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 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 M1 M2 D1 D2

Submitted

Achieved

Referred

Resubmitted

Outcome Grading Criteria Evidence Type

Deadline AchievementDate

AssessorSignature

P1Manipulate and simplify three algebraic expressions using the laws of indices and two using the laws of logarithms

submission20/11/2017

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

submission

5/2/2018

P3

Factorise by extraction and grouping of a common factor from expressions with two, three and four terms respectively

submission

15/01/2018

P4

Solve circular and triangular measurement problems involving the use of radian, sine, cosine and tangent functions

submission

26/02/2018

P5 Sketch each of the three trigonometric functions over a complete cycle

submission26/02/2018

P6Produce answers to two practical engineering problems involving the sine and cosine rule

submission23/3/2018

P7

Use standard formulae to find surface areas and volumes of regular solids for three different examples respectively

submission

27/03/2018

P8Collect data and produce statistical diagrams, histograms and frequency curves

submission21/05/2018

P9Determine the mean, median and mode for two statistical problems

submission21/05/2018

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

submission

25/06/2018

M1Solve a pair of simultaneous linear equations in two unknowns

examTBC

M2Solve one quadratic equation by factorisation an one by the formula method

examTBC

D1

Apply graphical methods to the solution of two engineering problems involving exponential growth and decay, analysing the solutions using calculus

exam

TBC

D2Apply the rules of definite integration to two engineering problems involving summation

exam

TBC

Unit Outcomes

Task Feedback Task First Submission Feedback Second Submission FeedbackP1

P2

P3

P4

P5

P6

P7

P8

P9

P10

M1

M2

D1

D2

Internal Verification of Assessment Decisions

Task Actions Required Internal Verifier Sign and DateP1

P2

P3

P4

P5

P6

P7

P8

P9

P10

M1

M2

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=√ ( y2× y3 )2

(ii) √ x4×x2 ¿ y

(iii) ( (x3 )3 (x−2 ))=√ y

(iv) As light passes through a medium the intensity of light at any depth in that medium is given by:

I=I 0 e(−αx)

where I is the intensity at a given depth, I 0 is the intensity at the surface, α is the coefficient of absorption (the negative sign signifies that the light intensity is decreasing

with an increase in depth) and x is the depth. This equation can be manipulated to calculate the depth at which the light intensity will reach a certain ratio of the surface intensity. Manipulate this equation to find the thickness of silicon that would absorb 99% of light if the coefficient of absorption is 0.3 x 106 m-1.

(v) The ratio of an output power to an input power is given by:-

X=10 log ( PoutP¿) dB

An amplifier has a gain of 12 dB and its input power is 8 mW. Manipulate the equation above to find its output power.

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 (N/cm )

8.51 8.07 7.80 7.47 7.23 6.78

Temperature t( ◦ C)

75 220 310 420 500 650

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 overa 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°.

B

A

C

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

B

C

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.

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)

20- 59 60-89 90-99 100-109

110-119

120-129

130-139

140-159

160-199

No. of weeks

2 3 6 8 12 8 5 4 4

c) The length in millimetres of a sample of bolts is as shown below. Draw frequency curves for the data.

Length(mm)

165 166 167 168 169 170 171 172 173 174

No. of bolts

5 14 18 28 36 29 29 24 19 15

Length(mm)

175 176 177

No. of bolts

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)

71-79 80-88 89-97 98-106 107-115

116-124

125-133

134-142

143-151

No. of weeks

1 3 6 8 12 8 5 4 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)

165 166 167 168 169 170 171 172 173 174

No. of bolts

5 14 18 28 36 29 29 24 19 15

175 176 1776 3 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 ∫ x7dx.

(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 = 266x + 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−( tT )]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+5 t 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.