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Assignment Briefs
02-27-2024

LO3 Illustrate the wide-ranging uses of calculus within different construction disciplines by solving problems of differential and integral calculus
Higher Nationals

Assignment Brief – BTEC

Higher National Diploma in Construction and the Built Environment / Civil Engineering

Student Name /ID Number

Unit Number and Title

Unit 8: Mathematics for Construction

Academic Year

2023-2024

Unit Assessor

Assignment Title

2/2: Calculus & other mathematical methods for solving

Issue Date

22/11/2023

Submission Date

17/01/2024

IV Name

Date

27/09/2023

Submission Format:

The submission comprises sets of Calculations and a written report (400 Words). Moodle links would be created for submission via ‘turnitin’ but you can still email your final draft to your tutor as backup.

You are encouraged to make use of images, drawings, tables etc. and other research material to buttress your points.

There should be a list of references showing the materials/ resources that you have cited within your work. This is to be presented using the Harvard system of referencing.

Unit Learning Outcomes:

LO3 Illustrate the wide-ranging uses of calculus within different construction disciplines by solving problems of differential and integral calculus

LO4 Use mathematical methods to solve vector analysis, arithmetic progression and dimensional analysis examples

Assignment Brief and Guidance:

This assignment is intended to explore the underpinning theories of mathematics in general and the Construction Industry in particular. It is important that engineers, technicians and managers are equipped with mathematical skills necessary to solve engineering and construction problems.

Below are some scenarios that engineers and technicians face in the construction industry. You have been asked to solve these problems to show competence for the new role in a construction firm that you have applied for.

Scenario 1

Use differential calculus techniques to solve the following as part of Maths skill test;

A cylindrical container which holds a volume of 2 litres (2000 cm^{3} ). Show that the surface area, A, is given by𝐴 = 2𝜋𝑟^{2} + 4000𝑟^{−1} . Find the dimensions so that the surface area is a minimum (minimizing the material used). What do you notice about your results?
Scenario 2

Use integral calculus techniques to solve the following problems as part of your maths skill test;
∎ ∫(3𝑒^{𝑡} + 2 cos 𝑡)𝑑𝑡

A beam of length 6 m has a uniform distributed load, w, given by

Where x is the distance along the beam. The total load P (N), and the moment about the origin, R (Nm), are given by

And

Determine P and R

Scenario 3.
You have been asked by the HR officer to do an estimate on how much would be spent on a new employee. On commencing employment, the employee would be paid a salary of £16,000 per annum and would receive annual increments of
£480. What will be his salary in the 5th year and calculate the total he will have received in the first 12 years using arithmetic progression.

Scenario 4.
The critical load, P of a steel column can be obtained from the equation

Where L is the length, E is the elastic modulus, I is the moment of inertia, n is a positive number and P is the load.

Transpose to make the load the subject of the formula
Apply dimensional analysis technique and prove that the formula has dimensional homogeneity
Determine the load (correct to 2 d.p) for n=1, E = 0.2 x 10^{12} N/m^{2} , I = 6.95 x 10^{-}
^{6} m^{4} and L = 1.07 m.

Scenario 5.
A pipeline is to be fitted under a road and can be represented on 3D Cartesian axes as below, with the x-axis pointing East, the y-axis North, and the z-axis vertical. The pipeline is to consist of a straight section AB directly under the road, and another straight section BC connected to the first. All lengths are in metres.

Calculate the distance AB
The section BC is to be drilled in the direction of the vector 3i + 4j + k. Find the angle between sections AB and BC.
The section of the pipe reaches ground level at point C (a, b, 0). Write down the vector equation of line BC. Hence find a and b.
Evaluate the effectiveness of this method in solving this problem
Learning Outcomes and Assessment Criteria

Pass

Merit

Distinction

LO3 Illustrate the wide-ranging uses of calculus within different construction disciplines by solving problems of differential and integral calculus

D2 Analyse differential calculus techniques in the determination of maxima and minima in construction industry-related problem.

P5 Use differential calculus techniques to solve functions which incorporate: ax^{n} , sine ax, cosine ax, log_{e} x, e^{ax} and methods including function of a function

M3 Apply the rules of integral calculus to determine solutions for complex construction related problems

P6 Use integral calculus techniques to determine indefinite and definite integrals of functions involving ax^{n} , sine ax, cosine ax, 1/x, and e^{ax}

LO4 Use mathematical methods to solve vector analysis, arithmetic progression and dimensional analysis examples

D3 Evaluate the effectiveness and relevance, to the solving of complex construction problems, of the mathematical technique of vector analysis

P7 Apply dimensional analysis to solve problems

M4 Solve construction problems using vector analysis

P8 Generalise answers from a contextualised arithmetic progression problems

Recommended Texts:

SINGH, K. (2011) Engineering Mathematics Through Applications . 2nd ed. Basingstoke: Palgrave Macmillan.

STROUD, K.A. and BOOTH, D.J. (2013) Engineering Mathematics . 7th ed. Basingstoke: Palgrave Macmillan

Brief Information About the first Learning Outcome
Calculus plays a crucial role in various construction disciplines by providing tools to analyze and solve problems related to rates of change, accumulation, and optimization. Let`s illustrate the wide-ranging uses of calculus in different construction disciplines through examples involving both differential and integral calculus:

Structural Engineering:

Differential Calculus: Structural engineers use differential calculus to analyze the behavior of structures under various loads. For example, determining the slope and deflection of a beam under a distributed load involves solving differential equations.
Integral Calculus: Calculating the total load-bearing capacity of a structure or finding the center of mass for irregular shapes involves using integral calculus.
Civil Engineering - Earthwork and Excavation:

Differential Calculus: Calculating the rate at which soil erodes or the rate of settlement in a construction site involves differential calculus.
Integral Calculus: Estimating the total volume of soil to be excavated or filled during earthwork operations requires integrating over the given area.
Construction Management:

Differential Calculus: Determining the rate of change of costs with respect to time or the rate of progress on a construction project involves differential calculus.
Integral Calculus: Calculating the total cost or total work done over a given time period uses integral calculus.
Geotechnical Engineering:

Differential Calculus: Analyzing the consolidation of soil and predicting settlement over time involves solving differential equations.
Integral Calculus: Estimating the total volume change in soil during compaction or consolidation requires integrating the rate of change.
Hydraulic Engineering:

Differential Calculus: Analyzing the rate of flow in pipes or channels, determining velocity profiles, and studying water pressure involve differential calculus.
Integral Calculus: Calculating the total volume of water flow, finding the discharge of a river, or determining the total hydrostatic force on a dam involves integral calculus.
Quantity Surveying:

Differential Calculus: Analyzing the cost function and determining the marginal cost for different quantities involves differential calculus.
Integral Calculus: Calculating the total cost of construction or the total quantity of materials required uses integral calculus.
These examples highlight how calculus is fundamental to understanding and solving problems in various construction disciplines. Whether analyzing structural stability, managing construction projects, or designing hydraulic systems, calculus provides the necessary mathematical framework for precise calculations and decision-making.

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