| University | Milton Keynes College |
|---|---|
| Subject | Unit 7: Calculus to solve engineering problems |
Unit number and title
Unit 7: Calculus to solve engineering problems
Learning aim(s)
C: Investigate the application of calculus to solve a defined specialist engineering problem
Assignment title
Investigate an engineering problem using calculus
Vocational Scenario or Context
You are working as a Junior Engineer for a small motor racing team. You have been given a proposed mathematical model to calculate the velocity of a car accelerating from rest in a straight line. The equation is:
v(t) = A(1 – e^(-t/t_max))
Where:
- v(t) is the instantaneous velocity of the car (m/s)
- t is the time in seconds
- t_max is the time to reach the maximum speed in seconds
- A is a constant.
The Team Manager has asked you to carry out an analysis of the model and produce a written report. As part of the analysis you have been given some data on a known model.
More Same code BTEC 7 CALCULUS Assessment Tasks
Tutors have set other Same code BTEC 7 CALCULUS tasks on this module too. They are listed below so you can see every version of the brief on one page.
BTEC Unit 7 Calculus Assignment: Integration Methods, Motion Analysis, Work & Cooling Law Applications
Unit number and title
Unit 7: Calculus to solve engineering problems
Learning aim(s)
B: Examine how Integral calculus can be used to solve engineering problems
Assignment title
Solving engineering problems that involve differentiation
Vocational Scenario or Context
You are working as an apprentice engineer at a company involved in the research, design production and maintenance of bespoke engineering solutions for larger customers.
Part of your apprenticeship is to spend time working in all departments, however a certain level of understanding needs to be shown before the managing director allows apprentices into the design team and so she has developed a series of questions on integration to determine if you are suitable.
Task 1
1. Integration Problems
The tasks are to:
a) Find the indefinite integral of the function ∫(3x² + 2x + 1)dx
b) Calculate the definite integral ∫14 (3x² + 2x + 1)dx
2. Motion with Uniform Acceleration
An object is moving with a uniform acceleration a, determine the functions for:
a) Velocity – given v = ∫a dt
b) Displacement – given s = ∫v dt
c) Calculate the values of v and s for:
i) a = 2 m/s², initial velocity u = 3 m/s, initial displacement s₀ = 0
ii) a = 9.81 m/s², initial velocity u = 0 m/s, initial displacement s₀ = 10 m
3. Work Done Calculation
The extension, x, of a material with an applied force, F, is given by F = kx.
a) Calculate the work done if the force increases from 100N to 500N using:
i) An analytical integration technique
ii) A numerical integration technique
[Note: the work done is given by the area under the curve]
b) Compare the two answers
c) Using a computer spreadsheet increase the number of values used for your numerical method
d) Analyse any affect the size of numerical step has on the result.
4. Periodic Function Analysis
For the function y = sin(θ), calculate the:
a) Mean
b) Root mean square (RMS)
Over a range of 0 to 2π radians.
[Note the trigonometric identity sin²θ = ½(1 – cos2θ)]
5. Substitution Method
A complex function can be modelled by the equation: y = 2x(3x² + 4)⁵
Find the indefinite integral of the function using a substitution method.
6. Integration by Parts
The acceleration of an object moving in a strange way has been modelled as a = t²eᵗ.
a) Use integration by parts to find an equation to model the velocity if v = ∫a dt.
b) Is the problem any different if you find ∫t³eᵗ dt?
7. Newton’s Law of Cooling
Newton’s laws of cooling proposes that the rate of change of temperature is proportional to the temperature difference to the ambient (room) temperature. And can be modelled using the equation:
dT/dt = -k(T – Troom)
This can also be written as:
dT/(T – Troom) = -k dt
Where:
T = temperature
Troom = room temperature
k = cooling constant
a) Integrate both sides of the equation and show that the temperature difference is given by:
T – Troom = (T0 – Troom)e-kt
b) Calculate T if the initial temperature is 90°C and Troom = 20°C, k = 0.1.
Checklist of evidence required
Your informal report should contain:
- Analysis
- Worked solutions to the problems
Each worked solution should be laid out clearly and contain brief explanations of the stages of the calculation to indicate your understanding of how calculus can be used to solve an engineering problem. Graphs should be well presented and clearly labelled and comparisons between methods should be accurate and well presented.
Criteria covered by this task:
7/B.D1 Evaluate, using technically correct language and a logical structure, the correct integral calculus and numerical integration solutions for each type of given routine and non-routine functions, including at least two set in an engineering context.
7/B.M2 Find accurately the integral calculus and numerical integration solutions for each type of given routine and non-routine function, and find the properties of periodic functions.
7/B.P4 Find the indefinite integral for each type of given routine function.
7/B.P5 Find the numerical value of the definite integral for each type of given routine function.
7/B.P6 Find, using numerical integration and integral calculus, the area under curves for each type of given routine definitive function.
Sources of information to support you with this Assignment
- mathsisfun.com/index.htm
- mathcentre.ac.uk/students/topics
Pay for a Custom Velocity Model Assignment Built Around This Brief
Task 1
Produce a report that contains written descriptions, analysis and mathematics that shows how calculus can be used to solve an engineering problem. The tasks are to:
- Use thinking methods to analyse the given engineering problem, e.g. break the problem down into a series of manageable elements, and produce a specification
- Prepare a valid proposal for solving the problem and present it
- Produce mathematical models for the identified elements of the problem
- Apply calculus methods to produce answers for each of the elements
- Bring the elements together in a formal presentation
In your proposal you need to outline the problem and the methods needed to solve it. You need to include how to:
1. Model Analysis
Identify the:
- units of the coefficient A
- physical meaning of A
- velocity of the car at t = 0
- asymptote of this function as t → ∞
2. Velocity Graph
Sketch a graph of velocity vs. time.
3. Position Function
Derive an equation x(t) for the instantaneous position of the car as a function of time. Identify the:
- value x when t = 0 s
- asymptote of this function as t → ∞
4. Position Graph
Sketch a graph of position vs. time.
5. Acceleration Function
Derive an equation for the instantaneous acceleration of the car as a function of time. Identify the:
- acceleration of the car at t = 0 s
- asymptote of this function as t → ∞
6. Acceleration Graph
Sketch a graph of acceleration vs. time.
7. Application to Specific Car
Apply your mathematical models to your allocated car. Use the given data for the 0 – 28 m/s and 400m times to calculate the:
- value of the coefficient A
- maximum velocity
- maximum acceleration
Want a Second Opinion on 7. Application? Ask a UK Tutor
Checklist of evidence required
Your informal report should contain:
- written descriptions
- analysis
- worked solutions to the problems
Each worked solution should be laid out clearly and contain brief explanations of the stages of the calculation to indicate your understanding of how calculus can be used to solve an engineering problem. Diagrams and sketches should be well presented and clearly labelled.
Criteria covered by this task:
7/C.D3 Critically analyse, using technically correct language and a logical structure, a complex engineering problem, synthesising and applying calculus and a mathematical model to generate an accurate solution.
7/C.M3 Analyse an engineering problem, explaining the reasons for each element of the proposed solution
7/C.M4 Solve accurately, using calculus methods and a mathematical model, a given engineering problem.
7/C.P7 Define a given engineering problem and present a proposal to solve it
7/C.P8 Solve, using calculus methods and a mathematical model, a given engineering problem
Sources of information to support you with this Assignment
- mathsisfun.com/index.htm
- mathcentre.ac.uk/students/topics
Other assessment materials attached to this Assignment Brief
| Candidate | Car model | t(0-28 m/s) (s) | t(400m) (s) | tmaxspeed (s) |
|---|---|---|---|---|
| 1 | 1994 Ford SVT Boss Mustang 10.0L Concept | 1.9 | 10.50 | 7.1 |
| 2 | 2002 Chevrolet Corvette (Lingenfelter 427 Twin Turbo) | 2.0 | 9.10 | 6.8 |
| 3 | Bugatti Veyron 16.4 Grand Sport Vitesse – [2012] | 2.5 | 9.21 | 10.2 |
| 4 | Caterham 7 620 R 2.0 L Supercharged – [2013] | 2.8 | 10.91 | 7.0 |
| 5 | Ariel Atom V8 500 – [2008] | 2.3 | 10.80 | 5.0 |
| 6 | Bugatti Veyron 8.0 litre W16 Super Sport – [2010] | 2.5 | 9.75 | 8.0 |
| 7 | Caparo T1 2.4 V8 – [2006] | 2.5 | 8.23 | 7.0 |
| 8 | Hennessey Venom GT – [2010] | 2.5 | 8.64 | 8.5 |
| 9 | Radical RXC 3.5 V6 Twin Turbo – [2014] | 2.6 | 10.46 | 7.0 |
| 10 | Ascari A10 5.0 V8 – [2006] | 2.8 | 10.36 | 7.8 |
| 11 | Lamborghini Veneno 6.5 V12 – [2013] | 2.8 | 9.92 | 8.5 |
| 12 | Porsche 918 Spyder 4.6 V8 – [2013] | 2.8 | 11.49 | 8.2 |
| 13 | Caterham 7 Superlight R500 – [2009] | 2.9 | 11.32 | 8.0 |
| 14 | Ferrari F12 tdf 6.3 V12 – [2015] | 2.9 | 10.18 | 7.4 |
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