Reference no: EM132710912

UNIT 28 Further Mathematics for Construction - Higher National Certificate/Diploma in Construction and the Built Environment

Learning Outcome 1: Apply instances of number theory in practical construction situations.

Task 1
a. Convert each number into denary,

• 11001.01

• 4D

b. calculate the following in both binary and denary

•11001+1001

Task 2
Apply de'Moivre's theorem or otherwise to solve for Zo and C from these expressions given below :
Z0=Z/Y and C=Z*Y
Where:
• Z is a complex number.
• Y is also a complex number.
• Re (Z0) >0 and Re (C) >0
Find Z0 and C when:
Z = 1 + 5 j,Y = 1 - 3 j

Task 3

a. Simplify the following equation:
G = 1 x e j^2Π x 2 x e^j0.5 x 0.5x e ^j0.75

b. Express the following expression in complex exponential form:

v=20sin (1000t-30°)

Task 4

Find a formula for cos (3θ) in terms of cos (θ) and sin (θ) using de Moivre's Theorem.

Learning Outcome 2: Solve systems of linear equations relevant to construction applications using matrix methods

Task 1

 a) Determine the vector Z when θ = Π/2, Z = R_θ x ( X - Y )

b) Determine the determinant of the matrix R_θ when θ = Π/4

c) Determine the inverse of R_θ When θ = Π/4

d) Solve the following equation for

Task 2

You have been asked by the structural engineering department to find the determinant and inverse of the following matrix

Task 4
You have been asked to the following set of equations that have been obtained from the structural engineering Department and verify your calculations using computer methods
2x₂ + x₃ = -8
x₁ - 2x₂ - x₃ = 0
- x₁ + x₂ + 2x₃ = 3

Learning Outcome 3: Approximate solutions of contextualised examples with graphical and numerical methods

Assignment Brief and Guidance

Task 1
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 if the bending moment is zero in the range between 3-4m and you have been asked to
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 ≤ 4m
b) Use the graph to estimate where the bending moment is zero
c) Use the bisection method to numerically estimate the exact location where the bending moment is zero
d) Newton-Raphson method to obtain the required location
e) Compare the results of the above method to determine which gives a best solution

Task 2
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       5.0       6.5       5.5       7.3       6.0       4.0

Common distance d =25m
You have been asked to estimate the area using the following methods and compare and comment on their difference and accuracy.
a) Trapezium Rule
b) Simpson's Rule

Task 3
The equation governing a body travelling in a water channel is given by the following equation
dv/dt = 1 - v²

Plot the velocity time graph for the object and determine the final velocity and the time taken to reach this velocity

Learning Outcome 4: Review models of construction systems using ordinary differential equations

Assignment Brief and Guidance

TASK 1

The equation of catenary is given by the following second order differential equation
y " = 5
x = 0, y = 100
x = 100, y = 100

Solve the above differential equation and plot the curve at 10m intervals.

Task 2
The differential equation governing the motion of a particle is given by the following differential equation
y ''+ 5 y = 0
t = 0, y = 20
Solve the above and plot the results and determine the amplitude and frequency of the oscillations

Task 3
A new series of tests is carried out and the equation modified to
y ''+ 2 y + 5 = 0
y = 20, t = 0

Use Laplace transforms or any other method to solve the new equations and plot the function and comment on the results.

Attachment:- Further Mathematics for Construction.rar