Reference no: EM132181170

Building Mathematics

Learning Outcomes

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

LO2 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 calculusAssignment Brief and Guidance

TASK 1

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.

Scenario 2

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

Conversion Table

METRIC TO ENGLISH			ENGLISH TO METRIC
From Mettle	To English	Multiply by	From English	To Metric	Multiply by
meters	yards	1.09	yards	meters	0.91
meters	feet	3.28	feet	meters	0.30
centimeters	Inches	039	inches	centimeters	2.54
kilometers	miles	0.62	miles	kilometers	1.61
grams	ounces	0.035	ounces	grains	28.35
kilograms	pounds	2.20	pounds	kilograms	0.45
liters	quarts	1.06	quarts	liters	0.95
liters	gallons	0.26	gallons	titers	3.78

A car driving at an average speed 65 miles/hour (note 1 mile=1760 yards)

Calculate : (a)
- speed in meters/second

- How long will it take to drive 100miles

- The car's fuel consumption averages 30miles/gallon convert this to liters/kilometer

- How much fuel is required in liters for the journey

(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 x ρ xV 2 x A
A = Area of the wing in meter²
ρ = Air density in Kg/meter³
A = Area of the wing k has no dimensions

Scenario 3
You have asked to investigate the following arithmetic sequences
1. An arithmetic sequence is given by b, 2b/3, b/3, 0.......

- 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, 1/2, 1/4 ........ determine
- 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 the sequence tends to this value S_n = _n=0Σ^n→∞ arⁿ

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

LO2 Investigate applications of statistical techniques to interpret, organise and present data by using appropriate computer software packages

TASK 2

Scenario 1

You have been asked to investigate the following data for a large building services company Revenue Number of customers

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 each 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) How could the assumption that the bulb life is a normal distribution be tested?

b) 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?

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

d) 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?

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

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

TASK 3

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 R sin(100Πt + α).

vi. Using appropriate spread sheet software, copy and complete the following table of values:

vii. Plot the graphs of x1 and x2 on the same axes using any suitable computer package or otherwise. Extend your table to include x1 + x2 and plot this graph on the same axes as the previous two. State the amplitude and frequency of the new wave.

viii. Using your answers from parts v and vii, what conclusions can be drawn about x1 + x2 and the two methods that were used to obtain this information?

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 ??-axis North, and the ??-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 ?? and ??.

TASK 4

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)

The bending moment, M of a beam is given by

M = 3000 - 550x - 20 x²

Plot the bending moment and determine where the bending moment is zero (P9)

(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)

The temerature, θ(°c) at time t(mins) of a body is given by 

θ = 300 + 100e^-0.1t

Evalute θ for t  = 0, 1, 2 and 5 

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.

log(P) + nlog(V) - log(C)

where P represnts pressure, V represents volume, C is a constant and n is an index. 

show that

PVⁿ = C

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 mathematic alminimum.

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)