Vectors
From the geometry
of this vector triangle, the magnitude of qp = 452 +552
= 71.06 m/s and the
55
= 50.71 but must
direction of qp = tan1
45
lie in the third quadrant; i.e., the required angle is
180 + 50.71 = 230.71
That is, the velocity of car
310 Basic Engineering Mathematics
Problem 8. A box contains 74 brass washers,
86 steel washers and 40 aluminium washers. Three
washers are drawn at random from the box without
replacement. Determine the probability that all
three are steel washers
Two bra
Introduction to differentiation
dy
12
8
= x3 + 4
dx
3
x
i.e.
1
Problem 10. If f (t ) = 4t + nd f (t )
t3
3
1
1
f (t ) = 4t + = 4t + 3 = 4t 1 + t 2
t3
t2
3 3 1
11
t 2
f
(t
)
=
(4)(1)t
+
Hence,
2
3 5
= 4t 0 t 2
2
3
3
i.e.
f (t) = 4 5 = 4
2 t5
2t 2
y = x 1
272 Basic Engineering Mathematics
The vertical component of the 15 N force is 15 sin 0 and
the vertical component of the 10 N force is 10 sin 90 .
The total vertical component of the two velocities,
The 15 m/s2 acceleration is drawn horizontally, shown
as
284 Basic Engineering Mathematics
vR =
Hence,
36.652 + (12.50)2
by Pythagoras theorem
i2 5 10 A
608
= 38.72 volts
tan =
i15 20 A
12.50
V
=
= 0.3411
H
36.65
= tan1(0.3411)
from which
= 18.83 or 0.329 radians.
Hence, vR = v1 + v2 = 38.72 sin(t 0.329) V
Pro
Vectors
a1 1 a2
Problem 12. Calculate the resultant of
(a) v1 v2 + v3 and (b) v2 v1 v3 when
v1 = 22 units at 140 , v2 = 40 units at 190 and
v3 = 15 units at 290
a1
a2
275
1.5 m/s2
(a)
2.6 m/s2
1458
1268
The vectors are shown in Figure 29.37.
a1 2 a2
1V
2a
Vectors
the horizontal component of the 20 m/s velocity is
20 cos90 = 0 m/s
and the horizontal component of the 15 m/s velocity is
15 cos195 = 14.489 m/s.
The total horizontal component of the three velocities,
3.
Calculate the magnitude and direction of
290 Basic Engineering Mathematics
Problem 3. The distance in miles travelled by
four salesmen in a week are as shown below.
Salesman
P
Q
R
S
Distance travelled (miles) 413 264 597 143
Use a horizontal bar chart to represent these data
diagrammatically
Equ
276 Basic Engineering Mathematics
The vector equation of the system shown in
Figure 29.39(a) is
ad = ab + bd
and that for the system shown in Figure 29.39(b) is
27.67
ad = ab + bc + cd
0
Thus, in vector equations of this form, only the rst and
last letter
Presentation of statistical data
A cumulative frequency distribution is a table giving
values of cumulative frequency for the values of upper
class boundaries and is shown in Table 31.6. Columns
1 and 2 show the classes and their frequencies. Column
3 lis
280 Basic Engineering Mathematics
30
26.5
19
iR 20 sin t 10 sin (t 3 )
20
i1 20 sin t
i2 10 sin (t )
3
10
90
19
2
180
270
3
2
360
2 angle t
10
20
30
Figure 30.4
y1 5 4
3. Express 12 sin t + 5 cos t in the form
A sin(t ) by drawing and measurement.
608
y 2
Chapter 33
Probability
33.1
Introduction to probability
(a)
The probability of selecting at random a man, p,
is given by the ratio
number of men
number in crowd
33.1.1 Probability
The probability of something happening is the likelihood or chance of it ha
274 Basic Engineering Mathematics
7.
8.
If velocity v1 = 25 m/s at 60 and
v2 = 15 m/s at 30 , calculate the magnitude
and direction of v1 + v2 .
Calculate the magnitude and direction of the
resultant vector of the force system shown in
Figure 29.31.
29.7
Mean, median, mode and standard deviation
Now try the following Practice Exercise
Practice Exercise 128 Quartiles, deciles
and percentiles (answers on page 354)
1.
The number of working days lost due to accidents for each of 12 one-monthly periods are
as
Methods of adding alternating waveforms
10 sin 120
26.46
= 0.327296
sin =
from which
= sin1 0.327296 = 19.10
= 0.333 rad
= 19.10
180
and
Hence, by cosine and sine rules,
iR = i1 + i2 = 26.46sin(t + 0.333)A
Now try the following Practice Exercise
Practic
Mean, median, mode and standard deviation
frequency values are represented vertically and variable
values horizontally. The mean value is given by the value
of the variable corresponding to a vertical line drawn
through the centroid of the histogram. The
Vectors
269
(iii) The resultant force is shown as R and is measured
as 18 N and angle is measured as 34 .
(iii) From the nose of 2 , 3 is drawn 15 units long at
an angle of 195 to the horizontal, shown as br.
Thus, the resultant of the two force vectors i
Chapter 30
Methods of adding
alternating waveforms
30.1 Combining two periodic
functions
There are a number of instances in engineering and science where waveforms have to be combined and where
it is required to determine the single phasor (called
the res
270 Basic Engineering Mathematics
i.e. the horizontal component of F = F cos , and
sin =
ab
from which, ab = 0b sin = F sin
0b
17.32 m/s
0
308
20
i.e. the vertical component of F = F sin .
Problem 4. Resolve the force vector of 50 N at an
angle of 35 to
Presentation of statistical data
classes to form a frequency distribution. To assist in
accurately counting members in the various classes, a
tally diagram is used (see Problems 8 and 12).
A frequency distribution is merely a table showing classes and the
292 Basic Engineering Mathematics
Now try the following Practice Exercise
centre during four one-week periods is as
shown.
Practice Exercise 123 Presentation of
ungrouped data (answers on page 352)
1.
The number of vehicles passing a stationary
observer o
302 Basic Engineering Mathematics
(vi) Determine the square root of (v).
3.
The gain of 90 similar transistors is measured
and the results are as shown. By drawing a
histogram of this frequency distribution, determine the mean, median and modal values of
Revision Test 12 : Vectors and adding waveforms
This assignment covers the material contained in Chapters 29 and 30. The marks available are shown in brackets at
the end of each question.
5N
1. State the difference between scalar and vector
quantities.
(2
282 Basic Engineering Mathematics
y1 5 5
y2 5 3
308
y2
54
/4 or 458
Figure 30.12
y1 5 2
(a)
y1 5
0
yR
a
150
30
y2
yR
y2 5 3
4
1358
b
458
y1 5 2
(b)
Figure 30.13
Figure 30.14
y R = 75.641 = 8.697
from which
Using the sine rule,
4
8.697
=
sin 150
sin
4 sin
Revision Test 12 : Vectors and adding waveforms
11. Given a = 3i + 3j + 5k, b = 2i 5j + 7k and
c = 3i + 6j 4k, determine the following: (i) 4b
(ii) a + b c (iii) 5b 3a
(6)
12. Calculate the magnitude and direction of the resultant vector of the displaceme
300 Basic Engineering Mathematics
The mean value is obtained by adding together the
values of the members of the set and dividing by the
number of members in the set. Thus,
mean value, x
2 + 3 + 7 + 5 + 5 + 13 + 1 + 7 + 4 + 8 + 3 + 4 + 3
=
13
65
=
=5
13
T
Presentation of statistical data
subdivided into the values in the table of percentages
shown above. A key is used (different types of shading
or different colour schemes) to indicate corresponding
percentage values in the rows of the table of percentages
Vectors
total horizontal component of the two forces,
H = F1 cos 1 + F2 cos 2
The vertical component of force F1 is F1 sin 1 and the
vertical component of force F2 is F2 sin 2 . The total
vertical component of the two forces,
The vertical component of the
Methods of adding alternating waveforms
waveform leads y1 = 3 sin A by 34 or 34
rad =
180
0.593 rad.
The sinusoidal expression for the resultant waveform is
y
458
Problem 2. Plot the graphs of y1 = 4 sin t and
y2 = 3 sin(t /3) on the same axes, over one
Methods of adding alternating waveforms
yR
y2 3
i2 5 10 A
i1 5 20 A
0
608
a
60
y1 4
281
60
iR
2i2
b
y2
Figure 30.11
By measurement, iR = 17 A and = 30 or 0.52 rad.
Hence, by drawing phasors,
Figure 30.8
The relative positions of i1 and i2 at time t = 0 ar