Reference no: EM132785982
Unit 8 Further Engineering Mathematics - BTEC Level 3 National Extended Diploma in Engineering
Assignment 1: Using sequences and series to solve engineering problems
Scenario:
You have applied for a work placement at a local company as part of your training as a quality control (QC) and in-service Engineer. The company Training Manager has explained that there are a number of roles available that involve applying mathematical skills and knowledge.
The Training Manager has said that they would like to see how you approach solving engineering problems before deciding on the most appropriate role for you. This is the second set of problems that they have provided. The problems are based on matrices, determinants and complex numbers.
Task 1
You have been asked to explore a range of engineering problems that will require you to solve problems based around sequences and series
To do this:
Your tutor will provide you with a set of data for you to use to complete the following activities. You need to:
1 Use your given sequences to answer the questionsuse series A, B and D
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Series A
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1, -1, -3, -5, -7 ...
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Series B
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4, 2, 1, 0.5, 0.25 ...
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Series C
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6, 7, 8, 9, 10 ...
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Series D
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4, 5, 7, 10, 14 ...
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Series E
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1, 10, 100, 1000, 10000 ...
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Series F
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2, 5, 8, 11, 14 ...
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Series G
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1, -1, 1, -1, 1 ...
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Series H
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1, 4, 9, 16, 25 ...
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(a) Identify which of your given sequences is
(i) an Arithmetic Progression (AP)
(ii) a Geometric Progression (GP)
(iii) neither an AP or a G
(b) Identify for your AP,
(i) the first term (a)
(ii) the common difference (d)
(iii) the 10th term
(iv) the sum of the first 10 terms
(c) Identify for your GP,
(i) the first term (a)
(ii) the common ratio (r)
(iii) the 8th term
(iv) the sum of the first 8 terms
(v) whether the GP is convergent,(give reasons).
(d) Explain why the other sequence is neither an AP or a GP
2 A tunnelling machine drills 500 metres. Use your values from the table to estimate the cost of drilling if the first metre costs £1000 and each extra metre costs £100.
3 A motor has 5 speeds varying from 10 rpm to 160 rpm in a Geometric Progression.
(a) Calculate the common ratio
(b) Make a table of the speeds.
4 A CNC machine centre costs £100,000. Its value depreciates at 1.0% per annum.
(a) What will be its value after 2 years?
(b) How long will it take to have a value of £50,000?
5 Use Pascal's triangle to expand your binomial expression A.
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A
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(1+x)3
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B
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(1-x)3
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C
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(1+x)4
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D
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(1-x)4
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E
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(1+2x)3
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F
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(1-2x)3
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6 Use thebinomial theorem to expand the same binomial expression as question 5.
7 Use the binomial theorem to expand your binomial expression up to and including the term in x^3and state the range of values of A for which the expansion is valid.
8 Write the first four terms of a Maclaurin seriesfor f(x)= e^x
9 Use your value of 1 to calculate a value for ? e?^xfrom the first four terms. Compare the answers to questions 7 and 8.
10 You have carried out an experiment to investigate the effect of the length of a pendulum on the time period of oscillation. Theory says that the pendulum should follow the rule T=2π√(L/g)
Where T is the time period (seconds)
L is the length of the pendulum (metre)
G is the acceleration due to gravity (g=9.81m/s2)
Calculate the expected percentage error in the time period calculation if your measurement of length is 1% high.
Attachment:- Further Engineering Mathematics.rar