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Qualification - HND in Construction and the Built Environment

Unit Name - Mathematics for Construction

Unit Number - Unit 8

Unit Code - K/615/1394

Assignment Title - Applying Principals of Mathematics in Sport Club Assessment

LO 1: Identify the relevance of mathematical methods to a variety of conceptualized engineering examples.

LO 2: Investigate applications of statistical techniques to interpret, organize and present data by using appropriate computer software packages.

LO 3: Use analytical and computational methods for solving problems by relating sinusoidal wave and vector functions to their respective engineering applications.

LO 4: Illustrate the wide-ranging uses of calculus within different engineering disciplines by solving problems of differential and integral calculus.

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You are working as a civil engineering technician in a construction company. An investor reaches your company requesting a design proposal for an international sport club. The investor doesn't know much information about the average member's fees in the area where the sport club is intended to be built. Therefore, he requested your company to give him an approximate estimation about the fees. As a part of the team who is working on the project, the project manager assigned you the following tasks:

Table 1: Number of associate people expected to join the sport club over the coming 3 years for the different ages and sex.

 Years Total Number of Associate People Females Males Kids 1 105 25 80 100 2 244 35 94 115 3 275 40 109 126

Part 1: (members regression and cost analysis)

A- State in your report what type of progression the student data follows.

B- Your report should include data for the number of people that will join the sport club at the end of the year 4, year 5, and year15. Plot the relationship between years and number of people joining the sport club, using the data in the table above in addition to the data founded.

C- Include in your report the total number of people that the sport club will be accommodated in the 10th year.

D- The sport club should have maintenance after each 10,000 associate people joining the sport club. You have to define in your report after how many years the first, second and third maintenance procedure will take place.

E- Assume that the expected average cost per member per year is 200JOD, the expected overhead cost is 100K JOD per year and the expected cost of employee's salary and maintenance of sports equipment is 300K JOD per year.

Problem 1. Use Table.1 to find the average cost/year for the first 3 years individually if no member left the sport club.

Problem 2. Use the dimension analysis to find a function to find the average profit/year and ensure that the derived function you found is dimensionally consistent.

Problem 3. Use your answers from parts A, B, C and D to give the investor an estimation and recommendation about the associate members fees. Noting that the investor does not have a technical background about the statistical terms, so you have to clarify for him in simple words what each term means and indicate.

F- The project manager requests data be provided by the statistic department about the average members' fee/year on the area where the sport club is intended to be built. The data provided by the statistical department is in the attached Excel file. The data is collected from 100 sport club. The members' fee /year is found to have a normal distribution probability function.

Problem 1. Find the range, mean and standard deviation of the data. You may use a software package for this part. However, you must show the equations applicable for this problem.

Problem 2. Draw a histogram of the data of bin width of 50 JOD. You may use a software package for this part.

Problem 3. What is the probability that at a given sport club the fee will be between 1000 and 1400 JOD.

Problem 4. Statistics of the statistical department state that the average members' fee on that particular area is less than 1250 ± 155JOD (μ ± σ). Your team suspects this statistic is an underestimation and consequently commence they're on study by studying 100 Sport club and finding that μsample = 1250. Based on this information conduct a hypothesis test to determine whether the statistical department's statistic is an underestimate or not at a 0.05 level of significance.

G- The marketing department modelled the probability of the ability of the members to pay more than 1250 JOD/year with a binomial probability distribution, where the probability of paying more than 1250 JOD/year is p = 0.73. For a population of 1000 members, if you choose a sample of 30 members, address the following:

Problem 1. What is the probability of having exactly 20 member that would pay more than 1250?

Problem 2. What is the probability of having no member that would pay more than 1250?

Part 2: (Design analysis)

Consider that the geometry of the arc for billboard for the sport club is following the hyperbolic equation: y = 10 - 0.658 cosh(x - 4.025). It is required to attach two torches for each direction for billboard. The purpose of the torch is to give clear vision of the cars parking it should not be hanged no more than 7m in height. Accordingly, you have to define in your report the rang of the value of x where the torch could be hanged without violating the maximum allowable height. The billboard width is 6.5m. P3 *If the design was changed for the billboard and the arc geometry function was changed to y = 2.5*(sin(30x) + 3.011) Redefine the range of x that was found in the first part.

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Part 3: Analysing Vibration Response of the slab and columns.

It is required to check the slab behaviour for the roof floor in the sport club under throwing the lifting and the movement of using the sports equipment, in order to accomplish that, a simulation was performed for the slab and the vibrations (displacements) in the columns with time were found to have the following relationship:
y1 = 0.08 sin (2.532 + 3.45t)cm

In your report you are expected to:

Problem 1. Define the maximum amplitude, frequency and phase angle of the displacement function.

Problem 2. You are expected to apply compound angle identities in order to separate the displacement function into a function in terms of sine and cosine.

It is considered that the movement of the water in the Olympiad swimming pool will create a vibration in the same direction of y1 of the following equation:

y2 = 5/8 sin(8.78t - 0.0475) cm

Problem 3. You are expected to model the combination between the two sine waves and find the displacement at time t=2 s, using both analytical and graphical methods.

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Part 4: Checking the structural safety of the beams.

In order to check the overall structural safety for the skylight slab of the Olympiad swimming pool, you were requested to check the strength of the beams and this requires making analysis for the beams and finding the internal forces in each beam. The following figure shows a segment of the skylight slab of the Olympiad swimming pool and it shows the points where the beams are attached.

In your analysis report you are expected to:

Problem 1. Represent beam AB in vector form and find the length of the beam using the formula for the vector resultant.

Problem 2. Represent the force in the beam in vector form, if each beam is carrying half the weight of the slab consider the weight is 6,000 kg, you have to find the force in each beam.

Part 5: Applying differential equations of the beams.

One of the beams in the sport club slab has the shear force V for a beam with length L is give V = dM/dx where M is the bending moment at distance x along the beam.

M = wo + 1/x e.ln(x2) + L/Π sin (Πx/L) Where wo and L are constants.

Problem 1. Find the sheer force V.

Problem 2. Find the value of M and V at the midpoint and the endpoint of the beam.

Part 6: Quantity Estimate.

The area of the terrace was designed to have the following shape the arc has an equation Y= 0.429X2 + 3.81X-4.97 It is required to make a quantity in order to determine the overall cost of construction material. Your manager Assessment Tasked you to calculate the area of the tile for the terrace using integral calculus.

Part 7: Assessment of the corrosion of the rods

The skylight roof of Olympiad swimming pool is support with steel rods, due to humidity conditions inside the pool those rods will rust, to analysis the effect of rust in the rods, an arbitrary rod was selected and tested, and it was founded that the change in the rod diameter with time due to the rust is following an exponential decay model and that the relation between the rate in change in diameter and the diameter itself can be written as:

dD/dt = -1/90 D(t)

Where D(t) is the diameter of the rod in meter and t is the time in year. The initial diameter of the rod is 0.5m.

Problem 1. Determine the diameter of the rod after 10 years.

Problem 2. This kind of rod will stay functional if its diameter is greater than 0.2m. Find how much years would the rod stay functional.

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Part 8: Locating Minimum and maximum Point.

One of the sports devices consists of a system of a rod of length l attached on the slab and mass m attached to a spring of stiffness K.

The potential energy, V, is given by: V=0.5kl²cos²θ+mglsinθ

where θ is an angle such that 0 ≤ θ ≤ π, k= 500 N/m, l = 10 m, m = 250 kg, g = 9.8 m/s2

A system is in equilibrium if dv/dθ = 0

Problem 1. Find the value(s) of θ for the system to be in equilibrium.

Problem 2. Sketch the graph of V against θ, marking all the points of maximum, minimum using the higher order derivative.

 Learning Outcomes and Assessment Criteria Learning Outcome Pass Merit Distinction LO1:      Identify               the relevance                                  of mathematical methods to      a      variety       of conceptualized engineering examples. P1:         Apply    dimensional analysis techniques to solve complex problems M1:                  Apply dimensional analysis to derive equations D1:               Present statistical data in a method that can be understood    by     a non-technical audience P2: Generate answers from contextualized arithmetic and geometric progression P3: Determine solution of equations using exponential, trigonometric and hyperbolic functions LO2:                     Investigate applications                of statistical techniques to interpret, organize and present data by using appropriate computer software packages. P4:     Summarize     data    by calculating mean, standard deviation, and simplify data into graphical form M2: Interpret the results      of      a statistical hypothesis                 test conducted from a given scenario P5: Calculate probabilities within both binomially distributed and normally distributed random variables LO3: Use analytical and computational methods for solving problems by relating sinusoidal wave and vector functions to their respective engineering applications. P6:       Solve       construction problems         relating         to sinusoidal functions M3:                  Apply compound angle identities to separate waves into component waves. D2 Model the combination of sine waves graphically and analyse the variation in results between graphical and           analytical methods. P7: Represent construction quantities in vector form, and apply                     appropriate methodology to determine construction parameters LO4: Illustrate the wide-ranging uses of calculus               within different engineering disciplines by solving problems of differential and integral calculus P8: Determine rates of change for algebraic, logarithmic and circular functions. M4:     Formulate predictions       of exponential growth                       and decay       models using  integration methods. D3              Analyze maxima and minima of   increasing    and decreasing functions         using higher                        order derivatives. P9: Use integral calculus to solve      practical     problems relating to engineering.

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