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EG4012 Coursework Assignment 1

This assessment is designed to assess your ability in the following module learning outcomes:

• apply differential and integral calculus in an engineering context and appreciate practical applications with the use of a suitable Computer Algebra System

• evaluate statistical data and probability both manually and with the application of computing software such as Excel

This assignment is designed to assess your ability to formulate simple mathematical problems in engineering and to use the computer algebra package Maple and Excel for solving them. You must use Excel for Tasks 1 and 2 and Maple for Task 3. Marks will be lost if you carry out calculations on paper or with a calculator and type in the results. Please make sure your answer to each question is very clear to the marker and use comments as necessary to describe important intermediate steps in the solutions. Marks will be given for the worksheets functioning correctly and will also be awarded for clarity of the work. Please do not share files, or email worksheets to each other. Any attempt to do so will be treated as academic misconduct. Be aware of the University rules on plagiarism: https://mykingston.kingston.ac.uk/myuni/academicregulations/Pages/plagiarism.aspx

The deadline for submitting this assignment is 7th January 2019. Late submissions within one week will be capped at 40%.

Please use new worksheets within a single spreadsheet in Excel as specified in Tasks 1 and 2, and a separate Maple worksheet file, so a total of two files should be submitted. Start each file by entering your name and ID number then save your files in your normal directory. Your files should be named as follows:

1. Excel file – as (your) ‘kunumberTask12.xlsx’, submitted via Canvas before the deadline.

2. Maple file – as (your) ‘kunumberTask3.mw’ file, submitted via Canvas before the deadline.

Important - once you have submitted your files to Canvas, do NOT open that file again until you have been given your mark for the coursework. This is a safeguard against the event that the files are not readable by the marker.

Feedback Date. You will receive initial feedback in the form of solutions to the questions soon after the due date and written feedback on your coursework in Teaching Week 16.

Further guidance. In the coursework we will be looking for clearly laid-out solutions to the problems posed, with fully worked solutions within the scope as indicated by the question.

1. Aircraft engines manufactured and used by WaterCoach Industries are known to have a probability of failing of 0.1 in a given period (which may be regarded, pessimistically, as the length of a typical flight). On any given aircraft this probability is independent of any other engine failures.

It may be assumed that a WaterCoach aircraft can land safely if at least half of its engines are working.

Which is safer a two-engine plane or a four-engine plane?

1. Answer the above question using Excel as your calculator. Explicitly calculate the probability of each type of plane crashing – highlight the cells in which you obtain these results and write a brief explanation of your calculations and conclusions in neighbouring cells. Credit will be awarded for an easy to read spreadsheet (assume that you are presenting it as a very brief electronic report).

2. Answer the above question by carrying out a simple Monte Carlo Simulation exercise – use a new worksheet inside your spreadsheet file. Simulate 2000 flights. Use a random number generator based on a uniform distribution to represent the operation/failure of each engine and calculate the number of crashes in your sample of 2000 flights. Explicitly calculate the proportion of flights which crash for each type of plane (this can be compared with the probabilities calculated earlier) – highlight the cells in which you obtain these results and write a brief explanation of your calculations and conclusions in neighbouring cells (ensure that this is in cells near the top of your spreadsheet – in the first 10 rows). Credit will be awarded for an easy to read spreadsheet (assume that you are presenting it as a very brief electronic report).

(Please note the actual probability of an aircraft engine failing in flight is reported to be less than 0.000001 – the reason for choosing a larger probability in the questions is so that sensible results may be obtained with only 2000 simulated flights.)

(30 marks)

2. A certain manufacturer of LED light bulbs finds that in normal usage their 10W bulbs have a lifespan which is approximately normally distributed with a mean of 15,000 hours and a standard deviation of 2,000 hours. Use the built in functions in Excel to help you perform the following calculations:

1. Find the probability that a bulb fails after 10,000 hours normal usage;

2. Find the probability that a bulb lasts for longer than 18,000 hours normal usage;

3. Find the probability that a bulbs lasts for between 14,000 and 16,000 hours.

4. The manufacturer wishes to advertise that their bulbs are guaranteed to last for longer than x hours. What is the highest value can they give for x if they wish to be sure that 98% of their bulbs will last longer than x hours?

Highlight the cells in which you obtain these results and write a brief explanation of your calculations and conclusions in neighbouring cells. Credit will be awarded for an easy to read spreadsheet (assume that you are presenting it as a very brief electronic report).

(15 marks)

3. During a manufacturing process a component is to be machined in the shape of a hollow torus (a torus is a ring-doughnut shape). The manufacturer wishes to model this torus to calculate the surface area as this determines the quantity of steel required for each component.

Use Maple to create plots of the graphs of the circles and (take care with the + and - signs) and display both on the same axes according to the following specifications (it is easier to use the mouse activated plot editing features in Maple rather than separate plot command options):

• Label the horizontal and vertical axes ‘x values’ and ‘y values’ respectively and make sure that the horizontal axis runs from -4 to 10 while the vertical axis runs from -7 to 7.

• For the curve use a green dotted line.

• For the curve use orange + signs (size 20).

• Give the plot the title ‘Torus Cross Section’.

• Create a legend which labels the curves, Cross Section 1 and Cross Section 2 respectively.

• Superimpose a grid on the plot (with gridlines at axes tick marks – this is the default).

(Marks are available for the specifications above – if you cannot plot everything requested then at least plot something with these specifications. Typing the command: with(plots); will activate the implicitplot command which you will need to plot the circles above.)

(For safety, to back-up your formatting amendments, export the plot (when complete) to a Graphics Interchange Format file and save it on your H:\ drive with your username as the file name.)

(30 marks)

The implicit definition of the circle may be split into 2 explicit formulae for y in terms of x (for the top half of the circle) and (for the bottom half of the circle). (You may wish to create appropriate functions for these.) Use Maple to obtain a formula for for each of these.

(10 marks)

If the curve is rotated around the x-axis it forms a ‘surface of revolution’ in the shape of the surface of a torus (a ring-doughnut shape). In fact the pair of circles which you have plotted represent the cross section shown if the torus is cut in half.

The area of a surface of revolution obtained when a curve between and is rotated about the x-axis is given by the value of the integral . Apply this formula to the given curves representing the circle to evaluate the total surface area of the torus.

(15 marks)