Engineering maths

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Higher National Certificate/Diploma Assessment

Qualification Pearson BTEC Higher Nationals for England (2024)

Unit number and title 4002. Engineering Mathematics

Assignment title Mathematical Methods and Statistical Techniques

Assessor Engineering Team

Academic year 1 Unit Code A/651/0708 Assignment 1 of 3

Internal Verifier Mike Shaw Verification Date 2nd September 2024

Issue Date 2nd September 2024 Final Submission Date No later than 31st August 2025

Submission Format All text elements of your submission should be word processed, mathematical solutions can be handwritten (neatly) and scanned into your document. Assignment Format

• Organisation: Use clear headings, paragraphs, and sub-sections, to ensure clarity and ease of reading. • Referencing: Support your work with research. Use Harvard referencing or an approved alternative. Ensure accuracy to

avoid plagiarism. • Bibliography: Include a bibliography listing all referenced sources at the end, following the same referencing style.

Assignment Structure

Your assignment MUST include the following sections:

• Cover Page: Your Course, Name, Unit Name and Assignment number/name • Contents Page: List tasks or questions with page numbers. • References: List all sources used, but do not use Wikipedia.

Submission Requirements

By submitting your assignment, you confirm the following:

• Originality: The work is your own, with all sources properly cited. • Plagiarism: You acknowledge that plagiarism and collusion are forms of academic misconduct and are strictly prohibited. • Plagiarism Detection: Your assignment will be submitted to TurnItIn, a plagiarism detection service, that compares your

work against databases, online sources, and other students' work. • False Declaration: Making a false declaration is academic misconduct.

Professional Conversation

• Selected assignments will require a video conference to assess understanding before final grades are issued.

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Vocational Scenario or

Context

You work as a Test Engineer for a global manufacturer of electrical and mechanical

components and systems. Your Line Manager is responsible for delegating to you and your

colleagues the testing of theory, principles, and hypotheses from several worldwide

company divisions. She has asked you to undertake a series of such evaluations.

Task 1

a) The power (P) dissipated in a resistor (R) which is subjected to a voltage (V) across its terminals is given by the well-known formula;

𝑃𝑃 = 𝑉𝑉2

𝑅𝑅

Since you are new to the company your new Line Manager is unsure of your capabilities,

and has asked you to use the dimensions of P and V to determine the dimensions of R.

b) A guitar string, made by your Musical Instruments division, has mass (m), length (𝑙𝑙) and tension (F). It is proposed by one of your junior colleagues that a formula for the period of vibration (t) of the string might be;

𝑡𝑡 = 2𝜋𝜋� 𝑚𝑚3𝑙𝑙 𝐹𝐹

Use dimensional analysis to show your colleague that this formula is incorrect.

c) An analogue-to-digital converter (ADC), manufactured by your Signals division, takes 20 voltage samples of a ramp waveform, measured in mV, as follows… 2, 4, 6, 8, 10 …40

Your Test colleague has asked you to assist by using a formula to calculate the sum of these 20 voltage samples.

d) A digital chip, made by your Microelectronics division, counts continuously in the sequence: 1024, 2048, 4096, 8192, …

You have been asked to use a formula to calculate the 9th count of the chip.

e) A series electrical circuit which you are testing features a capacitor (C) charging via a resistor (R) and a DC supply voltage (Vs). The voltage across the capacitor (Vc) may be described by the equation…

vc = vs �1 − e−𝑡𝑡 𝑅𝑅𝑅𝑅� �

where t represents time. Assuming that Vc is 1V after a time of 4 seconds, determine the approximate value of the capacitor. (cont.)

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NOTES  The value of R is determined from the number of characters in your email address. As

an example, if your email address is [email protected] then we count all the characters, including the dot, to be 20. Therefore, the value of R will be 20 MΩ.

 The value of the DC supply voltage (Vs) is determined from the year of your birth. As an example, if you were born in 1994 then we add 1 + 9 + 9 + 4 to arrive at a voltage of 23 V.

f) One of your commonly-used laboratory instantaneous test signal voltages (vs) is described

by the equation…

6sin 2 4sv ft ππ = − 

  where f = 1MHz and t represents time.

Make time (t) the subject of this formula, and hence determine the first point in time when the instantaneous signal voltage has a magnitude of +3V. Note: A colleague has reminded you that you need to have your calculator in radians mode (RAD) for this calculation, because the angle is given in radians (i.e. π is featured). Use suitable software to draw at least two cycles of this signal and annotate the drawing so that non-technical colleagues may understand it.

g) The curve assumed by a heavy power cable, manufactured by your Power division, is

described by the equation…

60cosh

60 xy  =  

  where x and y are horizontal and vertical positions respectively. Calculate;

(i) The value of y when x is 104. (ii) The value of x when y is 180.

h) You are testing a decorative clock, to possibly be manufactured by your Consumer

Electronics division, and attach a mass (m) to a string of length (𝑙𝑙) to form a simple pendulum. Assuming that the acceleration due to gravity (g) of the earth may have an influence on the period (t) of the pendulum swing, use dimensional analysis to find a formula for t which could possibly involve m, 𝑙𝑙, and g.

i) You are testing the voltage across a capacitor in an AC circuit. The instrument you are using indicates this voltage to have a magnitude of 100 V and a phase angle of 45 degrees. Convert this voltage into a complex number.

j) A set of data from a manufacturing robot has been expressed in matrix form, as follows;

𝐷𝐷 = �8 5 3 4�

Find the inverse of this matrix.

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Task 2

a) Your communications division manufactures wireless dongles for use in general computing. These dongles have a maximum allowed radiative power of +20dBm. A random sample of ten dongles was taken, and their transmit power was measured by a colleague, using a spectrum analyser. The results are as follows;

Sample 1 2 3 4 5 6 7 8 9 10

Power (+dBm) 18.1 19.2 18.4 18.1 19.9 18.1 17.4 19.1 18.1 17.4

(i) Calculate the mean transmit power for these samples. (ii) Calculate the mode of the samples. (iii) Calculate the median of the samples (iv) Determine the standard deviation for the samples.

b) You visit your Manufacturing division, which has a machine producing metal bolts. In a

tray of these bolts, 94% are within the allowable diameter tolerance value. The remainder exceed the tolerance. You withdraw six bolts at random from the tray. Determine the probabilities that;

(i) Two of the six bolts exceed the diameter. (ii) More than two of the six bolts exceed the diameter.

c) Your components division manufactures capacitors, and the mean capacitance of 400

capacitors you have selected is 100µF, with a standard deviation of 7µF. If the capacitances are normally distributed, determine the number of capacitors likely to have values between 90µF and 110µF.

Note: Use the z-table given in Appendix A when answering part c.

d) A colleague, who is a Fuels engineer, is testing the effects of an experimental fuel additive for petrol engines which your company is developing. She adds the same sample amount of additive to 100 full petrol tanks for the same model of car and records the number of miles per gallon (mpg) for each car after being driven around a test track at a constant speed, until the fuel runs out. She knows that such testing undertaken without the additive produces a mean mpg figure of 44. Collecting results with the additive, she notices that the mean mpg figure is 48 with a sample standard deviation of 13 mpg.

By interpreting the results of the testing, show whether you agree, or not, with her hypothesis that the fuel additive has influenced the number of miles per gallon for the cars.

Draw by hand, or use suitable software, to produce a graphic, suitable for a non-technical company executive, which represents the results of your analysis. Note: Use the z-table given in Appendix A when answering part d.

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Sources of information to

support you with this Assignment

Bird J. (2021) Higher Engineering Mathematics. 9th Ed. Routledge. Bird J. (2019) Science and Mathematics for Engineering. 6th Ed. Routledge. Glyn J. and Dyke P. (2020) Modern Engineering Mathematics. 6th edition. Pearson. Made Easy Editorial Board (2022) Engineering Mathematics for GATE 2023 and ESE 2023 (Prelims) – Theory and Previous Year Solved Papers. India: Made Easy Publications Pvt Ltd. Rattan K.S., Klingbeil N.W., and Baudendistel C.M. (2021) Introductory Mathematics for Engineering Applications. 2nd Ed. Wiley. Ram M. (2021) Recent Advances in Mathematics for Engineering. CRC Press. Teodorescu P., Stanescu N., and Pandrea N. (2013) Numerical Analysis with Applications in Mechanics and Engineering. Wiley-IEEE Press. Ram M. (2020) Mathematics in Engineering Sciences: Novel Theories, Technologies, and Applications. 1st Edition. CRC Press. Sobot, R. (2022) Engineering Mathematics by Example. 1st Ed. Springer. Stroud, K.A. and Booth, D.J. (2020) Engineering Mathematics. 8th Ed. Bloomsbury Publishing Urbano M. (2019) Introductory Electrical Engineering with Math Explained in Accessible Language. Wiley. Vick B. (2020) Applied Engineering Mathematics. CRC Press.

Relevant Learning Outcomes and Assessment Criteria

Pass Merit Distinction

L01 Apply a variety of mathematical methods to a range of engineering and manufacturing sector problems

LO1 and LO2

P1

Apply dimensional analysis techniques to solve complex engineering/manufacturing problems.

M1 Use three mathematical

concepts to solve engineering/ manufacturing problems, justifying your chosen methods.

D1 Present data as meaningful information using appropriate methods that can be understood by a nontechnical audience.

P2

Generate answers from engineering arithmetic and geometric progressions.

P3

Determine solutions of engineering equations using exponential, logarithmic, trigonometric, and hyperbolic functions.

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

P4

Investigate engineering data by calculating mean, mode, median, and standard deviation.

M2 Conduct an engineering

hypothesis test and interpret the results.

P5

Calculate probabilities within Poisson binomially and normally distributed engineering random variables.

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Appendix A

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0 0.00399 0.00798 0.01197 0.01595 0.01994 0.02392 0.0279 0.03188 0.03586

0.1 0.03983 0.0438 0.04776 0.05172 0.05567 0.05962 0.06356 0.06749 0.07142 0.07535

0.2 0.07926 0.08317 0.08706 0.09095 0.09483 0.09871 0.10257 0.10642 0.11026 0.11409

0.3 0.11791 0.12172 0.12552 0.1293 0.13307 0.13683 0.14058 0.14431 0.14803 0.15173

0.4 0.15542 0.1591 0.16276 0.1664 0.17003 0.17364 0.17724 0.18082 0.18439 0.18793

0.5 0.19146 0.19497 0.19847 0.20194 0.2054 0.20884 0.21226 0.21566 0.21904 0.2224

0.6 0.22575 0.22907 0.23237 0.23565 0.23891 0.24215 0.24537 0.24857 0.25175 0.2549

0.7 0.25804 0.26115 0.26424 0.2673 0.27035 0.27337 0.27637 0.27935 0.2823 0.28524

0.8 0.28814 0.29103 0.29389 0.29673 0.29955 0.30234 0.30511 0.30785 0.31057 0.31327

0.9 0.31594 0.31859 0.32121 0.32381 0.32639 0.32894 0.33147 0.33398 0.33646 0.33891

1 0.34134 0.34375 0.34614 0.34849 0.35083 0.35314 0.35543 0.35769 0.35993 0.36214

1.1 0.36433 0.3665 0.36864 0.37076 0.37286 0.37493 0.37698 0.379 0.381 0.38298

1.2 0.38493 0.38686 0.38877 0.39065 0.39251 0.39435 0.39617 0.39796 0.39973 0.40147

1.3 0.4032 0.4049 0.40658 0.40824 0.40988 0.41149 0.41308 0.41466 0.41621 0.41774

1.4 0.41924 0.42073 0.4222 0.42364 0.42507 0.42647 0.42785 0.42922 0.43056 0.43189

1.5 0.43319 0.43448 0.43574 0.43699 0.43822 0.43943 0.44062 0.44179 0.44295 0.44408

1.6 0.4452 0.4463 0.44738 0.44845 0.4495 0.45053 0.45154 0.45254 0.45352 0.45449

1.7 0.45543 0.45637 0.45728 0.45818 0.45907 0.45994 0.4608 0.46164 0.46246 0.46327

1.8 0.46407 0.46485 0.46562 0.46638 0.46712 0.46784 0.46856 0.46926 0.46995 0.47062

1.9 0.47128 0.47193 0.47257 0.4732 0.47381 0.47441 0.475 0.47558 0.47615 0.4767

2 0.47725 0.47778 0.47831 0.47882 0.47932 0.47982 0.4803 0.48077 0.48124 0.48169

2.1 0.48214 0.48257 0.483 0.48341 0.48382 0.48422 0.48461 0.485 0.48537 0.48574

2.2 0.4861 0.48645 0.48679 0.48713 0.48745 0.48778 0.48809 0.4884 0.4887 0.48899

2.3 0.48928 0.48956 0.48983 0.4901 0.49036 0.49061 0.49086 0.49111 0.49134 0.49158

2.4 0.4918 0.49202 0.49224 0.49245 0.49266 0.49286 0.49305 0.49324 0.49343 0.49361

2.5 0.49379 0.49396 0.49413 0.4943 0.49446 0.49461 0.49477 0.49492 0.49506 0.4952

2.6 0.49534 0.49547 0.4956 0.49573 0.49585 0.49598 0.49609 0.49621 0.49632 0.49643

2.7 0.49653 0.49664 0.49674 0.49683 0.49693 0.49702 0.49711 0.4972 0.49728 0.49736

2.8 0.49744 0.49752 0.4976 0.49767 0.49774 0.49781 0.49788 0.49795 0.49801 0.49807

2.9 0.49813 0.49819 0.49825 0.49831 0.49836 0.49841 0.49846 0.49851 0.49856 0.49861

3 0.49865 0.49869 0.49874 0.49878 0.49882 0.49886 0.49889 0.49893 0.49896 0.499

3.1 0.49903 0.49906 0.4991 0.49913 0.49916 0.49918 0.49921 0.49924 0.49926 0.49929

3.2 0.49931 0.49934 0.49936 0.49938 0.4994 0.49942 0.49944 0.49946 0.49948 0.4995

3.3 0.49952 0.49953 0.49955 0.49957 0.49958 0.4996 0.49961 0.49962 0.49964 0.49965

3.4 0.49966 0.49968 0.49969 0.4997 0.49971 0.49972 0.49973 0.49974 0.49975 0.49976

3.5 0.49977 0.49978 0.49978 0.49979 0.4998 0.49981 0.49981 0.49982 0.49983 0.49983

3.6 0.49984 0.49985 0.49985 0.49986 0.49986 0.49987 0.49987 0.49988 0.49988 0.49989

3.7 0.49989 0.4999 0.4999 0.4999 0.49991 0.49991 0.49992 0.49992 0.49992 0.49992

3.8 0.49993 0.49993 0.49993 0.49994 0.49994 0.49994 0.49994 0.49995 0.49995 0.49995

3.9 0.49995 0.49995 0.49996 0.49996 0.49996 0.49996 0.49996 0.49996 0.49997 0.49997

4.0 0.49997 0.49997 0.49997 0.49997 0.49997 0.49997 0.49998 0.49998 0.49998 0.49998