LO1 Identify the relevance of mathematical methods to a variety of conceptualised construction examples. (Class Test / Exam)

8 Mathematics for Construction

Unit Learning Outcomes

LO1 Identify the relevance of mathematical methods to a variety of conceptualised construction examples. (Class Test / Exam)

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.

TASK 2 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 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 fact 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 timebe 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.

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. Atwhattimedoeseachvibrationfirstreachadisplacementof−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: