxi, a distance of 11 km. Determine which is the

better taxi for him to use, and calculate the saving he would make using this one.

95---



Simultaneous equations can also be solved algebraically. For example, to solve

y = −3x + 4 and y = x + 18

we can make the equations equal to each other and solve.

−3x + 4 = x + 18

−6x + 8 = x + 36 Multiplying both sides by 2−7x = 28

x =

= −4

To find the value of y, substitute x = −4 into either of the original equations, say, the first.y = −3(−4) + 4

= 12 + 4= 16

∴ The solution is x = −4, y = 16.Solve each of the questions in Exercise 1-09 algebraically.

The ‘guess, check and refine’ method of solving equations can also be used to solve simultaneous equations. Consider the equations from question 7 in Exercise 1-09.

Burntrubber Taxis: C = 0.9k + 4.6Whiteknuckle Cabs: C = 1.2k + 2.5

Create a spreadsheet that calculates the values of C for both equations given a starting guess value for k. Enter a step size for increasing the values of k. Use it to find the value of k that gives the same value of C for both equations.

A B C D

1 SIMULTANEOUS EQUATIONS

2

3 Starting guess, k 8

4 Step size 0.5

5 (Burntrubber) (Whiteknuckle)

6 k Equation 1 (C) Equation 2 (C) Difference

7 8.0 11.80 12.10 -0.30

8 8.5 12.25 12.70 -0.45

9 9.0 12.70 13.30 -0.60

10 9.5 13.15 13.90 -0.75

11 10.0 13.60 14.50 -0.90

12 10.5 14.05 15.10 -1.05

13 11.0 14.50 15.70 -1.20

14 11.5 14.95 16.30 -1.35

15 12.0 15.40 16.90 -1.50

16 12.5 15.85 17.50 -1.65

17 13.0 16.30 18.10 -1.80

Investigation: Algebraic solution of simultaneous equations

12---

12---

287–

------

Spreadsheet activity: Guess, check and refine



Chapter review

Equations and functions1. Algebraic expressions 2. Scientific notation3. Formulas4. Solving equations5. Equations involving powers and roots6. Changing the subject of a formula7. Equations and formulas8. Linear functions9. Intersection of lines

This chapter, Equations and functions, revised and extended algebraic modelling concepts introduced in the Preliminary Course. New theory included equations involving powers, the ‘guess, check and refine’ method of solving equations and changing the subject of a formula, so be sure to include them in your topic summary. By now you should be competent in solving all types of equations, graphing and analysing linear functions (including lines of best fit) and finding the point of intersection of two lines.

Make a summary of this topic. Use the chapter outline above as a guide. An incomplete mind map has been started below. Use your own words, symbols, diagrams, boxes and highlighting. Make connections, look for general principles, and include personal observations and reminders. Use the questions in Your say below to think about your understanding of the topic. Gain a ‘whole picture’ view of the topic and identify any weak areas.

Topic summary

Equations and functions

Linear functions

EquationsAlgebra revision

FormulasIntersection of lines



� Have you satisfied the outcomes listed at the front of this chapter?� What was the most important thing that you learned?� How did you feel about the topic? Did you enjoy it?� What was new?� What are your weaknesses? What will you need to study more?� How will you revise and summarise this topic?


(a) 3f 2 × (b) (−4b3)2

(c) 2xy + 4y − 2y − 2xy (d) 2m2n � 4m

(e) 2f + 4f 2 + 3f − f 2 (f) 6gh × 4gh

2. Expand and simplify these expressions.(a) 3d(d + 4) + 8(d − 1) (b) z(4 + y) − y(4 − z)(c) πr(r + s) + πr2 (d) 4m(3m + y) − m(10m + 7y)

3. Express in scientific notation:(a) 830 000 (b) 47 100 (c) 0.000 000 162

4. Express in normal form:(a) 2.9 × 10−4 (b) 6.54 × 108 (c) 3 × 103

5. Evaluate and express your answers in scientific notation correct to 3 significant figures:(a) (3.75 × 108) � (4.6 × 103) (b)

6. The maximum distance d m that a ball covers if thrown with velocity v m/s is

d =

where g is the gravitational acceleration. Find d correct to 1 decimal place if v = 17.2 m/s and g = 9.8 m/s2.

7. The average blood pressure of a person aged y years, measured in millimetres of mercury (mm Hg), is

P = 110 +

Calculate the average blood pressure of a person aged:(a) 18 years (b) 29 years (c) 40 years

8. Make y the subject of the formula P = 110 + .

9. The distance (in metres) between Earth and another planet can be calculated using the formula d = ct

where c m/s is the speed of light and t seconds is the time it takes a radio signal to travel from Earth to the other planet and back. Calculate the distance (in kilometres) between Earth and Mars if c = 3 × 108 m/s and t = 520 s.

Your say: Reflecting about the topic � � � �

Chapter assignment

2 fh15h---------

2.9 10 7–×3

v2

g-----

y2---

y2---

12---



10. Solve these equations.

(a) 7(2x + 12) = 5x (b) = −2 (c) =

(d) = 12 (e) = −1 (f) = 8

11. Solve these equations correct to 1 decimal place.

(a) = 15 (b) 5a3 = 24 (c) 4x2 + 7 = 9

(d) d3 − 4 = −14 (e) 3k = 100 (f) 1.4p = 25

12. Make d the subject of the formula V = hd2 + 9.9.

13. The number of matchsticks needed to make the following pattern of s squares is m = 3s + 1. How many squares can be made from 82 matches?

14. Eve’s height (in centimetres) over her first 20 years can be represented by the formulah = 5.5a + 68

where a is her age in years. At what age (to the nearest year) did Eve’s height reach:(a) 100 cm? (b) 155 cm? (c) 167 cm?

15. The formula A = P(1 + r)n

shows the amount ($A) to which a principal ($P) will grow if invested for n years at a rate of r per annum compound interest, where r must be written as a fraction or decimal. By the ‘guess, check and refine’ method, determine how many whole years it will take for a principal of $8000 invested at 10% p.a. to double in value.

16. Of the functions listed below, which ones are:(a) linear? (b) quadratic? (c) cubic?

A. y = 2x2 − x3 B. y = −x + 4 C. y = x3

D. y = x2 − 6x + 16 E. y = x F. y = 3x2 + 5x

G. y = 2x3 + 4x2 + x + 1 H. y = I. y = x

17. Loads of different masses M kg are tied to a spring and the length of the spring L cm is measured.

(a) What is the dependent variable?(b) Find the linear function.(c) What is the vertical intercept and what does it represent?(d) How much does the spring’s length increase for each new kilogram of mass

added?(e) Calculate the length of the spring when the load is 12 kg.(f) Calculate the mass (to the nearest kilogram) of the load that will make the spring

150 cm long.

M (kg) 8 15 21 29 30 34

L (cm) 74.4 108 136.8 175.2 180 199.2

3k5

------ k8---+

4 p 6+3

---------------- p 10–2

----------------

2d2 18+3

---------------------- m 4–3 6 2r–

3h2

4---------

12---

1 2 3 4 5

14---

12---

x2

9-----



18. Find the linear function for this table of values.

19. The number of bacteria in a sample of milk is checked daily.

(a) Graph this data and construct a line of best fit.(b) Find the equation of the line.(c) What does the gradient represent?(d) Calculate the amount of bacteria per millilitre on the 7th day.(e) Milk is unfit for use when the number of bacteria exceeds 100 000/mL. When will

this occur?

20. The cost of manufacturing palm-sized computers is given by the functionC = 270p + 8400

where p is the number of units made and C is the cost in dollars. The revenue from selling these computers is given by the function

R = 410pwhere R is the sales in dollars.(a) Graph both functions on the same axes for values of p from 0 to 100.(b) Is p = 50 above or below the break-even point? What would happen if the

company made 50 palm-sized computers?(c) Calculate the cost and revenue from producing 50 palm-sized computers.(d) What are the values of p, C and R at the break-even point?(e) What is the vertical intercept of the cost function (C) and what does it represent?(f) How much does one palm-sized computer sell for?

x 12 20 36 44 60

y 15 17 21 23 27

Day, d 0 1 2 3 4 5

Bacteria, B (1000/mL) 3 12 25 32 47 51


equations and functions century year 12/01 equations and...equations and functions 1 1 equations and...

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