Reference no: EM132454054

Learning Outcomes

LO1. Select and evaluate the correct mathematical techniques for a range of construction' and civil engineering problems.
LO2. Analyse and model construction and civil engineering situations using a range of mathematical methods.
LO3. Analyse and demonstrate ability to evaluate construction and civil engineering situations using statistical analysis techniques.
LO4. Ability to demonstrate knowledge and understanding of the above under constraint conditions.

Introduction

You have been appointed to prepare an initial conceptual design and implementation of a bridge crossing a local a stream near the expressway that links an international airport in the major city in the UK. The Client who is a local authority needs the bride to carry a carriageway8m wide with side walkways 2.4m each. The structural design should take in consideration the aesthetical and environmental aspect of the design.

Assignment Requirement

On behalf of the client, you are Parted with the responsibility of preparing the all design mathematical calculations for the mathematical model to be used in proposed 3 options and forms, include construction information package that will allow the construction team to build the project. The package of information required includes the following:
• Select the correct mathematical techniques to evaluate a number of options.
• Analysis and model construction and civil engineering situations.
• Consideration of appropriate constraints involved.
• Evaluation and choice of the various statistical and analytical techniques.
• Ability to demonstrate knowledge of various mathematical techniques in developing the solution.

LO 1: Select and evaluate the correct mathematical technique for a range of construction and civil engineering problems.

Part 1

Question a) In the triangle ABC, A=53°, B=61° and the length a= 12.6 cm. Find the Unknown sides of the triangle below.

Question b) Which one of the following is the sine of the angle θ in the right-angled triangle DEF below?

i) 8/17

ii) 17/8

iii) 17/15

iv) 15/17

Part 2

Question a) State the amplitude and period of the waveform. y= 4cos(2θ+ 45°) And sketch the curve between 0° and 360°.

Question  b) State the amplitude and period of the wave form y= 6sin(t-30°) And sketch the curve between 0° and 360°.

Question c) Prove the trigonometric identity:

(sin²x(secx+ cosecx))/cosxtanx = 1+ tanx

LO2 Analyse and model civil engineering situations using range mathematical methods

Part 1

Question a) The following offsets are taken from a chain line to an irregular boundary towards right side of the chain line.

Chainage	0	25	50	75	100	125	150
Offset'm'	3.6	5	6.5	7.5	7.3	6	4

Common distance d =25 m

You have been asked to estimate the area using the following methods and compare and comment on their difference and accuracy.
i) Trapezium Rule
ii) Simpson's Rule

Question b) Solve the following logarithmic equation
(i) 2log₉(√x) - log₉ (6x - 1) = 0

Question c) determine the determinant of 3x3 matrix :

Question d) Determine the Inverse of 3x3 matrix (above)

Part 2

Question a) Solve the following Indical equation

(3²)^x2 X (3)^(2x-3) = 27

Determine the value for the (x)

LO3 Analyse and demonstrate ability to evaluate construction and civil engineering situation using statistical analysis techniques.

Part 1
Concrete blocks are tested and it is found that on average, 7% fail to meet the required specification. For a batch of 9 blocks,

Question a) Determine the probabilities that (a) three blocks and 2 Marks

Question b) Less than four blocks will fail to meet the specification

Part 2

You have been asked to investigate the following data for a large building services company

Revenue	Number of customers	Number of customers
Less than 5	27	22
5 and less than 10	38	39
10 and less than 15	40	69
15 and less than 20	22	41
20 and less than 30	13	20
30 and less than 40	4	5

Question a) Produce a histogram for each of the distributions scaled such that the area of each rectangle represents frequency density and find the mode

Question  b) Produce a cumulative frequency curve for each of the distributions and find the median, and interquartile range

Question  c) For each distribution find the: the mean, the range, the standard deviation

Part 3

In an experiment to determine the relationship between force and momentum, a force X, is applied to a mass, by placing the mass on an inclined plane, and the time Y, for the velocity to change t torn u m/s to v m/s is measured. The results obtained are as follows:

Force (N) 11.4 18.7 11.7 12.3 14.7 18.8 19.6 Time (s) 0.56 0.35 0.55 0.52 0.43 0.34 0.31 Determine the equation of the regression line of time on force, assuming a linear relationship between the quantities, correct to 3 significant figures.

LO 4

Ability to demonstrate knowledge and understanding of the above outcomes under constraint conditions

Part 1

Determine the surface area generated by a line 350mm long rotating around the x-axis at a distance of 400mm using Pappus theorem. And sketch the shape.

Part 2

Question a).The engineering department has developed the following equation for the bending moment of a beam and you have asked to investigate its behaviour

M(x) = x³ - 3x² - 4

The Beam is 4m long and the design team suspect the is problem it the bending moment in the range between 3-4m and you have been asked to

Question a) Plot the bending moment at 0.5m interval for the range 0 ≤ x ≤ 4m 0 ≤ x ≤ 4 and determine if the bending moment is zero in range 3m ≤ x ≤ 4 m

Question b) Use the graph to estimate where the bending moment is zero

Question c) Newton-Raphson method to obtain the required location

Question d) Compare the results of the above method to determine which gives a best solution

Part 3

Question a) Using the following integration formulas below calculate the centroid and first moment of area for the triangle below:

A = ₁∫²_-x+2∫^2x-1 dxdy

My = ₁∫²_-x+2∫^2x-1 xdydx

Mx = ₁∫²_-x+2∫^2x-1 ydydx

Having calculated the first Moment of area; calculate the second moment of area (known commonly as moment of Inertia.

Part 4

Question: Determine the volume area generated by a rectangle 300X200 mm rotating around the x-axis at a distance of 400mm using Pappus theorem. And sketch the shape.