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BTEC Extended Diploma in Engineering
Level 3
Unit 4
Mathematics for Engineering Technicians
Task Booklet
Student Name:
Assessor Name:
Dave Kirkness
Document Revision & Date:
Rev 2 (24/08/13)
Students to initial the box in the “Submitted” row below to show that an assignment is being
submitted for assessment.
Learner Declaration:
By initialling the submission box below, I confirm that the work submitted to complete the given
tasks is my own. I have indicated where research and other sources have been used to confirm the
conclusions reached within the submission and have listed those sources in a bibliography.
Outcome
Submitted
Achieved
Referred
Resubmitted
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
M1
M2
D1
D2
Outcome
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
Grading Criteria
Manipulate and simplify three
algebraic expressions using
the laws of indices and two
using the laws of logarithms
Solve a linear equation by
plotting a straight-line graph
using experimental data and
use it to deduce the gradient,
intercept and equation of the
line
Factorise by extraction and
grouping of a common factor
from expressions with two,
three and four terms
respectively
Solve circular and triangular
measurement problems
involving the use of radian,
sine, cosine and tangent
functions
Sketch each of the three
trigonometric functions over a
complete cycle
Produce answers to two
practical engineering
problems involving the sine
and cosine rule
Use standard formulae to find
surface areas and volumes of
regular solids for three
different examples
respectively
Collect data and produce
statistical diagrams,
histograms and frequency
curves
Determine the mean, median
and mode for two statistical
problems
Apply the basic rules of
calculus arithmetic to solve
three different types of
function by differentiation and
two different types of function
by integration
M1
Solve a pair of simultaneous
linear equations in two
unknowns
M2
Solve one quadratic equation
by factorisation an one by the
formula method
D1
Apply graphical methods to
the solution of two
engineering problems
involving exponential growth
and decay, analysing the
solutions using calculus
Evidence
Type
Deadline
21/11/2014
submission
6/2/2015
submission
16/01/2015
submission
27/02/2015
submission
27/02/2015
submission
02/04/2015
submission
27/03/2015
submission
22/05/2015
submission
22/05/2015
submission
26/06/2015
submission
TBC
exam
TBC
exam
TBC
exam
Achievement
Date
Assessor
Signature
Apply the rules of definite
integration to two
D2
engineering problems
involving summation
Unit Outcomes
Task Feedback
Task First Submission Feedback
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
M1
M2
TBC
exam
Second Submission Feedback
D1
D2
Internal Verification of Assessment Decisions
Task
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
M1
M2
Actions Required
Internal Verifier Sign and Date
D1
D2
BTEC Extended Diploma in Engineering
Mathematics for Technicians
Assignment booklet
Don’t forget that when submitting work you must declare which outcome you
are claiming. (P1, M3, D2, for example)
Don’t forget to put your name on all submitted work.
When requested, work must be submitted with the assignment facing sheet,
signed.
Make sure that you understand the work you have submitted. You may be
asked questions upon submission.
Work which is not reasonably presented might not be accepted.
P1 manipulate and simplify three algebraic expressions using the laws of
indices and two using the laws of logarithms
Manipulate and simplify the following expressions:
(i) 𝑥 = √(𝑦 2 × 𝑦 3 )2
(ii) √𝑥 4 × 𝑥 2 = 𝑦
(iii) ((𝑥 3 )3 (𝑥 −2 )) = √𝑦
(iv) Show how the addition of logarithms can be used to multiply:
4.2 × 6.1 × 5.5
(v) Show how the subtraction of logarithms can be used to divide:
13000
62
P2 solve a linear equation by plotting a straight-line graph using
experimental data and use it to deduce the gradient, intercept and
equation of the line
The following table gives the results of tests carried out to determine the
breaking stress σ of rolled copper at various temperatures, t:
Stress σ
8.51
(N/cm )
Temperature 75
t( ◦ C)
8.07
7.80
7.47
7.23
6.78
220
310
420
500
650
Plot a graph of stress (vertically) against temperature (horizontally). Draw the
best straight line through the plotted co-ordinates. Determine the slope of the
graph and the vertical axis intercept. Determine the equation of the line.
P3 factorise by extraction and grouping of a common factor from
expressions with two, three and four terms respectively
Factorise the following expressions:(i)
2bc + 4ab
(ii)
6x2y + 3xy
(iii)
ax + ay – az
(iv) 2x2 + 4y + 8zx
(v)
ax – ay + bx – by
(vi) 2ax + 6ay + 3bx + 3by
P4 solve circular and triangular measurement problems involving
the use of radian, sine, cosine and tangent functions
a) If the radius of a wheel on a vehicle is 0.5m, and the vehicle travels 2km,
how many revolutions has the wheel gone through? How many radians is this?
b) (i) Convert 7 radians per second into revolutions per minute.
(ii) Convert 1000 revolutions per minute into radians per second.
c) Find the length of an arc of a circle of radius 8.32cm when the angle
subtended at the centre is 2.24 rad. Calculate also the area of the sector
formed.
d) Use the tan ratio to calculate the length of the horizontal side in this right
angled triangle. Use the sine ratio to calculate the hypotenuse. The angle at
a is 40°.
e) Use the cosine ratio to calculate the length of the horizontal (adjacent) side
for this right angled triangle. The angle at a is 28°.
P5 sketch each of the three trigonometric functions over
a complete cycle
Sketch each of the three trigonometric functions over one cycle.
This would be best achieved on graph paper, and by using a calculator to find
the values of each function at intervals of, say, 10°.
P6 produce answers to two practical engineering problems
involving the sine and cosine rule
For P6 answer questions 1 and 2 below.
Q1.
The triangle represents the relative positions of three transmitting stations.
In order to calculate the signal delay between the stations it is necessary to
calculate the distances between them. In triangle ABC, the angle at B = 23°, the
angle at C = 47° and length AB = 10km. Use the sine rule to solve this triangle.
A
C
B
Q2.
Triangle ABC represents part of a system of struts which forms part of a
design for a football stadium. In triangle ABC, AB = 6.5m, BC = 9.0m and AC =
7.5m. Use the cosine rule to find the internal angles.
B
A
C
P7 use standard formulae to find surface areas and volumes of regular solids
for three different examples respectively
For P7 for this unit, find the surface areas and volumes of:1) A sphere of radius 100mm.
2) A pyramid of height 70mm and base 50mm.
3) A cone of height 60mm and base 20mm radius.
P8 collect data and produce statistical diagrams, histograms and
frequency curves
For P8 complete the following tasks a), b) and c).
a) You are asked to inspect a batch of rejected components. You are asked to
produce a report which shows the proportion of the sample which comes into
each of the following categories:(i) Incorrect dimensions
(ii) Broken
(iii) Wrong colour
(iv) Incomplete
(v) Wrong material
To be “OK” a component should have the following properties:
It should be made from blue sheet plastic, of 3 mm thickness, and 100mm
square. It should have a circular hole in the middle, 20mm in diameter. There is
a 1mm tolerance for all dimensions.
You tested a batch of 100 components. You found 20 that were longer than
101mm. 12 were shorter than 99mm. 15 were wider than 101mm. 10 were
green. 6 were white. 12 did not have the hole removed. 5 were cardboard. 10
were badly cracked. You found 10 where the hole had been partly punched
out, but where the unwanted material had not completely come away from
the square blank.
Produce (a) a pie chart and (b) a bar chart showing the information.
b) The quantity of electricity used by an office over a 52 week period is shown
below. Show the information as a histogram.
Usage
(kWh)
No. of
weeks
20- 59
60-89
90-99
2
3
6
100109
8
110119
12
120129
8
130139
5
140159
4
160199
4
c) The length in millimetres of a sample of bolts is as shown below. Draw
frequency curves for the data.
Length
(mm)
No. of
bolts
165
166
167
168
169
170
171
172
173
174
5
14
18
28
36
29
29
24
19
15
Length
(mm)
No. of
bolts
175
176
177
6
3
2
P9 determine the mean, median and mode for two statistical problems
Question 1.
The quantity of electricity used by an office over a 50 week period is shown
below. For example, for six of the fifty weeks the usage was between 89 and
97 kWh. Determine the mean, mode and median. Explain which of these
measures would be most useful to the accounts manager.
Usage
(kWh)
No. of
weeks
71-79
80-88
89-97
98-106
1
3
6
8
107115
12
116124
8
125133
5
134142
4
143151
3
Question 2.
The length in millimetres of a sample of bolts is as shown below. Calculate the
mean, mode and median. Which of these measures would be most useful in
setting a machine to cut the bolts to length?
Length
(mm)
No. of
bolts
165
166
167
168
169
170
171
172
173
174
5
14
18
28
36
29
29
24
19
15
175
6
176
3
177
2
P10 Apply the basic rules of calculus arithmetic to solve three different types
of function by differentiation and two different types of function by
integration
(a) Differentiate the equation θ = 9t² – 2t³ with respect to t.
(b) Differentiate the equation y = 3 sin 5t with respect to t.
(c) Differentiate the equation y = 2e6t with respect to t.
(d) Determine ∫ 𝑥 7 𝑑𝑥.
(e) Determine ∫ (5sin 3t – e3t) dt.
M1 solve a pair of simultaneous linear equations in two unknowns
Solve this pair of simultaneous equations:-
7x – 2y = 26
6x + 5y = 29
M2 solve one quadratic equation by factorisation and one by the formula
method
Solve the equation x² – 4x + 4 = 0 by factorization.
Solve the equation 2x² – 7x + 4 = 0 by the formula method.
D1 apply graphical methods to the solution of two engineering problems
involving exponential growth and decay, analysing the solutions using
calculus
For D1 answer the following two questions:-
(a)
In an experiment involving Newton`s law of cooling, the temperature
θ (°C) of a body at any moment in time is given by:-
θ = θ0 e-kt
where θ0 is the temperature at t = 0 seconds.
If k = 1.485 x 10–2 and θ0 = 100°C, draw a graph which shows as accurately as
possible the value of θ between t = 100s and t = 110s.
From your graph estimate the rate of cooling at t = 105 seconds and use an
appropriate method of calculus to check your result.
(b)
The charging characteristic for a series capacitive circuit is:
𝑣 = 𝑉 [1 − 𝑒
𝑡
−( )
𝑇 ]
T is the time constant and is given by T = CR.
If C = 100nF, R = 47kΩ and V = 5V plot the charging curve over the range 0 to
20 ms.
From your graph estimate the rate of charging at 6ms.
Differentiate the charging equation to find the rate of charging at 6 ms and
compare this with your estimation.
D2 apply the rules for definite integration to two engineering
problems that involve summation.
Apply the rules for definite integration to answer the following two questions:(a)
The velocity of a body in metres per second is given as v = 3.5t² + 1.5t – 10.
Draw a graph showing velocity against time for values of t between 0s and 10s.
From your graph approximate the distance travelled by the body between
t =1s and t =3s.
Use an appropriate method of integration to calculate the distance and
compare this answer with your previous approximation.
(b)
The power (in watts) from an engine is given by the equation 𝑃 = 80𝑡 1.3 + 5𝑡
where 𝑡 is the time in seconds.
Draw a graph of power against time for the engine. From the graph
approximate the energy produced between 3 and 7 seconds.
Using an appropriate method of integration to calculate the energy produced,
and compare the answer with your approximation.
Bibliography :
Bird, John. Basic Engineering Mathematics (4th Edition).
Jordan Hill, GBR: Newnes, 2005. p 92.
http://site.ebrary.com/lib/canterbury/Doc?id=10127881&ppg=105