## Higher National Certificate/Diploma in Construction and the Built Environment

HNC/HND Construction and the Built Environment 2

Higher National Certificate/Diploma in Construction and the Built Environment

Scenario

You are employed by a large design and build contractor, as a design build technician. Your supervisor has

recognised that this mathematical unit would benefit and support you both in terms of completing your Higher

National studies but also in understanding how mathematical techniques could be applied to varying construction

situations.

Your supervisor also recognises that some advanced mathematical techniques will be needed in future studies and

therefore has developed a series of tasks to support you in the understanding of techniques included within tasks

are advanced , it is hoped that all tasks will be attempted.

Unit Learning Outcomes

LO1 Identify the relevance of mathematical methods to a variety of conceptualised construction examples

L02 Investigate applications of statistical techniques to interpret, organise and present data by using appropriate

computer software packages

LO3 Use analytical and computational methods for solving problems by relating sinusoidal wave and vector

functions to their respective construction applications.

LO4 Illustrate the wide -ranging uses of calculus within different construction disciplines by solving problems of

differential and integral calculus

Assignment Brief and Guidance

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Scenario 1

You have been contracted as a mathematical consultant to solve and confirm a number of mathematical

problems/solutions for projects on a major contract

1. A building services engineer is to design a water tank for a project. The tank has a rectangular area

of 26.5m2. With the design specifics of the width being 3.2m shorter than the length, calculate the

length and width to 3 significant figures for resource requirements.

2. As an employee of company JR construction you have received a letter regarding the project your

company is working on. It has a penalty clause that states the contactor will forfeit a certain some of

money each day for late completion. (i.e. the contractor gets paid the value of the original contract

less any sum forfeit). If she is 5 days late she receives £4250 and if she is 12 days late she

receives £2120. Calculate the daily forfeit and determine the original contract.

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Scenario 2

You have asked to convert various dimensional parameters using the following table

(a)

(i) In fluid dynamics the formula for drag force is given by

FD = óu2 CD A

where

FD is Drag Force

is density

u is velocity

CD is drag coefficient and A is area

Show that this equation is dimensionally correct.

(ii) The power P delivered to a pump depends on the specific weight w of the fluid pumped, the

height h to which the fluid is pumped, and the fluid flow rate q. Use dimensional analysis to

determine an equation for power. Below are the listed dimensions.

Variable Dimension

P ML2T -3

w ML-2T -2

h L

q L3T -1

(b)

Determine the units of the lift produced by an aircraft wing. The lift is directly proportional the product of the

air density, the air speed over the wing and the surface area of the wing.

Lift = k  V 2  A

A = Area of the wing in meter2

= Air density in Kg/meter3

A = Area of the wing

k has no dimensions

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Scenario 3

You have asked to investigate the following arithmetic sequences

1. An arithmetic sequence is given by b,

2b

,

b

, 0…….

3 3

• Determine the sixth term

• State the kth term

• If the 20th term has value of 15 find the value of b and the sum of the first 20 terms

2. For the following geometric progression 1,

1

,

1 ………

determine

2 4

• The 20th term of the progression

• The value of the sum when the number of terms in the sequence tends to infinity and explain why

n→

the sequence tends to this value S = ar

n

n

n=0

3. Solve the following Equations for x :

(a) 2Log (3x) + Log (18x) = 27

(b) 2LOGe(3x) + LOGe(18x) = 9

(c) Solve the following Hyperbolic Equations for the variables involved:

(i) Cosh(X) + Sinh(X) = 5

(ii) Cosh(2Y) – Sinh(2Y) = 3

(iii) Cosh(K) * Sinh(K) = 2

(iv) Cosh(M) / Sinh(M) = 2

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L02 Investigate applications of statistical techniques to interpret, organise and present data by

using appropriate computer software packages

Scenario 1

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

Revenue Number of customers

Number of customers

January July

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

a) Produce a histogram for either of the distributions scaled such that the area of

each rectangle represents frequency density and find the mode.

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

interquartile range.

c) For each distribution find the:

• the mean

• the range

• the standard deviation

Scenario 2

(A) In the new Epiphyte Engineering factory 5000 light bulbs Type A are installed. Their lengths of life are

normally distributed with a mean of 360 days and a standard deviation of 60 days.

a) If it is decided to replace all bulbs at one specified time, what interval must be allowed between

replacements if not more than 10% of bulbs should fail before replacement?

b) What practical considerations might dictate such a replacement policy?

c) The supplier offers a new type of bulb, Type B, that has a mean life of 450 days and the same

standard deviation (60 days) as the present type. If these bulbs were to be used how would the

replacement time be affected?

d) Determine whether the new type of bulb is preferable given that is costs 25% more than the

HNC/HND Construction and the Built Environment 7

existing Type A. Present and explain your conclusions.

f) A rival supplier now offers a third type of bulb, Type C, that has a mean life of 432 days and a

standard deviation of 45 days. If these bulbs were to be used how would the replacement time be

affected?

How should the Type C bulb compare for costs if it is to be adopted? Present and explain your

conclusion.

(B) A simple random sample of 10 people from a certain population has a mean age of 27 years. Can we

conclude that the mean age of the population is not 30 years? The variance of the populate ages is

known to be 20. Test your chosen hypothesis at a 5% level of significance using both a two tailed test

and a one tailed test and explain your conclusions.

L03 Use analytical and computational methods for solving problems by relating sinusoidal wave and

vector functions to their respective construction applications

Scenario 1

A support beam, within an industrial building, is subjected to vibrations along its length; emanating from two

machines situated at opposite ends of the beam. The displacement caused by the vibrations can be modelled

by the following equations.

x1 = 3.75 sin (100πt + 2𝜋/9)

𝑥2 = 4.42 sin (100𝜋𝑡 − 2𝜋/ 5 )

i. State the amplitude, phase, frequency and periodic time of each of these waves.

ii. When both machines are switched on, how many seconds does it take for each machine to produce

its maximum displacement?

iii. At what time does each vibration first reach a displacement of −2𝑚𝑚?

iv. Use the compound angle formulae to expand 𝑥1and 𝑥2 into the form 𝐴 sin 100𝜋𝑡 ± 𝐵 cos 100𝜋𝑡, where

A and B are numbers to be found.

v. Using your answers from part iv, express x1 + x2 in a similar form. Convert this expression into the

equivalent form Rsin(100πt + ∝).

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Scenario 2

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.

i. Calculate the distance AB.

The section BC is to be drilled in the direction of the vector 3i+ 4j+ k

ii. Find the angle between the sections AB and BC.

The section of pipe reaches ground level at the point (a,b,0).

iii. Write down a vector equation of the line BC. Hence find a and b.

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L04 Illustrate the wide-ranging uses of calculus within different construction disciplines by solving

problems of differential and integral calculus

SCENARIO 1

You have asked to investigate the following :

(a i)

Plot the bending moment and determine where the bending moment is zero .

(a ii) Investigate and state the range of values where the above Bending Moment Function ls maximum or

Minimum, decreasing or increasing or neither.

(b)

Determine the range of the temperature for positive t

(c) Note that in the thermodynamic system provided herein, the expression given is equated to 0 to solve

the problem given to be solved.

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Also determine the rate of change of V when P changes at regular intervals of 10 N/mm2 from 60 to

100N/m2 and the variable n=2.

Scenario 2

Righton Refrigeration specialises in the production of environmental engineering equipment. The

cost of manufacture for a particular component, £C, is related to the production time (t)minutes, by

the following formula

C=16t-2+2t

Investigate the variation of cost over a range of production times from 1 minute to 8 minutes:

a) Plot the cost function over the given range

b) Explain how calculus may be used to find an analytical solution to this problem of optimisation.

c) Use calculus to find the production time at which the cost is at a turning point.

d) Show that the turning point is a mathematicalminimum.

Discuss whether there would still be a minimum cost of production.

Scenario 3

The heat flow within a building is increasing or decreasing exponentially E to power 3t in line with

temperature difference which is t degrees ( C) with the outside surroundings.

Estimate and explore the growth rate graphically when the temperature difference changes from –

20degrees to + 20 degrees (C)

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Learning Outcomes and Assessment Criteria

Pass Merit Distinction

LO1 Identify the relevance of mathematical methods to a variety ofconceptualised

construction examples

D1 Present statistical data in a

method that can be understood by

a non – technical audience.

P1 Apply dimensional analysis techniques

to solve complex problems.

arithmetic and geometric progressions.

P3 Determine the solutions of equations

using exponential, trigonometric and

hyperbolic functions.

M1 Apply dimensional analysis to derive

equations

LO2 Investigate applications of statistical techniques to interpret, organise and present

data by using appropriate computer software packages

P4 Summarise data by calculating mean

and standard deviation, and simplify data

into graphical form.

P5 Calculate probabilities within both

binomially distributed and normally

distributed random variables.

M2 Interpret the results of a statistical

hypothesis test conducted from a given

scenario.

LO3 Use analytical and computational methods for solving problems by relating

sinusoidal wave and vector functions to their respective construction applications.

D2 Model the combination of sine

waves graphically and analyse the

variation between graphical and

P6 Solve construction problems relating to analytical methods.

sinusoidal functions.

P7 Represent construction quantities in

vector form, and apply appropriate

methodology to determine construction

parameters.

M3 Apply compound angle identities to

separate waves into distinct component

waves.

LO4 Illustrate the wide -ranging uses of calculus within different construction

disciplines by solving problems of differential and integralcalculus

D3 Analyse maxima and minima of

increasing and decreasing

functions using higher order

derivatives.

P8 Determine rates of change for

algebraic, logarithmic and circular

functions.

P9 Use integral calculus to solve practical

problems relating to engineering.

M4 Formulate predictions of exponential

growth and decay models using

integration methods.