Higher National Certificate/Diploma in Construction and the Built EnvironmentAssignment Brief Unit Number and TitleUNIT: 28 Further Mathematics for ConstructionAcademic Year2020/21Unit TutorDr. Shivan TOVIAssignment TitleFurther MathsIssue Date24th Oct, 2020Submission Date19th December, 2020IV Name & DateDr Victor Oke 12th Oct,2019 Submission FormatThis assignment is to be a time-controlled assessment.You are to complete the following tasks, submitting your working calculations (including anydiagrams or sketches) with your completed solution clearly indicated. All work is to be submittedon A4 sheets. Environment Unit Learning OutcomesLO1 Apply instances of number theory in practical construction situations.Assignment Brief and GuidanceTask 1a. Convert each number into denary,• 11001.01• 4Db) calculate the following in both binary and denary• 1101+1001Task 2Apply de’Moivre’s theorem or otherwise to solve for Zo and C from these expressions givenbelow :Z0=Z/Y and C=Z*YWhere:• Z is a complex number.• Y is also a complex number.• Re (Z0) >0 and Re (C) >0Find Z0 and C when:Task 3a. Simplify the following equation:b. Express the following expression in complex exponential form:v=20sin (1000t-30°)Task 4Find a formula for cos (3θ) in terms of cos (θ) and sin (θ) using de Moivre’s Theorem.L02 Solve systems of linear equations relevant to construction applications using matrixmethods Z j Y j 1 5 , 1 3G e e e 1 2 0.5 j j j 2 0.5 0.75 Assignment Brief and GuidanceTask 1ea) Determine the vector Z when,ab) Determine the determinant of the matrix whenc) Determine the inverse ofd) Solve the following equation forTask 2You have been asked by the structural engineering department to find the determinant andinverse of the following matrixaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaamethsolveBuilt ETask 4You have been asked to the following set of equations that have been obtained from thestructural engineering Department and verify your calculations using computer methods*Please access HN Global for additional resources support and reading for this unit. For further guidanceand support on report writing please refer to the Study Skills Unit on HN Global. Link towww.highernationals.com cos sin 1 0, ,sin cos 0 1R X Y 2 Z R X Y R 4R When = 441 0R X 3 0 22 0 20 1 1 2 31 2 31 2 32 82 02 3x xx x xx x x 5 HNC and the Built Environment LO3 Approximate solutions of contextualised examples with graphical and numericalmethodsAssignment Brief and GuidanceTask 1The engineering department has developed the following equation for the bending moment of abeam and you have asked to investigate its behaviourThe Beam is 4m long and the design team suspect the is problem if the bending moment is zeroin the range between 3-4m and you have been asked toa) Plot the bending moment at 0.5m interval for the range anddetermine if the bending moment is zero in rangeb) Use the graph to estimate where the bending moment is zeroc) Use the bisection method to numerically estimate the exact location where the bendingmoment is zerod) Newton-Raphson method to obtain the required locatione) Compare the results of the above method to determine which gives a best solutionTask 2The following offsets are taken from a chain line to an irregular boundary towards right sideof the chain line.chainage 0 25 50 75 100 125 150Offset ‘m’ 3.6 5.0 6.5 5.5 7.3 6.0 4.0Common distance d =25mYou have been asked to estimate the area using the following methods and compare andcomment on their difference and accuracy.a) Trapezium Ruleb) Simpson’s RuleTask 3The equation governing a body travelling in a water channel is given by the following equationPlot the velocity time graph for the object and determine the final velocity and the time taken toreach this velocity M x x x 3 2 3 40 4 x m 0 4 x 3 4 m x m dv 1 v2dt LO4 Review models of construction systems using ordinary differential equationsAssignment Brief and GuidanceTASK 1The equation of catenary is given by the following second order differential equationSolve the above differential equation and plot the curve at 10m intervals.Task 2The differential equation governing the motion of a particle is given by the following differentialequationSolve the above and plot the results and determine the amplitude and frequency of theoscillationsTask 3A new series of tests is carried out and the equation modified toUse Laplace transforms or any other method to solve the new equations and plot the functionand comment on the resultsAssignment Brief and GuidanceSolve the above differential equation and determine the frequency and amplitude of thevibrations ” 50, 100100, 100yx yx y ” 5 00, 20y yt y ” 2 5 020, 0y yy t Please note that Example Assessment Briefs are for guidance and support only.They can be customised and amended according to localised needs and requirements. Allassignments must still be moderated as per the internal verification process.Learning Outcomes and Assessment CriteriaPass Merit DistinctionLO1 Apply instances of number theory in practical construction situations.P1 Apply addition and multiplication methods to numbers that are expressed in different basesystems.P2 Solve engineering problems using complex number theory.P3 Perform arithmetic operations using the polar and exponential form of complex numbers.M1 Deduce solutions of problems using de Moivre’s Theorem.D1 Test the correctness of a trigonometric identity using de Moivre’s Theorem.LO2 Solve systems of linear equations relevant to construction applications using matrixmethodsP4 Ascertain the determinant of a 3×3 matrix.P5 Solve a system of three linear equations using Gaussian elimination.M2 Determine solutions to a set of linear equations using the inverse matrix method.D2 Validate all analytical matrix solutions using appropriate computer software.LO3 Approximate solutions of contextualised examples with graphical and numericalmethodsP6 Estimate solutions of sketched functions using a graphical estimation method.P7 Identify the roots of an equation using two different iterative techniques.P8 Determine the numerical integral of construction functions using two different methods.M3 Solve construction problems and formulate mathematical models using first-order differentialequations.D3 Critique the use of numerical estimation methods, commenting on their applicability and theaccuracy of the methods.LO4 Review models of construction systems using ordinary differential equationsP9 Determine first-order differential equations using analytical methods.P10 Determine second- order homogeneous and non-homogenous differential equations usinganalytical methods.P11 Calculate solutions to linear ordinary differential equations using Laplace transforms.M4 Evaluate how different models of construction systems use first-order differential equationsto solve structural or environmental problems.D4 Evaluate first- and second-order differential equations when generating the solutions toconstruction situations. HNC/HND Construction and the Built Environment

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