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Cork Institute of Technology
Bachelor of Engineering in Building Services Engineering – Award
(NFQ Level 7)
Summer 2007
Mathematics
(Time: 3 Hours)
Instructions Answer a total of four questions with at least one question from each Section All questions carry equal marks.
Examiners: Dr. M. Smyth Mr. D. Leonard Ms. M. Harley
Section A
Q1a Find the Laplace transforms of the following, expressing your answer as a single fraction. (i) 3 ( t t^2 − 2 t + 1) (ii) e −^3 t ( t^2 −5) (4 Marks) Q1b Deduce the Inverse Laplace Transform of the expressions
(i)
s s
s
(ii) ( 1 )( 2 )
2
2
s s
s s (iii) (^2)
s s s
(15 Marks) Q1c Use Laplace Transforms to solve the differential equation
5 x ′( ) t + x t ( ) = 3 t given that x (0) = 0 Evaluate x ′(20) (6 Marks)
Q2 The differential equation governing a free oscillator is given by y ′′( ) t +100 ( ) y t = 0 where y ( t ) is the displacement. Solve this differential equation if the displacement is zero initially and the velocity is 15 ms- initially. What is the period and amplitude of the oscillation? (7 Marks) In the presence of a damping force, the equation becomes
y ′′( ) t + ky t ′( ) +100 ( ) y t = 0 where k is a damping constant. Solve this differential equation for the case where k = 4 and subject to the same conditions i.e. displacement is zero and the velocity is 15 ms-1^ initially. Determine the period of the oscillation in this case and the duration of the oscillations. Draw a sketch to illustrate your solution, labelling the axes appropriately. (9 Marks) State the value of k which will produce critical damping and solve the differential equation in this case, subject again to the conditions y (0) = 0 and y ′(0)^ = 15 Draw a sketch to illustrate your solution, labelling the axes appropriately. Determine the maximum displacement in this case and the time taken to reach this maximum. (9 Marks)
Q3a According to Newton's law of cooling, the rate of cooling of a object is directly proportional to the difference in temperature between the object and that of it’s
surroundings i.e. k ( (^) S ) dt
d
= − where θ is the temperature of the object,
θ (^) S is the temperature of the surroundings and k is a constant A hot object is immersed into a liquid which is maintained at 20 oC. After 20 s the temperature of the object is 87 oC and after 60 s the temperature is 29 oC Determine (i) k (ii) when the the temperature reaches 24 oC (iii) the initial temperature of the hot object
(10 Marks)
Q3b A tank is in the form of an inverted cone of base diameter 2m and height 5m. If h is the height of liquid in the tank at any instant t show that the volume of liquid is
given by 75
h^3 V
=. The tank is emptied through a valve of diameter 10 mm. If the
velocity of the emerging liquid is given by 2 gh , show that
h
dt
dh
h^2 = − 2. 77 × 10 −^3 where g = 9.81ms-
Determine (i) the time taken to empty the tank if it was full initially (ii) the height of liquid in the tank after 1 hour (iii) the time taken to empty half of this conical tank (15 Marks)
Section B
Q4a A survey of CIT students indicated that they spent an average of €82.35 on texts and stationary with a standard deviation €7. Determine the probability that a class of 32 BIS3 students spent less than € on texts and stationary. (5 Marks) Q4b A transport company requires batteries that last on average 20,000 km before replacement. A battery company claims that it can supply batteries that meet this requirement. 35 batteries are purchased and the transport company find that they last an average of 19,600 km with a standard deviation of 1200 km. Test the suppliers claim against the transport company concern that batteries last less than 20,000 km at a 5% and 1% level of significance. (5 Marks)
Q4c A process is producing resistors of nominal resistance 300.0MΩ
8 samples were investigated. The results for Sample No.1 were 300.6, 301.0, 298.8, 300.2, 300.4. Calculate the mean and range of this sample. Mean and range values for the other samples are given in the table.
Sample No. 1 2 3 4 5 6 7 8
X 299.6^ 300.0^ 299.3^ 300.0^ 300.1^ 301.0^ 301.
R 2.4 3.4 1.9 2.8 2.9 3.6 4.
Construct mean and range control charts and comment on them. (10 Marks) Estimate the standard deviation. If the specifications for the resistors are quoted as 300.
- 4.0 MΩ determine the process capability index. Comment on your answer. Are the specifications being met? (5 Marks)
Q6b Solve the set of equations using Gaussian Elimination
Check your answer. (10Marks)
Q7a The relationship between T and t for the data below is V = Eekt where E, and k are constants. Rewrite this equation in linear form Use the Least Squares Method (working to 3 decimal places ) to estimate E and k. Hence estimate t when V = 0.
V 0.082 0.064 0.056 0.046 0.
t 10 20 30 40 50
(15 Marks)
Q7b The following table shows lecture attendance and performance in the final exam of a group of students
Lecture attendance 80 65 68 35 89 62 Final exam mark 65 42 58 25 48 51
Evaluate the linear correlation coefficient and the coefficient of determination and comment on your answers.
(10 Marks)
The coefficient of linear correlation, 2 2 2 2
∑ (^ ). ∑ ( )
x Nx y N y
xy Nxy r
a - 2 b + 3 c - 2 d = - a + 4 b - 2 c + 3 d = 18 2 a + 3 b - 4 c + d = 11 3 a - b - c + 2 d = - 4
