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Qualification - Higher National Certificate/Diploma in Engineering
Unit Name - Engineering Maths
Assignment Title - Engineering Maths
Unit Number - Unit 2
Learning Outcome 1: Identify the relevance of mathematical methods to a variety of conceptualised engineering examples.
Learning Outcome 2: Investigate applications of statistical techniques to interpret, organise and present data by using appropriate computer software packages.
Learning Outcome 3: Use analytical and computational methods for solving problems by relating sinusoidal wave and vector functions to their respective engineering applications.
Learning Outcome 4: Examine how differential and integral calculus can be used to solve engineering problems.
Assignment Brief
Scenario:
As an electrical and electronic engineer you always encounter equations, problems and data that need to be analysed in order to understand the results in a practical way.
You are working as an engineer and you have completed many tasks in a job that you were asked to carry out. A report must be prepared for you senior in order to analyse the results that you have obtained in your experiments. You must analyse the results carefully using methodologies that you have learnt in engineering maths. Your report must be clear and presented in a professional way as this will have a major impact on how well you impress your superiors and obtain the promotion that you have been working for a long time. All the tasks must be finalised on time and the analysis must be thorough and convincing.
a- The owner of the Ches Tahoe restaurant is interested in how much people spend at the restaurant. He examines 10 randomly selected receipts for parties of four and writes down the following data. Find the mean and standard deviation
44, 50, 38, 96, 42, 47, 40, 39, 46, 50
b- Show and explain what is the normal distribution approximation for the binomial distribution where n = 20 and p = .25 (i.e. the binomial distribution displayed in Figure 1 of Binomial Distribution)?
c- The average life of a motor is 10 years and the standard deviation is 2 years. Given that the manufacturer can replace only 3% of the motors for faulty reasons, how long should the guarantee be for if the lives of the motors represent a normal distribution?
d- The manufacturer of metal pistons found that on average 12% of the pistons are rejected because they are oversize or undersize. What is the probability that in any group of 10 pistons:
1- Not more than 2 pistons are rejected
2- At least 2 pistons are rejected
3- how would you interpret these results in terms of profitability and loss of revenues
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The lifetime of a particular type of light-bulb has been shown to follow a Normal distribution with mean lifetime of 1000 hours and standard deviation of 125 hours. Three bulbs are found to last 1250, 980 and 1150 hours. Convert these values to standardized normal scores.
a- A system of solar panels produces a daily average power P that changes during the year. It is maximum on the 21st of June (day with the highest number of daylight) and equal to 20 kwh/day. We assume that P varies with the time t according to the sinusoidal function P(t) = a cos [b(t - d)] + c , where t = 0 corresponds to the first of January, P is the power in kwh/day and P(t) has a period of 365 days (28 days in February). The minimum value of P is 4 kwh/day.
1- Find the parameters a, b, c and d.
2- Sketch P(t) over one period from t = 0 to t = 365.
3- When is the power produced by the solar system minimum?.
4- The power produced by this solar system is sufficient to power a group of machines if the power produced by the system is greater than or equal to 16 kwh/day. For how many days, in a year, is the power produced by the system sufficient?
b- The figure shown in the diagram below is showing a heavy box which is suspended by two wires where OA is T1 and OB is T2. Represent the two forces T1 and T2 by their vector components using the directions given in the diagram.
c- The two forces shown in the diagram below show two forces F1 and F2 and their values are 20lb and 30lb respectively. These two forces act on an object P as shown in the figure. What is the resultant force acting on the object?
d- Find the angle between the vectors P = 4i + 0j + 7k and Q = -2i + j + 3k.
e- Using the formula for compound angles prove that sin ( x - π ), sin ( x + π ) and cos ( x + π/x) are all the same.
f- Analyze the components of the expression and compare it to the analytical values
g- Evaluate the compound angle expression sin ( a + b ) if the angles a and b are obtuse angles and with the following values: Sin a = 3/5 and b = 5/13.
a- If two resistors with resistances R1 and R2 are connected in parallel as shown in the figure below, their electrical behaviour is equivalent to a resistor of resistance R such that 1/R = 1/R1 + 1/R2. If R1 changes with time at a rate r = dR1/dt and R2 is constant, express the rate of change dR/dt of the resistance of R in terms of dR1/dt R1 and R2.
b- If an object with a speed v giving by the equation v ( t ) = 1 + 4t + 3t2 where t is the time taken to move per minute, what is the travelled distance during the third minute?
c- A wire of mild steel has a radius of 0.5 mm and a length of 3 m. If the young modulus Y = 2.1 x 1011 N/m3 and the wire is stretched by a force of 40 N determine:
1- The longitudinal stress 2- The longitudinal strain 3- The elongation
d- If the switch ‘S' is at position ‘1' sufficiently long enough before it is moved to position ‘2' and after this it is kept in position 2 as shown in the figure below determine:
1- Values of the instantaneous inductor voltage vL and the instantaneous current through the inductor iL
2- Values of the inductor voltage vL and the current through the inductor iL just after the switch changes
3- The rate of change of iL at t = 0
If the equation y = x3 - 6x2 + 9x Find the stationary points of the curve and determine whether these points are maxima or minima and prove this.
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