Notesale: Turn your study into money

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---

Chapter 5

Percentages
5
...
The
use of percentages is very common in many aspects
of commercial life, as well as in engineering
...

For this chapter you will need to know about decimals
and fractions and be able to use a calculator
...
e
...

• Interest rates indicate the cost at which we can borrow money
...
5% interest
rate for a year, it will cost you 6
...
If you repay
the loan in 1 year, how much interest will you have
paid?
• A pair of trainers in a shop cost £60
...
How much will you
pay?
• If you earn £20 000 p
...
and you receive a 2
...
What will be its cost?
When we have completed his chapter on percentages
you will be able to understand how to perform the above
calculations
...
For example, the fraction
100
and is read as ‘forty per cent’
...


DOI: 10
...
00005-3

5
...
2
...

Problem 1
...
015 as a percentage
To express a decimal number as a percentage, merely
multiply by 100, i
...

0
...
015 × 100%
= 1
...

Problem 2
...
275 as a percentage
0
...
275 × 100%
= 27
...
2
...

Problem 3
...
5% as a decimal number
6
...
5
= 0
...


34 Basic Engineering Mathematics
Problem 4
...
5% as a decimal number
17
...
5% =
100
= 0
...
2
...

Problem 5
...
5%
5
Problem 6
...
3157889
...
32% correct to 2 decimal places
Problem 7
...
Is the second
mark better or worse than the first?

57/79 =

57 57
5700
=
× 100% =
%
79 79
79

= 72
...
13% correct to 2 decimal places
Hence, the second test is marginally better than the
first test
...


5
...
4 To convert a percentage to a fraction
A percentage is converted to a fraction by dividing by
100 and then, by cancelling, reducing it to its simplest
form
...


Express 75% as a fraction

75
100
3
=
4
75
The fraction
is reduced to its simplest form by can100
celling, i
...
dividing both numerator and denominator
by 25
...

37
...
5% as a fraction

37
...

1
...
0032

2
...
734

3
...
057

4
...
374

5
...
285
6
...

7
...
25% as a decimal number
...

16
5
9
...

8
...
Express as percentages, correct to 3 significant
figures,
19
11
7
(b)
(c) 1
(a)
33
24
16
11
...

12
9
5
6
(a)
(b)
(c)
(d)
21
17
9
11
12
...

13
...
25% as a fraction in its simplest
form
...
Express 56
...

15
...

Decimal number Fraction Percentage
0
...
5
× 32
100
= 4 minutes

12
...

Alternatively, if the time is reduced by 12
...
5% = 87
...
e
...
5
× 32
100
= 28 minutes

87
...
A 160 GB iPod is advertised as
costing £190 excluding VAT
...
5%, what will be the total cost of the iPod?
17
...
25
100
Total cost of iPod = £190 + £33
...
25
VAT = 17
...
175 ×
£190 = £223
...
3
...

Problem 13
...
3

Further percentage calculations

5
...
1 Finding a percentage of a quantity
To find a percentage of a quantity, convert the percentage
to a fraction (by dividing by 100) and remember that ‘of’
means multiply
...
Find 27% of £65
27
× 65
27% of £65 =
100
= £17
...
In a machine shop, it takes 32
minutes to machine a certain part
...
5%
...
94444
...
Express 47 minutes as a percentage
of 2 hours, correct to 1 decimal place
Note that it is essential that the two quantities are in the
same units
...
2% correct to
1 decimal place

36 Basic Engineering Mathematics
5
...
3 Percentage change
Percentage change is given by
new value − original value
× 100%
...
A box of resistors increases in price
from £45 to £52
...
6% = percentage change in cost

% change =

Problem 16
...
The nearest speed available on the
machine is 412 rev/min
...


Calculate 43
...


2
...


3
...


A block of Monel alloy consists of 70%
nickel and 30% copper
...
2 g
of nickel, determine the mass of copper in the
block
...


An athlete runs 5000 m in 15 minutes 20
seconds
...
5%
...


8
...
5%
iron and the remainder aluminium
...
8 kg
mass of the alloy
...


A computer is advertised on the internet at
£520, exclusive of VAT
...
5%, what is the total cost of the computer?

10
...

11
...
Express this as a percentage of the
whole day, correct to 1 decimal place
...
Express 408 g as a percentage of 2
...

13
...
Calculate the percentage
pay increase, correct to 3 significant figures
...
A metal rod 1
...
6 mm
...

15
...
5% of a length of wood is 70 cm
...
A metal rod, 1
...
Calculate the
percentage increase in length
...
42 grams
(c)

147% of 14
...


4
...
Determine the percentage that
is unsatisfactory
...


Express
(a) 140 kg as a percentage of 1 t
...
4

More percentage calculations

5
...
1 Percentage error
Percentage error =

error
× 100%
correct value

(b) 47 s as a percentage of 5 min
...
4 cm as a percentage of 2
...


Problem 17
...
5 mm
...
What is the percentage error in the
measurement?
error
× 100%
correct value
64
...
5
150
=
× 100% =
%
63
63
= 2
...
A couple buys a flat and make an
18% profit by selling it 3 years later for £153 400
...
e
...

new value
× 100
Original cost =
100 + % change

The percentage measurement error is 2
...
38% error
...
The voltage across a component in
an electrical circuit is calculated as 50 V using
Ohm’s law
...
4 V
...
4 − 50
× 100%
=
50
...
4
40
=
× 100% =
%
50
...
4
= 0
...
79% too
low, which is sometimes written as −0
...


In this case, it is a 35% reduction in price, so we
new value
use
× 100, i
...
a minus sign in the
100 − % change
denominator
...
5
× 100
=
100 − 35
149
...
An electrical store makes 40% profit
on each widescreen television it sells
...
e
...

new value
Dealer cost =
× 100
100 + % change

new value
× 100%
100 ± % change

Problem 19
...
50 in a sale for a
DVD player which is labelled ‘35% off’
...
4
...


5
...
3 Percentage increase/decrease and
interest
New value =

100 + % increase
× original value
100

Problem 22
...
25% interest per annum
...
25
× £3600
100
106
...
0625 × £3600

Value after 1 year =

annum
...

8
...
5%
...
Calculate the increase he will actually receive per
month
...


Five mates enjoy a meal out
...
They add a 12
...
How much does each pay?

= £3825
Problem 23
...
5%
...
What is the new price?
100 + 6
...
5
=
× £2, 400 = 1
...
In December a shop raises the cost of a 40
inch LCD TV costing £920 by 5%
...
What is the sale price of
the TV?
11
...
What did he pay originally for the
business?

1
...
The
length is incorrectly measured as 36
...

Determine the percentage error in the measurement
...
A drilling machine should be set to
250 rev/min
...
Calculate the
percentage overspeed
...


When a resistor is removed from an electrical circuit the current flowing increases from
450 μA to 531 μA
...


13
...
Determine the masses of the three
elements present
...


In a shoe shop sale, everything is advertised
as ‘40% off’
...


Over a four year period a family home
increases in value by 22
...

What was the value of the house 4 years ago?

14
...

Determine the percentage of each of these
three constituents correct to the nearest 1%
and the mass of cement in a two tonne dry
mix, correct to 1 significant figure
...


An electrical retailer makes a 35% profit on
all its products
...


The cost of a sports car is £23 500 inclusive
of VAT at 17
...
What is the cost of the car
without the VAT added?

7
...
75% per

15
...
How
much ore is needed to produce 3600 kg of
iron?
16
...
5 ± 8% mm
...

17
...
If
the efficiency of the engine is 75%, determine
the power input
...
The marks available are shown in brackets at the
end of each question
...


Convert 0
...


(2)

2
...
4375 to a mixed number
...


Express

4
...

32
Express 0
...


(2)
(2)

5
...
0572953 correct to 4 significant
figures
...


Evaluate
(a) 46
...
085 + 6
...
07
(b) 68
...
34

(4)

7
...
37 × 1
...


Evaluate 250
...
1 correct to 1 decimal
place
...
2 × 12
(2)

9
...
Evaluate the following, correct to 4 significant
(3)
figures: 3
...
73 + 1
...
Evaluate 6
...
54 correct to 3 decimal
places
...
Evaluate
−
correct to 4 significant
0
...
065
figures
...
Evaluate 6 − 4 as a mixed number and as a
7
9
decimal, correct to 3 decimal places
...
Evaluate,
correct


1
...
673
√
4
...
If a = 0
...
85, c = 0
...
7 and
e = 0
...
Evaluate the following, each correct to 2 decimal
places
...
22 × 0
...
8 × 12
...
692
(4)
(b)
√
17
...
98
21
...
6 km = 1 mile, determine the speed of
45 miles/hour in kilometres per hour
...
The area A of a circle is given by A = πr 2
...
73 cm, correct
to 2 decimal places
...
The potential difference, V volts, available at battery terminals is given by V = E − I r
...
23, I = 1
...
60
(3)

23
...
20, v = 10
...
81, given that
W v2
B=
(3)
2g

4 1 3
14
...

(3)

24
...
25% as a fraction in its simplest
form
...
Evaluate
form
...
Evaluate resistance, R, given
=
+
+
R
R1 R2
1
when R1 = 3
...
2 k and
R3
R3 = 13
...

(3)

25
...
5% of a length of wood is 70 cm
...
A metal rod, 1
...
Calculate the percentage increase in length
...
A man buys a house and makes a 20% profit when
he sells it three years later for £312 000
...

The n’th term is: a + (n − 1)d
n
Sum of n terms, Sn = [2a + (n − 1)d]
2

Geometric progression:
If a = first term and r = common ratio, then the geometric progression is: a, ar, ar2 ,
...

4
...

10
...

16
...


Exercise 5 (page 11)

19 kg
2
...
479 mm
−66
5
...
−225
−2136
8
...
£10 7701
−4
11
...
5914
189 g
14
...
$15 333
89
...

3
...

5
...

9
...

9
...

17
...

(a) 8613 kg (b) 584 kg
(a) 351 mm (b) 924 mm
(a) 10 304 (b) −4433
6
...

(a) 8067 (b) 3347
10
...

4
...

8
...


(a) 48 m (b) 89 m
(a) 1648 (b) 1060
18 kg

1
...
14
7
...
88
8
...
1016/B978-1-85617-697-2
...


3 4 1 3 5
, , , ,
7 9 2 5 8
9
10
...
1
15
16
18
...


6
...

21
...
2
3
...
11
8
...

13
...

18
...

11
...

19
...

8
...

16
...


4
...
2

5
...
5

5
12
1
9
...

23
4
...
15

3
28
8
10
...


15
...
400 litres
22
...

15

1
...
59
6
...
−1

2
5

Exercise 6 (page 13)

11
...
7

(a) £1827 (b) £4158

Exercise 3 (page 6)
1
...

5
...

9
...
2

11
...
4
20
17
12
...
−

3
...
2

1
6

3
4
1
9
...
4

13
20
1
10
...


Answers to practice exercises
Exercise 14 (page 25)

Chapter 3

1
...
571
5
...
96
8
...
0871

Exercise 8 (page 17)
1
...
23
8
...


13
20
21
6
...
(a) 1
50
5
...
0
...


4
...
6

7
16

(e) 16

17
80

1
...

7
...

13
...
182
2
...
122
3
...
82
0
...
0
...
2
...
273
8
...
256
9
...
30366
6
1
...
3
...
37
...
2 × 10
14
...
767 ×10
15
...
32 ×106

12
...
6875 13
...
21875 14
...
1875

Exercise 16 (page 27)
1
...
4667

Exercise 9 (page 18)
1
...
18
5
...
297

2
...
785
3
...
38
6
...
000528

2
...
3
6
...
3

4
...
27

3
...
54
7
...
52 mm

4
...
83

13
14

3
...
458

6
...
7083

7
...
2
...
3

1
3

10
...
0776

1
...
9205
5
...
4424
9
...
6992

2
...
7314
6
...
0321
10
...
8452

3
...
9042
7
...
4232

4
...
2719
8
...
1502

Exercise 18 (page 28)

4
...
47
...
385
...
582
...
9 6
...
82
7
...
1
8
...
6
0
...
0
...
1
...
53
...
84 14
...
69
15
...
81 (b) 24
...
00639 (b) 0
...
(a) 8
...
6˙
2400

1
...
995
5
...
6977
9
...
520

Exercise 12 (page 23)
3
...
62
7
...
330

4
...
832
8
...
45

Exercise 13 (page 24)
1
...
25
2
...
0361 3
...
923 4
...
296 × 10−3
5
...
4430 6
...
197 7
...
96 8
...
0549
9
...
26 10
...
832 × 10−6

2
...
782
6
...
92
10
...
3770

3
...
72
7
...
0

4
...
42
8
...
90

Exercise 19 (page 29)
1
...

7
...


Chapter 4

2
...
1
...
12
...


Exercise 17 (page 27)

Exercise 11 (page 20)

1
...
797
5
...
42
9
...
59

1
21
9
...
567
5
...

5
...

13
...

18
...
40
3
...
13459
4
...
9
6
...
4481 7
...
36 × 10−6
9
...
625 × 10−9
10
...
70

Exercise 15 (page 25)

3
125

15
...
28125

1
...
3
5
...
3

341

A = 66
...
144 J
14 230 kg/ m3

2
...

8
...


C = 52
...
407 A
628
...
1 m/s

3
...

9
...


R = 37
...
02 mm
224
...
526 

Exercise 20 (page 30)
1
...

7
...

12
...
27
2
...
1 W
3
...
61 V
F = 854
...
I = 3
...
T = 14
...
96 J
8
...
77 A 9
...
25 m
A = 7
...
V = 7
...
53 h (b) 1 h 40 min, 33 m
...
h
...
02 h (d) 13
...
£556 2
...
264 kg 4
...
14
...
(a) 0
...
(a) 440 K (b) 5
...
0
...
173
...
5
...
37
...
128
...
0
...
0
...
0
...
38
...
(a) 21
...
2% (c) 169%
13
5
9
11
...

13
...

20
16
16
1
15
...
25, D = 25%, E = 0
...
60, H = 60%, I = 0
...
(a) 2 mA (b) 25 V 2
...
685
...
83 lb10 oz
5
...
1 litres (b) 16
...
29
...
584
...
$1012

Exercise 28 (page 46)
Exercise 22 (page 36)
1
...

5
...

10
...


21
...
9
...
4 t (b) 8
...
67%
14 minutes 57 seconds
37
...
39
...
7%
15
...
60 m

(c) 20
...

(c) 5
...

8
...

12
...

16
...
5%

2
...
8 g
£611
38
...
3
...
20 days
3
...
18 (b) 6
...
3375 4
...
(a) 300 × 103 (b) 0
...

5
...

13
...

15
...


2
...
18%
3
...
£175 000
£260
6
...
£9116
...
£50
...
60 10
...
70 11
...
7
...
6 kg, B 0
...
5 kg
54%, 31%, 15%, 0
...
5 mm, 11
...
600 kW

Chapter 6

1
...
±5

2
...
±8

3
...
100

5
...
64

4
...
1

Exercise 30 (page 50)
1
5
...
8
3
...

9
1
or 0
...
5 11
...
100 8
...

100
12
...
36
14
...
1 16
...
5 or
18
...

or 0
...
1
3
243
2
1
...
39

Exercise 24 (page 41)
1
...
3
...
47 : 3
1
4
...
5 hours or 5 hours 15 minutes
4
6
...
12 cm
8
...


1
3 × 52

5
...
1 : 15 2
...
25% 4
...
6 kg
5
...
3 kg 6
...
−
18
9
...


1
3
7 × 37

6
...
−1
14
...
±
2

3
...
−3
15
...


1
210 × 52

8
...

45
9

2
3

344 Basic Engineering Mathematics
Chapter 10

Chapter 11

Exercise 39 (page 69)
1
...

5
...

9
...

13
...

17
...

21
...

25
...


x 2 + 5x

+6
+9
4x 2 + 22x + 30
a 2 + 2ab + b2
a 2 − 2ac + c2
4x 2 − 24x + 36
64x 2 + 64x + 16
3ab − 6a 2
2a 2 − 3ab − 5b2
7x − y − 4z
x 2 − 4x y + 4y 2
0
4ab − 8a 2
2 + 5b2
4x 2 + 12x

Exercise 42 (page 75)

2
...

6
...

10
...

14
...

18
...

22
...

26
...


2x 2 + 9x

+4
− 12
2 pqr + p2 q 2 + r 2
x 2 + 12x + 36
25x 2 + 30x + 9
4x 2 − 9
r 2 s 2 + 2rst + t 2
2x 2 − 2x y
13 p − 7q
4a 2 − 25b2
9a 2 − 6ab + b2
4−a
3x y + 9x 2 y − 15x 2
11q − 2 p
2 j2 +2 j

Exercise 40 (page 71)
2(x + 2)
p(b + 2c)
4d(d − 3 f 5)
2q(q + 4n)
bc(a + b2 )
3x y(x y 3 − 5y + 6)
7ab(3ab − 4)


2x y x − 2y 2 + 4x 2 y 3
3x
17
...
(a + b)(y + 1)
22
...

3
...

7
...

11
...

15
...
0 19
...
( p + q)(x + y)
23
...

4
...

8
...

12
...

16
...
1

2
...
1

7
...
2
16
...
−4

3
...

2
13
...
−3

12
...
6

9
...
−2

2
...
4 + 3a
6
...
10y 2 − 3y +
9
...


1
− x − x2
5

1
7

1
4

8
...
5

2
...
−4

6
...
−4

8
...
−10

12
...
9

17
...
±12

22
...
−15t
12
...
2
...
2

5
...
2

10
...
3

14
...
−6

18
...
4

20
...
±3

24
...

4
...

6
...
8 m/s2
3
...
472
(a) 1
...
30 m/s2

Exercise 45 (page 80)
1
...
45◦ C
7
...
0
...
50
8
...
30
6
...
3
...
d = c − e − a − b

1
− 4x
3

10
...
2x + 8x 2
3
...

− 4x
2

5
...
R =
I
c
7
...
v =

y
7
v −u
4
...
y = (t − x)
3
y−c
8
...
x =

Answers to practice exercises
I
PR
E
11
...
C = (F − 32)
9
9
...
L =

XL
2π f

12
...
x = a(y − 3)
14
...
64 mm

1
2π CX C

*

1
√

Z2 −

14
...
1 × 10−6

aμ
ρCZ 4 n

2

Chapter 13
Exercise 47 (page 87)

Exercise 49 (page 92)

S −a
a
1
...
x =

yd
d
(y + λ) or d +
λ
λ

3
...
D =

AB 2
5E y

5
...
R2 =

R R1
R1 − R

E −e
E − e − Ir
or R =
−r
I
I

y

ay
8
...
x = 
2
4ac
(y 2 − b2 )

7
...
R =
πθ
√
Z 2 − R2
, 0
...
L =
2π f
10
...
u =




xy
1
...
r =
(1 − x − y)
c
5
...
b =
2( p2 + q 2 )
9
...
L =

8S 2
3d

Q
, 55
mc
+ d, 2
...
L =
μ−m


x−y
6
...
R = 4

uf
, 30
u− f


2dgh
, 0
...
v =
0
...
v =

x = 4, y = 2
x = 2, y = 1
...
5, n = 0
...

4
...

8
...

12
...

16
...
a = N 2 y − x

Exercise 48 (page 89)

1
...

5
...

9
...

13
...


1
...

5
...


p = −1, q = −2
a = 2, b = 3
x = 3, y = 4
x = 10, y = 15

2
...

6
...


x = 4, y = 6
s = 4, t = −1
u = 12, v = 2
a = 0
...
40

Exercise 51 (page 96)
1
1
1
...
p = , q =
4
5
5
...
x = 5, y = 1
4

1
1
2
...
x = 10, y = 5
1
6
...
1

Exercise 52 (page 99)
1
...

5
...

8
...
2, b = 4
u = 12, a = 4, v = 26
m = −0
...
00426, R0 = 22
...
I1 = 6
...
62
4
...
a = 12, b = 0
...
F1 = 1
...
5

Exercise 53 (page 100)
1
...
x = 5, y = −1, z = −2

2
...
x = 4, y = 0, z = 3

346 Basic Engineering Mathematics
5
...

9
...

11
...
x = 1, y = 6, z = 7
x = 5, y = 4, z = 2 8
...
5, y = 2
...
5
i1 = −5, i2 = −4, i3 = 2
F1 = 2, F2 = −3 F3 = 4

Exercise 57 (page 109)
1
...

6
...

10
...
191 s 2
...
345 A or 0
...
619 m or 19
...
066 m
1
...
165 m
12 ohms, 28 ohms

3
...

7
...


7
...
0133
86
...
4 or −4
4
...
5 or 1
...

10
...

16
...
−2 or −

2
3

2
...
0 or −
3
8
...
−3 or −7
14
...
−3
20
...
5

1
...
2 or −2

2
1
2
1
2
...

12
...

18
...
−1 or 1
...

or −
2
5
2
or −3
28
...
1 or −
3
7
4
1
29
...

2
3
27
...

4
21
...
4 or −7

31
...
x 2 + 5x + 4 = 0
35
...
2 or −6

or −
or

or −

Chapter 15
Exercise 59 (page 112)

1
3

1
3
1
1
6
...
2
8
...
1 10
...
2
12
...
100 000 14
...

32
1
16
...
01 17
...
e3
16
1
...
x 2 + 3x − 10 = 0
34
...
x 2 − 1
...
68 = 0

2
...
3

4
...


Exercise 60 (page 115)

Exercise 55 (page 106)
1
...
732 or −0
...
1
...
135
5
...
443 or 0
...
x = 0, y = 4 and x = 3, y = 1

2
...
137 or 0
...
1
...
310
6
...
851 or 0
...
log 6
5
...
log 15
6
...
log 2
7
...
log 3
8
...
log 10 10
...
log 2
12
...
log 16 or log24 or 4 log2
14
...

3
...

7
...

11
...


0
...
137
2
...
719
3
...
108
0
...
351
1
...
081
4 or 2
...
562 or 0
...

4
...

8
...

12
...
296 or −0
...
443 or −1
...
434 or 0
...
086 or −0
...
176 or −1
...
141 or −3
...
0
...
1
...
b = 2 20
...
x = 2
...
t = 8
21
...
x = 5

Exercise 61 (page 116)
1
...
690 2
...
170 3
...
2696 4
...
058 5
...
251
6
...
959 7
...
542 8
...
3272 9
...
2

Answers to practice exercises
Chapter 16

Chapter 17

Exercise 62 (page 118)
1
...

3
...

5
...
1653
(a)
5
...
55848
(a) 48
...
739

(b)
(b)
(b)
(b)
6
...
4584
0
...
40444
4
...
7 m

(c)
(c)
(c)
(c)

22030
40
...
05124
−0
...
2
...
(a) 7
...
7408
8 3
3
...
2x 1/2 + 2x 5/2 + x 9/2 + x 13/2
3
1 17/2
1
+ x
+ x 21/2
12
60

1
...
1 V
(c) Horizontal axis: 1 cm = 10 N, vertical axis:
1 cm = 0
...
(a) −1 (b) −8 (c) −1
...
14
...
(a) −1
...
4
5
...
3
...
05
3
...
1
...
30
4
...

2
...

7
...

14
...


(a) 0
...
91374 (c) 8
...
2293 (b) −0
...
13087
−0
...
−0
...
2
...
816
...
8274 8
...
02
9
...
522 10
...
485
1
...
3
13
...
9 15
...
901 16
...
095
a
t = eb+a ln D = eb ea ln D = eb eln D i
...
t = eb D a
 
U2
18
...
W = PV ln
U1

Exercise 68 (page 140)
1
...
75, 0
...
75, 2
...
75;
1
Gradient =
2
2
...
(a) 6, −3 (b) −2, 4 (c) 3, 0 (d) 0, 7
3
...
(a) 2, − (b) − , −1 (c) , 2 (d) 10, −4
2
3
3
18
3
3
5
6
...
(a) and (c), (b) and (e)
8
...
(1
...
(1, 2)

11
...
4 (d) l = 2
...
P = 0
...
5

13
...
(a) 40◦ C (b) 128 
2
...
5 V

Exercise 66 (page 127)
1
...
5◦C

3
...
25 (b) 12 (c) F = 0
...
99
...
(a) 29
...
31 × 10−6 s
4
...
993 m (b) 2
...
(a) 50◦ C (b) 55
...
30
...
(a) 3
...
46 s
8
...
45 mol/cm3
10
...
(a) 7
...
966 s

(d) 89
...
−0
...
73
5
...
5 m/s (b) 6
...
7t + 15
...
m = 26
...
63
7
...
31 t (b) 22
...
09 W + 2
...
(a) 96 × 109 Pa (b) 0
...
8 × 106 Pa

348 Basic Engineering Mathematics
1
1
(b) 6 (c) E = L + 6 (d) 12 N (e) 65 N
5
5
10
...
85, b = 12, 254
...
5 kPa, 280 K
9
...
(−2
...
2), (0
...
8); x = −2
...
6
10
...
2 or 2
...
75 and −1
...
3 or −0
...
(a) y (b) x 2 (c) c (d) d

2
...
(a) (b) x (c) b (d) c
x
x
1
y
5
...
a = 1
...
4, 11
...
y = 2x 2 + 7, 5
...
x = 4, y = 8 and x = −0
...
5
2
...
5 or 3
...
24 or 3
...
5 or 3
...
(a) y (b)

8
...
a = 0
...
6 (i) 94
...
2

Exercise 75 (page 162)
1
...
0, −0
...
5
2
...
1, −4
...
8, 8
...
x = 1
4
...
0, 0
...
6
5
...
7 or 2
...
x = −2
...
0 or 1
...
x = −1
...

2
...

4
...

6
...

9
...
0012 V2 , 6
...
0, b = 0
...
7, b = 2
...
53, 3
...
0, c = 1
...
y = 0
...
24x
T0 = 35
...
27, 65
...
28 radians

Exercise 72 (page 156)
x = 2, y = 4
x = 3
...
5
x = 2
...
2
a = 0
...
6

Exercise 76 (page 167)
1
...
27◦54
3
...
100◦6 52
◦


◦


5
...
86 49 1 7
...
55◦ 8
...
754◦
9
...
58◦22 52

Exercise 77 (page 169)
1
...
obtuse 3
...
right angle
5
...

3
...

7
...
x = 1, y = 1
4
...
x = −2, y = −3

Exercise 73 (page 160)
1
...
−0
...
6
3
...
9 or 6
...
−1
...
1
5
...
8 or 2
...
x = −1
...
75, −0
...
x = −0
...
6
8
...
63 (b) 1 or −0
...
(a) 102◦ (b) 165◦ (c) 10◦ 18 49
7
...
3◦ (h) 79◦ (i) 54◦
8
...
59◦ 20
10
...
51◦
12
...
326 rad 13
...
605 rad 14
...
(a) acute-angled scalene triangle
(b) isosceles triangle (c) right-angled triangle
(d) obtuse-angled scalene triangle
(e) equilateral triangle (f ) right-angled triangle

Answers to practice exercises
2
...
DF, DE
4
...
122
...
φ = 51◦, x = 161◦
7
...
a = 18◦ 50 , b = 71◦10 , c = 68◦ , d = 90◦,
e = 22◦ , f = 49◦, g = 41◦
9
...
17◦

11
...
sin A = , cos A = , tan A = , sin B = ,
5
5
4
5
3
4
cos B = , tan B =
5
3
8
8
3
...
sin X =
113
113
15
15
8
5
...
(a) sin θ =
(b) cos θ =
25
25
7
...
434 (b) −0
...
(a) congruent BAC, DAC (SAS)
(b) congruent FGE, JHI (SSS)
(c) not necessarily congruent
(d) congruent QRT, SRT (RHS)
(e) congruent UVW, XZY (ASA)
2
...

4
...

9
...


2
...
4
...
36
...
8660 (b) −0
...
5865
42
...
15
...
73
...
7◦56
◦

◦

◦
31 22 10
...
29
...
20◦21
0
...
1
...
x = 16
...
18 mm 2
...
79 cm
3
...
25 cm (b) 4 cm
4
...
(a) 12
...
619 (c) 14
...
349
(e) 5
...
275
2
...
831 cm, ∠A = 59
...
96◦
(b) DE = 6
...
634 cm, GH = 10
...
810 cm, KM = 13
...
125 cm, NP = 8
...
346 cm, QS = 6
...
Constructions – see similar constructions in
worked problems 30 to 33 on pages 179–180
...
6
...
9
...

4
...

9
...

13
...
36
...
48 m
3
...
5 m 4
...
1 m
5
...
0 m
6
...
50 m 7
...
8 m
8
...
43 m, 10
...
60 m

9 cm
2
...
9
...
81 cm 5
...
21 m
6
...
18 cm
24
...
82 + 152 = 172
(a) 27
...
20
...
35 m, 10 cm
12
...
7 nautical miles
2
...
24 mm

Chapter 22
Exercise 87 (page 198)

Exercise 83 (page 185)
40
40
9
9
1
...
(a) 42
...
22◦ (b) 188
...
47◦
2
...
08◦ and 330
...
86◦ and 236
...
(a) 44
...
21◦ (b) 113
...
12◦

350 Basic Engineering Mathematics
4
...
α = 218◦41 and 321◦19
6
...
5
2
...
30
4
...
1, 120
6
...
3, 90
8
...
, 960◦ 10
...
4, 180◦ 12
...
40 Hz
14
...
1 ms
15
...
leading 17
...
p = 13
...
35◦, R = 78
...
7 cm2
2
...
127 m, Q = 30
...
17◦ ,
area = 6
...
X = 83
...
62◦, Z = 44
...
8 cm2
4
...
77◦, Y = 53
...
73◦ ,
area = 355 mm2

Exercise 92 (page 210)
Exercise 89 (page 203)
1
...
04 s or 40 ms (c) 25 Hz
(d) 0
...
62◦) leading 40 sin 50πt

1
...

5
...


193 km 2
...
6 m (b) 94
...
66◦, 44
...
4 m (b) 17
...
163
...
9 m, EB = 4
...
6
...
37 m
32
...
31◦

2
...
37 Hz (c) 0
...
54 rad (or 30
...
(a) 300 V (b) 100 Hz (c) 0
...
412 rad (or 23
...
(a) v = 120 sin100πt volts
(b) v = 120 sin (100πt + 0
...
i = 20 sin 80πt −
6
i = 20 sin(80πt − 0
...
3
...
488) m
7
...
75◦ lagging
(b) −2
...
363 A (d) 6
...
423 ms

Chapter 23
Exercise 90 (page 207)
1
...
1 mm, c = 28
...
A = 52◦2 , c = 7
...
152 cm,
area = 25
...
D = 19◦48 , E = 134◦12 , e = 36
...
E = 49◦ 0 , F = 26◦ 38 , f = 15
...
6 mm2
5
...
420 cm,
area = 6
...
811 cm, area = 0
...
K = 47◦ 8 , J = 97◦ 52 , j = 62
...
2 mm2 or K = 132◦52 , J = 12◦8 ,
j = 13
...
0 mm2

Exercise 93 (page 212)
1
...
42◦, 59
...
20◦ 2
...
23 m (b) 38
...
40
...
05◦
4
...
8 cm 5
...
2 m
6
...
3 mm, y = 142 mm 7
...
13
...

2
...

4
...

6
...

8
...
83, 59
...
83, 1
...
61, 20
...
61, 0
...
47, 116
...
47, 2
...
55, 145
...
55, 2
...
62, 203
...
62, 3
...
33, 236
...
33, 4
...
83, 329
...
83, 5
...
68, 307
...
68, 5
...
294, 4
...
(1
...
960)
(−5
...
500)
4
...
884, 2
...
353, −5
...
(−2
...
207)
(0
...
299)
8
...
252, −4
...
04, 12
...
04, 12
...
51, −32
...
51, −32
...
47
...

3
...

7
...


Answers to practice exercises
Exercise 103 (page 234)

Chapter 25
Exercise 96 (page 221)
1
...
t = 146◦

351

2
...
(i) rhombus (a) 14 cm2 (b) 16 cm (ii) parallelogram
(a) 180 mm2 (b) 80 mm (iii) rectangle (a) 3600 mm2
(b) 300 mm (iv) trapezium (a) 190 cm2 (b) 62
...
35
...
(a) 80 m (b) 170 m 4
...
2 cm2
5
...
1200 mm
7
...
560 m2
2
9
...
4 cm
10
...
43
...
32

1
...

7
...

11
...

16
...

20
...
2376 mm2
3
...
1709 mm
6
...
(a) 106
...
9 cm2
2
21
...
17
...
07 cm2
(a) 59
...
8 mm
12
...
2 cm
8
...
48 cm 14
...
5◦ 15
...
698 rad (b) 804
...
10
...
24%
19
...
8 mm
7
...
(a) 2 (b) (3, −4)
2
...
Circle, centre (0, 1), radius 5
4
...

2
...

5
...


482 m2
(a) 50
...
9 mm2 (c) 3183 mm2
2513 mm2
4
...
19 mm (b) 63
...
01 cm2 (b) 129
...
5773 mm2
2
1
...
1932 mm2 2
...
(a) 0
...

4
...

8
...

12
...

15
...

19
...
2 m3
2
...
8 cm3
3
2
(a) 3840 mm (b) 1792 mm
972 litres
6
...
500 litres
3
9
...
3 cm3 (b) 61
...
44 m
(a) 2400 cm3 (b) 2460 cm2 11
...
04 m
1
...
8796 cm3
4
...
9 cm2
2
...
28060 cm3 , 1
...
22 m by 8
...
62
...
4
...
80 ha

2
...
3
...
45
...
259
...
2
...
47
...
38
...
12730 km 7
...
13 mm

Exercise 106 (page 246)
1
...
1 cm3 , 159
...
7
...
81 cm2
3
...
1 cm3 , 113
...
5
...
3 cm
6
...
(a) 268 083 mm3 or 268
...
06 cm2
8
...
53 cm
9
...
09 × 1012 km3 10
...
(a) 0
...
481 (c) 4
...
(a) 210◦ (b) 80◦ (c) 105◦
4
...

2
...

6
...
(a)

5890 mm2 or 58
...
55 cm3 (b) 84
...
13
...
393
...
32 cm3
(i) (a) 670 cm3 (b) 523 cm2 (ii) (a) 180 cm3
(b) 154 cm2 (iii) (a) 56
...
8 cm2
(iv) (a) 10
...
0 cm2 (v) (a) 96
...

9
...

13
...
5 cm3 (b) 142 cm2
(vii) (a) 805 cm3 (b) 539 cm2
(a) 17
...
0 cm
8
...
3 m , 25
...
6560 litres
12
...
7 cm3
657
...
77 m (c) £140
...
69 cm
5
...
72 N at −14
...
15 m/s at 29
...
28 N at 16
...
6
...
56◦
◦
15
...
33 to the 10 N force
21
...
22◦ S

Exercise 115 (page 276)

Exercise 108 (page 255)
1
...

5
...


6
...

8
...

11
...
403 cm3 , 337 cm2
4
...
55910 cm3 , 6051 cm2

1
...
0 N at 78
...
64 N at 4
...
(a) 31
...
81◦ (b) 19
...
63◦

Exercise 116 (page 277)
1
...
5 km/h at 71
...
4 minutes 55 seconds, 60◦
3
...
79 km/h, E 9
...
8 : 125

2
...
2 g

Chapter 28

Exercise 117 (page 277)

Exercise 110 (page 259)
1
...
5 square units 2
...
7 square units 3
...
33 m
4
...
70 ha
5
...
42
...
147 m3

3
...
42 m3

Exercise 112 (page 263)
1
...
5 A
3
...
093 As, 3
...
49
...
5 kPa

1
...

5
...

9
...
5j − 4k
3
...
4j − 6
...

4
...

8
...


4i + j − 6k
5i − 10k
−5i + 10k
20
...
(a) 2
...
(a) 31
...
4
...
5◦ )
2
...
9 sin(ωt + 0
...
5 sin(ωt − 1
...
13 sin(ωt + 0
...
A scalar quantity has magnitude only; a vector
quantity has both magnitude and direction
...
scalar
3
...
vector 5
...
scalar
7
...
scalar
9
...

2
...

4
...


17
...
00◦ to the 12 N force
13 m/s at 22
...
40 N at 37
...
43 N at 129
...
31 m at 21
...
4
...
5◦ )
2
...
9 sin(ωt + 0
...
5 sin(ωt − 1
...
13 sin(ωt + 0
...
4
...
5◦ )
2
...
9 sin(ωt + 0
...
5 sin(ωt − 1
...
13 sin(ωt + 0
...
11
...
324)
5
...
73 sin(ωt − 0
...
11 sin(ωt + 0
...
8
...
173)
i = 21
...
639)
v = 5
...
670)
x = 14
...
444)
(a) 305
...
2t − 0
...
21 sin(628
...
818) V (b) 100 Hz
(c) 10 ms
8
...
83 sin(300πt + 0
...
667 ms

1
...

4
...

6
...


Chapter 31
Exercise 122 (page 288)
1
...
(a) discrete (b) continuous (c) discrete (d) discrete

Exercise 123 (page 292)
1
...
5, 4
...

2
...
5
...
6 equally spaced horizontal rectangles, whose
lengths are proportional to 35, 44, 62, 68, 49 and
41, respectively
...
5 equally spaced horizontal rectangles, whose
lengths are proportional to 1580, 2190, 1840, 2385
and 1280 units, respectively
...
6 equally spaced vertical rectangles, whose heights
are proportional to 35, 44, 62, 68, 49 and 41 units,
respectively
...
5 equally spaced vertical rectangles, whose heights
are proportional to 1580, 2190, 1840, 2385 and 1280
units, respectively
...
Three rectangles of equal height, subdivided in the
percentages shown in the columns of the question
...

8
...

Little change in centres A and B, a reduction of
about 8% in C, an increase of about 7% in D and a
reduction of about 3% in E
...
A circle of any radius, subdivided into sectors having angles of 7
...
5◦, 52
...
5◦ and 110◦,
respectively
...
A circle of any radius, subdivided into sectors having angles of 107◦, 156◦, 29◦ and 68◦ , respectively
...
(a) £495 (b) 88
12
...
There is no unique solution, but one solution is:
39
...
4 1; 39
...
6 5; 39
...
8 9;
39
...
0 17; 40
...
2 15; 40
...
4 7;
40
...
6 4; 40
...
8 2
...
Rectangles, touching one another, having midpoints of 39
...
55, 39
...
95,
...

3
...
5–20
...
0–21
...
5–21
...
0–22
...
5–22
...
0–23
...

4
...

5
...
95 3; 21
...
95 24; 22
...
95 46;
23
...
Rectangles, touching one another, having midpoints of 5
...
5 and 43
...
The
heights of the rectangles (frequency per unit class
range) are 0
...
78, 4, 4
...
33, 0
...
2
...
(10
...
45 9), (11
...
45 31), (12
...
45, 50)
8
...

9
...
05–2
...
10–2
...
15–2
...
20–2
...
25–2
...
30–2
...

(b) Rectangles, touching one another, having midpoints of 2
...
12,
...

(c)

Using the frequency distribution given in the
solution to part (a) gives 2
...
145 13;
2
...
245 37; 2
...
345 48
...


Chapter 32
Exercise 125 (page 300)
1
...
33, median 8, mode 8
2
...
25, median 27, mode 26

Answers to practice exercises
Exercise 138 (page 324)
1
...
(a) 0
...
5 V
3
...
9 V/s
4
...
635 Pa/m

Chapter 35

5 4
x +c
4
2 3
x +c
3
...
(a) x 5 − x 2 + c
5
2

2
...
(a)
6
...

8
...


3x 2
2

− 5x + c

2
u2
ln x + c
(b)
− ln u + c
3
2
√
√
18 √ 5
14
...
(a)

Exercise 139 (page 328)
1
...
(a) −6 cos x + c
2
3 2x
12
...
(a)

355

4t 3
1
+c
(b) − + 4t +
t
3

Exercise 140 (page 330)

θ3
3

+c

1
...
5 (b) 0
...
(a) 105 (b) −0
...
(a) 6 (b) −1
...
(a) −0
...
833

5
...
67 (b) 0
...
(a) 0 (b) 4

7
...
248

8
...
2352 (b) 2
...
(a) 19
...
457

10
...
2703 (b) 9
...
proof
5
...
5
8
...
67

2
...
7
...
2
...
32
7
...
140 m

4
...
33 Nm

Index
Acute angle, 165
Acute angled triangle, 171
Adding waveforms, 278
Addition in algebra, 62
Addition law of probability, 307
Addition of fractions, 10
numbers, 1, 18
two periodic functions, 278
vectors, 267
by calculation, 270
Algebra, 61, 68
Algebraic equation, 61, 73
expression, 73
Alternate angles, 165, 191
Ambiguous case, 207
Amplitude, 199
Angle, 165
Angle, lagging and leading, 200
types and properties of, 165
Angles of any magnitude, 196
depression, 191
elevation, 191
Angular measurement, 165
velocity, 202
Annulus, 226
Arbitrary constant of integration, 325
Arc, 231
Arc length, 233
Area, 219
Area of common shapes, 219, 221
under a curve, 330
Area of circle, 222, 233
common shapes, 219
irregular figures, 257
sector, 222, 233
similar shapes, 229
triangles, 205
Arithmetic, basic, 1
Average, 299
value of waveform, 260
Axes, 130
Bar charts, 289
Base, 47
Basic algebraic operations, 61
BODMAS with algebra, 71
fractions, 13
numbers, 6

Boyle’s law, 46
Brackets, 6, 68
Calculation of resultant phasors, 281,
283
Calculations, 22, 28
Calculator, 22
addition, subtraction, multiplication
and division, 22
fractions, 26
π and e x functions, 28, 118
reciprocal and power functions, 24
roots and ×10 x functions, 25
square and cube functions, 23
trigonometric functions, 27
Calculus, 313
Cancelling, 10
Cartesian axes, 131
co-ordinates, 214
Charles’s law, 42, 142
Chord, 230
Circle, 222, 230, 233
equation of, 236
properties of, 230
Circumference, 230
Classes, 293
Class interval, 293
limits, 295
mid-point, 293, 295
Coefficient of proportionality, 45
Combination of two periodic functions,
278
Common factors, 69
logarithms, 111
prefixes, 53
shapes, 219
Complementary angles, 165
Completing the square, 105
Cone, 245
frustum of, 252
Congruent triangles, 175
Construction of triangles, 179
Continuous data, 288
Co-ordinates, 130, 131
Corresponding angles, 165
Cosine, 27, 183
graph of, 195
Cosine rule, 205, 281
wave, 195

Cross-multiplication, 75
Cube root, 23
Cubic equation, 161
graphs, 161
units, 240
Cuboid, 240
Cumulative frequency distribution,
293, 297
curve, 293
Cycle, 199
Cylinder, 241
Deciles, 304
Decimal fraction, 216
places, 13, 18
Decimals, 16
addition and subtraction, 19
multiplication and division, 19
Definite integrals, 328
Degrees, 27, 165, 166, 232
Denominator, 9
Dependent event, 307
Depression, angle of, 191
Derivatives, 315
standard list, 321
Derived units, 53
Determination of law, 147
involving logarithms, 150
Diameter, 230
Difference of two squares, 103
Differential calculus, 313
coefficient, 315
Differentiation, 313, 315
from first principles, 315
of ax n , 315
of e ax and ln ax, 320
of sine and cosine functions, 318
successive, 322
Direct proportion, 40, 42
Discrete data, 288
standard deviation, 302
Dividend, 63
Division in algebra, 62
Division of fractions, 12
numbers, 3, 4, 19
Divisor, 63
Drawing vectors, 266

358 Index
Parabola, 156
Parallel lines, 165
Parallelogram, 219
method, 267
Peak value, 199
Pentagon, 219
Percentage component bar chart, 289
error, 36
relative frequency, 289
Percentages, 33
Percentile, 304
Perfect square, 105
Perimeter, 171
Period, 199
Periodic function, 200
plotting, 238
Periodic time, 200
Phasor, 280
Pictograms, 289
Pie diagram, 289
Planimeter, 257
Plotting periodic functions, 238
Polar co-ordinates, 214
Pol/Rec function on calculator, 217
Polygon, 210
frequency, 293, 296
Population, 289
Power, 47
series for e x , 119
Powers and roots, 47
Practical problems
quadratic equations, 108
simple equations, 77
simultaneous equations, 96
straight line graphs, 141
trigonometry, 209
Precedence, 6, 71
Prefixes, 53
Presentation of grouped data, 292
statistical data, 288
Prism, 240, 242
Probability, 306
laws of, 307
Production of sine and cosine waves,
198
Proper fraction, 9
Properties of circles, 230
triangles, 171
Proportion, 40
Pyramid, 244
volumes and surface area of frustum
of, 252
Pythagoras’ theorem, 181
Quadrant, 230
Quadratic equations, 102

by completing the square, 105
factorization, 102
formula, 106
graphically, 156
practical problems, 108
Quadratic formula, 106
graphs, 156
Quadrilaterals, 219
properties of, 219
Quartiles, 303
Radians, 27, 165, 166, 232
Radius, 230
Range, 295
Ranking, 299
Rates of change, 323
Ratio and proportion, 40
Ratios, 40
Reciprocal, 24
Rectangle, 219
Rectangular axes, 131
co-ordinates, 131
prism, 240
Reduction of non-linear laws to linear
form, 147
Reflex angle, 165
Relative frequency, 289
velocity, 276
Resolution of vectors, 269
Resultant phasors, by drawing, 280
horizontal and vertical components,
283
plotting, 278
sine and cosine rules, 281
Rhombus, 219
Right angle, 165
Right angled triangle, 171
solution of, 188
Sample, 289
Scalar quantities, 266
Scalene triangle, 171
Scales, 131
Sector, 222, 230
area of, 233
Segment, 230
Semicircle, 230
Semi-interquartile range, 304
Set, 289
Short division, 4
Significant figures, 17, 18
Similar shapes, 229, 256
triangles, 176
Simple equations, 73
practical problems, 77
Simpson’s rule, 258

Simultaneous equations, 90
graphical solution, 155
in three unknowns, 99
in two unknowns, 90
practical problems, 96
Sine, 27, 183
graph of, 195
Sine rule, 205, 281
wave, 198, 260
mean value, 260
Sinusoidal form A sin(ωt ± α), 202
SI units, 53
Slope, 134
Solution of linear and quadratic
equations simultaneously, 110
Solving right-angled triangles, 188
simple equations, 73
Space diagram, 276
Sphere, 246
Square, 23, 219
numbers, 23
root, 25, 48
units, 219
Standard deviation, 302
discrete data, 302
grouped data, 303
Standard differentials, 321
form, 56
integrals, 326
Statistical data, presentation of, 288
terminology, 288
Straight line, 165
equation of, 135
Straight line graphs, 132
practical problems, 141
Subject of formulae, 83
Subtraction in algebra, 62
Subtraction of fractions, 10
numbers, 1, 18
vectors, 274
Successive differentiation, 322
Supplementary angles, 165
Surface areas of frusta of pyramids and
cones, 252
of solids, 247
Symbols, 28
Tally diagram, 293, 296
Tangent, 27, 183, 230
graph of, 195
Terminating decimal, 17
Theorem of Pythagoras, 181
Transposition of formulae, 83
Transversal, 165
Trapezium, 220
Trapezoidal rule, 257

Index
Triangle, 171, 219
Triangles, area of, 205
congruent, 175
construction of, 179
properties of, 171
similar, 176
Trigonometric functions, 27
Trigonometric ratios, 183
evaluation of, 185
graphs of, 195
waveforms, 195
Trigonometry, 181
practical situations, 209
Turning points, 156

Ungrouped data, 289
Units, 53
Upper class boundary, 293
Use of calculator, 22
Vector addition, 267
subtraction, 274
Vectors, 266
addition of, 267
by calculation, 267
by horizontal and vertical
components, 269
drawing, 266
subtraction of, 274
Velocity, relative, 276

Vertical axis intercept, 133
bar chart, 289
component, 269, 283
Vertically opposite angles, 165
Vertices of triangle, 172
Volumes of common solids, 240
frusta of pyramids and cones, 252
irregular solids, 259
pyramids, 244
similar shapes, 256
Waveform addition, 278
y-axis intercept, 135
Young’s modulus of elasticity, 143

359

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