Download Building Services Engineering Exam: Math & Computing, Summer 2005, BEng and more Exams Mathematics for Computing in PDF only on Docsity!
Cork Institute of Technology
Bachelor of Engineering in Building Services Engineering-Award
(National Diploma in Engineering in Building Services Engineering – Award)
(NFQ Level 7)
Summer 2005
Mathematics & Computing
(Time: 3 Hours)
Instructions Answer five questions. All questions carry equal marks.
Examiners: Dr. N. Hewitt Mr. T. Dunne Ms. M. Harley
Q1a Find the Laplace transforms of the following, expressing your answer as a single fraction.
(i) t^3 − 3 t 2 − 2 (ii) 12 e^3 cos( 5 t )
− t
(4 Marks)
Q1b Deduce the inverse Laplace transforms of the expressions
(i) (^2) ( 4 )
s +
(ii) 2 10
s s
s (iii) ( 3 )( 16 )
2
2
s s
s s
(8 Marks)
Q1c Use Laplace Transforms to solve the differential equation
2 h ′( t )+ h ( t )= 10 t given h (0) = 0
(8 Marks)
Q2 The displacement x cm. of a damped oscillator is given by
x ′′^ ( t )+ kx ′( t )+ 25 x ( t )= 0 where k is a damping constant.
Solve this differential equation for the cases where (i) k = 0 (ii) k = 2 and subject to the conditions x ( 0 )= 0 , x ′( 0 )= 20. Determine the period and duration of the oscillations. Draw sketches to illustrate your solutions, labelling the axes appropriately. State the value of k which will produce critical damping and solve the differential equation in this case, subject again to the conditions x ( 0 )= 0 , x ′( 0 )= 20. For this value of k determine the maximum displacement and sketch x ( t ).
Q3a The acceleration of an object at time t (s) is given by
v^2
dt
dv
Find an expression for the velocity v if the initial velocity is 25 ms-^. If the object is initially at P find an expression for its displacement from P at any subsequent time t. Evaluate the velocity and displacement after 10s. (10 Marks)
Q3b A cylindrical tank has a base diameter 1.5m and height 8m. Initially the tank is full and it is then emptied through a circular port at the base of diameter 25mm. If h is the height of the liquid in the tank then the velocity of the emerging liquid is
given by 0. 65 2 gh ms-1^ show that h dt
dh (^) 4 = − 8 × 10 − and hence determine
(i) the time taken to empty the tank (ii) the depth of liquid after 1 hour ( g = 9.81ms-2) (10 Marks)
Q4a Brass terminals have a mean mass of 0.25 kg and a standard deviation from this mean of 0.02 kg Find the proportion of terminals having a mass between 0.22 and 0.30kg The brass terminals are packed in boxes of 50 and 200 of these boxes are delivered to one company. Determine the number of boxes in this delivery which would have an expected weight less than 12.3 kg and the weight of the heaviest 10% of the boxes in this delivery. (6 Marks)
Q4b A manufacturer guarantees a lifetime of at least 1600 hours for a fluoresent light bulb it produces. The mean life-time of a sample of 100 bulbs is found to be 1570 hours with a standard deviation of 120 hours. Determine the legitimacy of the manufacturers claim. (3 Marks)
Q4c As part of a quality control procedure of a firm producing discs, samples are selected at hourly intervals and their diameters are measured. The results for samples 1 2 and 3 are given below: Sample 1: 60.1 61.6 60.3 60.9 64. Sample 2: 60.9 62.3 63.0 64.5 63. Sample 3: 60.5 61.2 64.3 60.9 61.
Determine the mean and range for these samples The results for the other samples (correct to 1 d.p.) are given in the table.
Sample No. 1 2 3 4 5 6 7 8
X 4.1^ 4.6^ 3.7^ 3.8^ 3.
R 60.8^ 61.9^ 61.1^ 62.5^ 61.
Use the data to estimate the mean diameter. Construct mean and range control charts and comment on these.
