Recent Question/Assignment

1``Qualification Pearson BTEC Level 3 National Extended Diploma in Engineering
Pearson BTEC Level 3 National Extended Diploma in Electrical/Electronic Engineering
Pearson BTEC Level 3 National Extended Diploma in Mechanical Engineering
Unit or Component number and title
Unit 7: Calculus to solve engineering problems
Learning aim(s) (For NQF/RQF only) A: Examine how differential calculus can be used to solve engineering problems
Assignment title Solving engineering problems that involve differentiation
Assessor Helen Christison
Hand out date 6th December 2021
Hand in deadline 6th January 2022
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 differentiation to determine if you are suitable.
Produce a report that contains written descriptions, analysis and mathematics that shows how calculus can be used to solve engineering problems as set out below.
1 The equation for a distance, s(m), travelled in time t(s) by an object starting with an initial velocity u(ms-1) and uniform acceleration a(ms-2) is:
s=ut+1/2 at^2
Plot a graph of distance (s) vs time (t) for the first 10s of motion if u=10ms^(-1) and a=5ms^(-2).
Determine the gradient of the graph at t=2s and t=6s.
Differentiate the equation to find the functions for
Velocity (v=ds/dt)
Acceleration (a=dv/dt=(d^2 s)/?dt?^2 )
Use your result from part c to calculate the velocity at t=2s and t=6s.
Compare your results for part b and part d.
2 The displacement of a mass is given by the function
y=sin 3t .
Draw a graph of the displacement y(m) against time t(s) for the time t=0s to t=2s.
Identify the position of any turning points and whether they are maxima, minima or points of inflexion.
Calculate the turning points of the function using differential calculus and show which are maxima, minima or points of inflexion by using the second derivative.
Compare your results from parts b and c.
3 The equation for the instantaneous voltage across a discharging capacitor is given by v=V_O e^(-t/t), where V_O is the initial voltage and t is the time constant of the circuit.
Draw a graph of voltage against time for V_O=12V and t=2s, between t=0s and t=10s.
Calculate the gradient at t=2s and t=4s.
Differentiate v=12e^(-t/2) and calculate the value of dv/dt at t=2s and t=4s.
Calculate the second derivative of the instantaneous voltage ((d^2 v)/?dt?^2 ).
4 The same capacitor circuit is now charged up to 12V and the instantaneous voltage is v=12(1-e^(-t/2) ).
Differentiate v with respect to t to give an equation for dv/dt.
Calculate the value of dv/dt at t=2s and t=4s.
Find the second derivative ((d^2 v)/?dt?^2 ).
5 The gain of an amplifier is found to be G=20 log?(10V_out ),:
The tasks are to find equations for:
dG/(dV_Out )
(d^2 G)/?dV_Out?^2
6 The displacement, y(m), of a body in damped oscillation is y=2e^(-t) sin?3t.
Use the Product Rule to find an equation for the velocity of the object if v=dy/dt.
7 The velocity of a moving vehicle is given by the equation v=(2t+3)^4
Use the Chain Rule to determine an equation for the acceleration when a=dv/dt.
8 A communication signal is given by the function y=sin?t/t
Derive an equation for dy/dt using the Quotient Rule.
9 A company is required to fence off a square/rectangular area around a robot arm to comply with health and safety law. They have 750m of fencing available.
Find the maximum square/rectangular area they can fence off?
10 You plan to make a simple, open topped box from a piece of sheet metal by cutting a square – of equal size – from each corner and folding up the sides as shown in the diagram:

If l=200mm and w=150mm calculate:
The value of x which will give the maximum volume
The maximum volume of the box
Comment of the value obtained in part b.
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. Your explanation should be detailed in response to questions 9 and 10 to show how the variables are optimised in each case. Graphs should be well presented and clearly labelled and comparisons between methods should be accurate and well presented.