National 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) x2 = (152 x 153)2

(ii) √(124 ÷ 122)

(iii) ((x3)3)(1÷x2)

(iv) 4log 20 + 2log 30 +3 log 10

(v) 5 log 20 – 6 log 202

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.

P3factorise 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) Convert 7 radians per second into revolutions per minute.

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.

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
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.

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.

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 / 177
6 / 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.

(d) Determine

(e) Determine

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 decay of voltage across a capacitor is given by v = 250e –t/3.

Draw a graph showing the natural decay curve over the first 6 seconds. From the graph, estimate the rate of decay at t = 2 seconds.Use an appropriate method of calculus to find the rate of change of voltage at t = 2 seconds, and analyse the result.

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.

Calculate the distance travelled by the object between t = 1s and t = 3s.

(b)

If the velocity,in metres per second,of a point in a vibrating system is described by the equation

v = 5sin 100πt

Calculate the distance travelledby the point between t = 0 seconds and t = 0.014 seconds.

Bibliography :

Bird, John. Basic Engineering Mathematics (4th Edition).

Jordan Hill, GBR: Newnes, 2005. p 92.