Unit 8 Further Engineering Mathematics, Pearson BTEC International Level 3 qualifications in Applied Science

Assignment - Using sequences and series to solve engineering problems

Learning Outcome 1: Examine how sequences and series can be used to solve engineering problems.

Assignment Brief: You have applied for a work placement at a local company as part of your training as a Mechanical 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.

The first set of problems that they have provided are based on sequences and series, which they would like you to solve using the supplied data.

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:

Use the table1 which contains a set of data for you to use to complete the following activities. You need to:

Task 1

Question 1. Use your given sequences to answer the questions

Series A |
1, -1, -3, -5, -7 ... |

Series B |
4, 2, 1, 0.5, 0.25 ... |

Series C |
6, 7, 8, 9, 10 ... |

Series D |
4, 5, 7, 10, 14 ... |

Series E |
1, 10, 100, 1000, 10000 ... |

Series F |
2, 5, 8, 11, 14 ... |

Series G |
1, -1, 1, -1, 1 ... |

Series H |
1, 4, 9, 16, 25 ... |

(a) Identify which of your given sequences is

(i) an Arithmetic Progression (AP)

(ii) a Geometric Progression (GP)

(iii) neither an AP or a GP

(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). Explain why the other sequence is neither an AP or a GP

Question 2. A tunnelling machine drills 500 metres. Use your values from the table to estimate the cost of drilling if the first metre costs £X and each extra metre costs £Y.

Question 3. A motor has 5 speeds varying from X rpm to Y rpm in a Geometric Progression.

(a) Calculate the common ratio

(b) Make a table of the speeds.

Question 4. A CNC machine centre costs £100,000. Its value depreciates at X% per annum.

(a) What will be its value after Y years?

(b) How long will it take to have a value of £50,000?

Question 5. Use Pascal`s triangle to expand your binomial expression.

A |
(1 + x)3 |

B |
(1 - x)3 |

C |
(1 + x)4 |

D |
(1 - x)4 |

E |
(1 + 2x)3 |

F |
(1 - 2x)3 |

Question 6. Use the binomial theorem to expand the same binomial expression as question 5.

Question 7. Use the binomial theorem to expand your binomial expression and state the range of values of x for which the expansion is valid.

A |
(1 + x)-2 |

B |
(1 + x)-3 |

Question 8. Write the first four terms of a Maclaurin series for f(x) = ex

Question 9. Use your value of x to calculate a value for ex from the first four terms. Compare the answers to questions 7 and 8.

Question 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

Where T is the time period (seconds)

T = 2Π√(L/g)

L is the length of the pendulum (metre)

G is the acceleration due to gravity (g = 9.81 m/s2)

Calculate the expected percentage error in the time period calculation if your measurement of length is X% high.