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Qualification - Higher National Diploma in Construction and the Built Environment
Unit Number - Unit 28
Unit Name - Further Mathematics for Construction
Assignment Title - Further Maths
Learning Outcome 1: Apply instances of number theory in practical construction situations.
Learning Outcome 2: Solve systems of linear equations relevant to construction applications using matrix methods
Learning Outcome 3: Approximate solutions of contextualised examples with graphical and numerical methods
Learning Outcome 4: Review models of construction systems using ordinary differential equations
LO1
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 j2Π x 2 x ej0.5 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.
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LO2
Task 1
1252_Mathematics for Construction.jpg
a) Determine the vector Z when θ = Π/2, Z = Rθ x ( X - Y )
b) Determine the determinant of the matrix Rθ when θ = Π/4
c) Determine the inverse of Rθ 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
LO3
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
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LO4
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.
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