BTEC HND in Civil Engineering: Analytical Methods Assignment

This document presents a comprehensive assignment solution for BTEC HND in Civil Engineering, focusing on analytical methods for engineers. The assignment covers a range of topics including solving simultaneous equations using the matrix method, sketching functions, and performing factorisation. It also delves into algebraic manipulations like partial fractions and algebraic division. The second task addresses practical applications, such as calculating forces in ropes, determining volumes, and finding room dimensions. Trigonometry, including sinusoidal waves and logarithmic relationships, is also explored. The third task involves calculus, focusing on finding gradients, optimisation problems related to poster design, and static force analysis. Finally, the assignment incorporates statistical concepts such as calculating mean and standard deviation, probability calculations involving different scenarios, and the application of probability to exam scenarios. The document includes an executive summary, table of contents, and references, providing a complete and detailed solution.

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Unit / Module Analytical Methods for Engioneers
Assessment Assignment-Individual
Lecturer Mr. Vidura Sonnadara
Student Name L.Y sachth madushanka ID CE/M/02/15
Date Handed Over Date Due
Initial Submission Date Re-Submission Date
Assessment
Criteria /Grading
Opportunity
Offered
Met
Initially?
Yes/no
Met On
Re-
Submissi
on?
Yes/no
Assessor’s Feedback on submission
LO 1
Decision
LO 2
LO 3
LO 4
M1
M2
M3
D1 Assessor’s Feedback On Re-Submission( if required)
D2
D3
Assessed by :
Evaluation on
Submission : Achieves Assessment Criteria
Resubmit
Signature: Date Assessed :
Evaluation on Re-
Submission : Achieves Assessment Criteria
Resubmit
Signature: Date Assessed :
Important Information for Students

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 The criteria each task relates to are shown against that task where possible.
 Please note that plagiarism is treated as a serious offence and therefore the work you produce
must be individual and original although may work in groups in some instances (Please refer to
Student Handbook on Plagiarism & Cheating).
 All sources of information must be referenced using “Harvard referencing” where a reference
list/Bibliography should be included at the end of the assignment.
 Please note that the submission date given for this assignment is the final date that you can hand
over the assignment. A one week grace period maybe extended with prior approval from the
Program Manager – Undergraduate. However, note that submissions handed in one week
late will be graded lower (Please refer to the Student Handbook on Assessments - Late
Submissions & Penalties).
 Assignments returned to students for re-working must be re-submitted within 10 days (Please
refer to Student Handbook on Assignments – Re-submission).
 Failure to re-submit the previously marked assignment with the re-submitted assignment will
mean that results cannot be released for the respective unit.
 Please read, follow and reference both the contents of the unit outline and the grading criteria
of the assignment before completing this assignment.
Plagiarism:
While research and discussion are an essential part of an assignment, the deliberate copying of someone else’s
work or unacknowledged copying from printed or electronic sources is NOT permitted. You may be subject to
disciplinary procedure if you do this. You should sign this sheet to show that you comply with these regulations.
Student’s Signature: …………………………… .............................................. Date: .…………………
Student’s Comments:
Assessment has been internally verified for use.
Internal Verifier Name :
Signature : Date
Verified :
Task 01
a) Use matrix method to solve the following simultaneous equations

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3x – 5y = 204
4x + 5y = 412 (LO 1.4, D3)
b) Sketch the following functions of a single variable for the domain −3 ≤ x ≤ 3
I. f (x) = −x4
II. f (x) = 2 − x − x2
III. f (x) = 1/x
(LO 1.2, M1)
c) Solve the following factorisation
i. 2x2 – 5x -3 = 0
ii. 7x2 – 20x -3 = 0
(LO 1.1, M2)
d) Divide the following algebraic repression in to partial fractions
e) Write the remainder and quotient in the following algebraic division
(LO 1.1, M3)
Task 02
a) A weight of 4 Tonnes is suspended on two ropes as shown. Calculate the length of the
ropes and the forces in them.

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(LO 2.3, D1)
b) A conical vessel of internal radius 14 cm and height 36 cm is full of water. If this water
is poured into a cylinder with internal radius 21 cm, find the height to which the water
rises in the cylinder.
(LO 2.3, D1)
b) The length of a room is 1.5 times its breadth. The cost of carpeting it at Rs 150 per
square meter is Rs 14400 and the cost of white washing the four walls at Rs 5 per square
meter is Rs 625. Find the dimensions of the room.
(LO 2.1, D3)
c) If the equation of a sinusoidal wave is given by
y = 4.5 sin (2ωt + 3Π/4) then find the displacement and the acceleration of the wave?
d) When a gas is compressed, it is found that the Pressure & Volume has a relationship as
PVn = C , where C and n are constants.
When P = 400 × 103 & V = 0.0004 m3 , the C = 45.7,
Then find the value of C, by using the knowledge of logarithms?
(LO 2.2 , M1)
Task 03
a) Find the gradient of the curve y = (x3+1)(x2+2x+5) at the point x = 1. (LO 4.2, M3)
b) A printer need to make a poster that will have a total area of 320 in2 and will have 1 inch
margins on the sides, a 1.75 inch margin on the top and a 3.5 inch margin on the bottom.
What dimensions will give the largest printed area? (LO 4.2, M3)
c) Show that if the coefficient of friction between the block and the plane is 0.25, the force
required to just start the block moving up the 40o incline is F= 1.38 W while the force
required to hold the block from sliding down the plane is F= 0.487 W

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(LO 4.1, D2)
Task 04
a) The breaking strength of 150 spot welds was measured in Newton and grouped into
bands of 20 N as shown.
Calculate the mean and the standard deviation
Range f
160 - 180 2
180 - 200 6
200 – 220 10
220 – 240 28
240 – 260 50
260 – 280 31
280 – 300 15
300 – 320 8
(LO 3.1, LO 3.2, M1)
b) 3 boxes contain balls as follow,
(Box A: 6 red, 4 black), (Box B: 4 red, 6 black), (Box C: 5 red, 6 black) respectively.
One box is selected randomly and a ball is taken from the selected box randomly, if the
ball is Red, find the probability that it’s drawn from the Box A?
C) Student takes his exam in four subjects, A,B,C and D. He estimates of passing each
subject as P(A) = 4/5 , P(B) = 3/4, P(C) = 5/6, P(D) = 2/3.
To qualify for the next semester he should pass subject A and 2 other subjects at least.
Find the probability of qualifying him? (Clearly show the solving steps)
(LO 3.3, D3)
Grading Information

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LO 1
 Algebra: linear, simultaneous and quadratic equations, laws of indices and
logarithms, common and Naperian logarithms, indicial equations, direct and
inverse proportion, inequalities, functional notation and manipulation of algebraic
problems
 Graphical representation: functions, points of intersection between two graphs,
graph sketching (straight line, polynomial, exponential and logarithmic), fit lines to
experimental data using least squares method
 Space, time and motion: plot space/time and velocity/time diagrams, determine
displacement, velocity and acceleration. Laws of motion, momentum, impulse
and projectiles
 Matrices: multiplication, transposition and inversion, applications
LO 2
 Trigonometry: basic trigonometric ratios and their inverses, trigonometric ratios for
the four quadrants, solution of triangles, calculation of areas and volumes of solids
 Determine co-ordinates: in 2-D and 3-D geometry
 Other functions: trapezoidal and Simpson’s rule
LO 4
 Trigonometric methods: to solve problems such as static forces, relative motion,
frameworks, metrology, friction torque, electrical and mechanical energy problems
 Calculus: to differentiate and integrate simple equations and demonstrate
applications of calculus (refer to Delivery guidance on delivering calculus to
construction disciplines)
M1 Identify and apply strategies to find appropriate solutions
 Effective judgments have been made
M2
Select/design and apply appropriate methods/ techniques
 Relevant theories and techniques have been applied
 A range of methods and techniques have been applied
M3
Present and communicate appropriate findings
 The appropriate structure and approach has been used.
 A range of methods of presentation have been used and technical language has
been accurately used.
D1
Use critical reflection to evaluate own work and justify valid conclusions
 Conclusions have been arrived at through synthesis of ideas and have been
justified
D2
Take responsibility for managing and organizing activities
 Autonomy/ independence have been demonstrated.
 Substantial activities or investigations have been planned, managed and
organized.
 The importance of interdependence has been recognized and achieved
D3
Demonstrate convergent / creative thinking
 Ideas have been generated with self evaluation and decisions taken.
 Problems have been solved with innovative and creative thoughts.
 Effective thinking has taken place in unfamiliar contexts

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Acknowledgement
I would like to express my deepest gratitude to all those who helped me to complete this assignment.
A special gratitude I give to my lecturer Mr. Vidura Sonnadara who gave me this valuable
opportunity to do this on the topic of Analytical Methods for Engineers and encouraged me to learn.
after I would like to thanks MIS HIMASHA GAMAGE advise to how to create a
assement
furthermore I would like to thank my parents with much appreciation for give their fullest cooperation to
me.
besides I would like to take this opportunity to mention about ICBT SOUTHERN CAMPUS which
provides me to good learning conditions for learning well.
also I would like to deliver my whole hearted thanks to all my friends who helped me to complete these
severe tasks within limited days.
finally I would like to thanks again all who helped me in various ways towards me to finish this
assignment with successfully.
Executive summary
Construction industry is very complex industry in the world .that Assignment is analytical
method of civil engineers .this lesson is very important lesson in the construction area. we got a
knowledge to how to use deflection , integration ,matrix’s part and probability in construction
site. This Assignment part uses to easy our work. That Assignment is very important to our
work site

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Table of Contents
Acknowledgement…………………………………………….. 07
Executive Summary…………………………………………..07
List of Figures…………………………………………………09
Introduction……………………………………………………10
Task 1…………………………………………………………..11
Task 2…………………………………………………………..21
Task 3…………………………………………………………..30
Task 4…………………………………………………………..36
Conclusion……………………………………………………..41
Reference list…………………………………………………..41

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List of Figures
Figure 1- graft…………………………………………………………12
Figure 2- graft………………………………………………………….15
Figure 3- graft………………………………………………………….16
Figure 4 -cone…………………………………………………………..23
Figure 5- cylinder………………………………………………………24
Figure 6 – room…………………………………………………………25
Figure 7- sinusoidal wave ………………………………………………27
Figure 8 – poster……………………………………………………….31
Figure 9 – plane…………………………………………………………32

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Introduction
We want a study to who is a civil engineer A civil engineer plans, designs and directs civil
engineering projects, such as roads, railways, airports, bridges, harbors, canals, dams, irrigation
systems, pipelines and power-plants. Civil engineers are involved in research, planning,
administration supervision, implementation and project management. They may do work in the
following fields: construction; communication engineering; environmental engineering
hydraulics and water resources; soil mechanics; building technology building administration.
Next we want a study dangers this job
Falls from elevated surfaces/levels or from ladders; falling into a cellar, shaft, trench
or open pit; falling while working on a project site – from cliffs, slopes, into pits,
tunnels, etc.
Injury/death as a result of collapse of an excavation, trench, floor or wall of a building or
auxiliary structure; collapse/buckling of components in a structure that is being built;
collapse and slide of piles of materials or of stored building equipment; landslides
Risk of being hit by heavy mechanical equipment/vehicles working on a sites
Electrocution as a result of unintentional contact with “live” electric wires in buildings,
during inspection and supervision rounds
Contact with and exposure to extreme temperatures (during outdoors work, at extreme
temperatures, etc.); or cold/frost bites
Potential exposure to noxious dust while staying at the construction site
Musculoskeletal system injuries, esp. those stemming from work postures, prolonged
This assignment about analytical method of civil engineers ..this assignment knowledge use to
we can find a slope some places .ect…… another one is we can find a stretchers forces (tension
,completion) and other important one is we got a knowledge probability .it very used full our
work

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Task 01
a)
=
2*2 2*1 2*1
A X B
AX = B
A X=
IX =
X =
= = =
= *
=
X= X=88
Y= Y=12 Final is answer X=88 / Y=12

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b)
f(x)= -
we get a f(x) = y and a get limit is -3
y =
x -3 -2 -1 0 1 2 3
y -81 -16 -1 0 -1 -16 -81
y = y = y = y = y = y =
y= -( y = y = -( y = y = y =
y = -81 y = -16 y=o y = -1 y = -16 y = -81
Figure 1 – graft

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ii) f(x) = 2 – x -
we get a f(x) = y and a get limit is -3
y = 2 – x -
x -3 -2 -1 0 1 2 3
y -4 0 2 2 0 -4 -10
y = 2 – x - y = 2 – x - y = 2 – x - y = 2 – x -
y = 2 – (-3) - y = 2 – (-2) - y = 2 –(-1) - y = 2 – 0 -
y = 2 +3 -9 y = 2+2-4 y = 2 +1 -1 y = 0
y = -4 y=0 y = 2
y = 2 – x - y = 2 – x - y = 2 – x -
y = 2 – 1 - y = 2 – 2 - y = 2 – 3 -
y = 2-1-1 y = 2-2-4 y = 2 -3-9
y = 0 y = -4 y = -10
Y = 2 – x - x²
= -1 – 2x
When; = 0
-1 -2x = 0
X = -
X = -
= 2 + -

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=
= Figure 2 – graft

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iii) f(x) =
we get a f(x) = y and a get limit is -3
y =
x -3 -2 -1 0 1 2 3
y -0.3 -0.5 -1 1 0.5 0.3
y = y = y = y = y = y = y =
y = y = y = y = y = y = y =
y = -0.3 y = -0.5 y = -1 y= y = 1 y = 0.5 y = 0.3
Figure 3 – graft

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C) i)
2x (x-3) + 1 (x-3) =0
(2x+1) (x-3) =0
2x+1=0 or x-3=0 x = - x = 3
Final is answer x = - / x = 3
ii) -20x-3 =0
7x² - 21x + x-3 = 0
7x(x-3) + (x-3) = 0
(7x + 1) (x-3) = 0
7x + 1 = 0 or x-3 = 0
x = or x = 3

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D)
=AX+B + +
=
4 +X) + C(2X+1) +DX
4
Equal Equal constant add to the x=- Equal X
4=2A -7=c 4 =DX 15=B+2C+D
A=2 4 +15 )-7=D(- ) 15=B-7(2)+30
D=30 B= -1
Final answer is =
=AX+B + +
= 2x – 1 - +

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e)
2x + 6
Polynomial Function = Quotient x Divisor + Remainder
2x³ + 3x + 6 = (2x+6) x (x² - 3x + 2) + (17x – 6)
Quotient = 2x + 6
Divisor = x² - 3x + 2
Remainder = (17x – 6) Final answer is =(2x+6) ( =17x-6
2 +3x+6
2 +4x
6 -x+6
6
17x-6

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Task 2
10m
A 45⁰ 30⁰ B
F1 F2
45⁰ 60
4T C
A
c b
B C
Sin equation ; a
= = K
Appling sin equation to ABC
=
AC =
AC =
AC =

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AC = m
AC = 5.176 m
=
BC =
BC =
BC =
BC = 7.320m

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coneFigure 4 –(b1)
1 4cm
V
36cm
V = x h
= x 14² x 36
3= 7389.03 cm
Cylinder full of water, the cone was filled with water equals the volume. Therefore,
Cylinder full of water = 7389.025 cm3

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cylinderFigure 5 –
12cm
r
h
v = r²h
=
h = =
h = 5.34 cm
Cylinder water volume height = 5.333 cm
v = volume of cylinder
h= height of cylinder
r= radios of cylinder

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Figure 6 – a room
b)
h
B
1.5B
Ground Area =1.5B x B
=1.5B²
Cost for the 1m² = Rs.150.00
Total cost =Rs.14400.00
Carpet area =
1.5B² =
B² = =
B =
B = 8m
White washing area of wall = 2(h*8) + (h*1.5)*8*2
=16h+24h = 40h
Cost =40h

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White paint on the wall
h= heght
Rate per 1=R.s 5
Total rate = R.s 625
5*40h=625
40h=625/5
h= 3.1m
Room ground length Breath = 8m
Room ground Length= 1.5*8=12m
WallHeight = 3.1m

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Figure 7 – sinusoidal wave
C)
Y= 4.5 sin(2 + )
Find a displacement we can get a t=0
T=0
Y=4.5 sin (2 +
Y=4.5 sin (2 +
Y=4.5 sin (
Y=0.18m
When we can get a accretion we want a this equation two time deviation
Y=4.5 sin (2 +
4.5 Cos (2 + 2
= 4.5 Cos (2 + 2 this is velocity in sinusoidal wave
Next time deviation
4.5 cos (2 + 2
= 4.5 - sin (2 + 2 2
= 4.5 *4 - sin (2 +
= - 4.5 *4 sin (2 +
= -18 this is accretion in sinusoidal wave

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We can get a accretion we want a get a t=0
T=0
= -18
= -18
= -18 ( wave accretion
Accretion = - 0.74

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d) P = C P=Pressure V=Volume C and n = constant
P= 400*
V = 0.0004
C = 45.7
P = C
= C/P
=
n =
n log 0.004 = log
n(-3.39) = log 45.7- log 400*
n(-3.39) = -3.34
n =
n = 1.16

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Task 3
A) y = (x³ + 1) (x² + 2x + 5)
y = + 2 + 5 + + 2x + 5
=5 + 8 + 15 + 2x + 2
When x = 1 , gradient of the curve
= + + + 2(1) + 2
= 5 + 8 + 15 + 2 + 2
= 32

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poster-Figure 8
1.75''B
1''1''
yA
3.5''
x
Area = 320 m2
X.Y = 320
A = (x – 2) ( y – 3.5 – 1.75)
A = (x-2) (y – 5.25)
A= xy – 5.25x – 2y + 10.5
A = x ( ) – 5.25x – 2( ) + 10.5
A=320 – 5.25x - + 10.5
= 0-5.25 + + 0
=
= 0
640 = 5.25
x =
x = 11.04''

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Therefore,
y =
= 28.98' inch'
Largest printed area = Largest Area = A
= (Y -5.25) (X -2)
= (28.982 – 5.25) (11.041 - 2)
= 23.732 x 9.041
= 214.561 Inch2

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(c)
R p
F
90
50
40 w 90
When the coefficient of friction if P
P = R 
differentiation
40°
R = wsin50 + Fsin40 
differentiation
40
P + Fcos40 = wcos 50
P = wcos50 - Fcos40 
 n
P = 0.25R
P = 0.25 (wsin50 + Fsin40 ) 
By  ,

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0.25(wsin50 + Fsin40 ) = wcos50 - Fcos40
(sin50 + Fsin40 ) = wcos50 - Fcos40
Fsin40 + wsin50 = 4wcos50 – 4Fcos40
F(sin40 + 4cos40 ) = 4wcos50 – wsin50
F =
F = w
F = 0.487w
When the movement is upwards.

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R
F
P 50
40
W
When the coefficient of friction the friction force is P
P = R 
differentiation
40
Fcos40 = P + wcos50
P = Fcos40 – wcos50 
Differentiation
40
R = wsin50 + Fsin40 
= wsin50 + Fsin40
P = (wsin50 + fsin40 ) 
by  ,

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(wsin50 + Fsin40 ) = Fcos40 – wcoc50
Wsin50 + Fsin40 = Fcos40 – 4wcos50 + wsin50
F(4cos40 – sin40 ) = 4wcos50 + wsin50
F =
F =
F = 1.38w

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Task 4
(a)
Range Mid Value
(
f D =(x-a) ft ft²
160-180 170 2 -60 -120 7200
180-200 190 6 -40 -240 9600
200-220 210 10 -20 -200 4000
220-240 230 28 0 0 0
240-260 250 50 20 1000 20000
260-280 270 31 40 1240 49600
280-300 290 15 60 900 54000
300-320 310 8 80 640 51200
a = Assume mean
a = 230
Mean = a +
= 230 +
= 230 + 21.47
= 251.47
.

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Standard deviation =
=
=
=
= 29.04

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(b) A B C
R – 6 R – 4 R – 5
B – 6 B – 6 B – 6
If the ball is red =R P(A)=1/3 P(R\A) = 6/10
Drawn from box A =A P(B) =1/3 P (R\B) = 4/10
Drawn from box B =B P(C) = 1/3 P(R\C) = 5/11
Drawn from box C =C
P(A\R)=
=
=
=

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c)
P(A) = , p(C) =
P(B) = , p(D) =
To qualify this student he should pass B, C along with A
B, D along with A
C, D along with A and
B, C & D along with
I passing B, C & falling D
P(B) x P(C) x (1-p(D))
= x x
=
II passing C, D & falling B
P(c) x p(D) x (1-p(B))
= x x
=

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III passing B, D & falling C
p(B) x p(D) x (1-p(C))
= x x
=
IV passing B, D & C
p(B) x p(C) x p(D)
= x x
=
A = + +
= + + +
=
=
Probability = x p(A)
= x
=
=

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Conclusion
This assignment is mainly focused on various method of analytical method of civil engineers
Task 1- I get a knowledge about matrix , graft , factorization , partial fraction ,algebra
Task 2 – this task I study a farces , volume .area ,deflection and log redeem
Task 3 - I get a knowledge about gradient of curve , deflection and forces coefficient of friction
Task 4 - this task I study I get a knowledge probability , mean and standard deviation
Reference list
Mr. Vidura Sonnadara 2015 theory note
. john bird (1983). engineering mathametics pocket book. 4th ed. london: newnes. 543
h.k dass (1998). advance engineer mathametic. 3rd ed. london: s.chand and company. 1358