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Cork Institute of Technology
Bachelor of Engineering in Biomedical Engineering - Award
(National Diploma in Engineering in Biomedical Engineering - Award)
(NFQ Level 7)
Autumn 2005
Mathematics & Computing
(Time: 3 Hours)
Answer question 1 from Section A. Answer two questions from Section B. Answer two questions from section C. Use a separate answer book for each section. The weighting of Computing: Maths is 15:
Examiners: Prof. P. McHugh Mr. D. Tallon Ms. M. Brennan Dr. T. Creedon Mr. G. Mc Sweeney
Section A
Q1. (a) If the population of a city is approximately 160,000 and the rate of annual growth is 3%
write a program to determine how long it would take for the population to reach 200,000. Design the program so that the user is prompted to enter the current population, the rate of growth and the designated growth. When executed your program might look something like this:
Enter the current population 160000 Enter rate of growth 3 Enter projected population 200000
After 1 year population is 164800 After 2 years population is 169744 After 3 years population is 174836 After 4 years population is 180081 After 5 years population is 185483 After 6 years population is 191047 After 7 years population is 196778 After 8 years population is 202681 You can implement this using any loop of your choice. (10 marks)
(b) Write a program
which prompts a user to enter ten numbers which are then stored in an array.
The program should calculate the total and average of the values entered. When
executed the output might look something like this:
Please enter number 1 4 Please enter number 2 9 Please enter number 3 8 Please enter number 4 6 Please enter number 5 7 Please enter number 6 8 Please enter number 7 6 Please enter number 8 7 Please enter number 9 8 Please enter number 10 8
Total: 71 Average: 7 (10 marks)
(b) The temperature in a workshop and the time taken to complete a standard task are recorded in the following table: Temperature X
Time Y 10.2 11.8 11.9 9.6 12.6 13.7 10.
Calculate the correlation coefficient and interpret your result. (8 marks)
Section C
Q5. (a) Use Euler’s Method to find the approximate value of y ( 0. 2 )for the solution of
y '^ = ( x + y − 1 )^2 given that h = 0. 1 and y ( 0 )= 2. (4 marks)
(b) Test whether the differential equation
( 1 3 2 2 ) y 2 xy^3 dx
dy
is exact.
Hence determine the general solution of the equation. (7 marks)
(c) Use the method of Undetermined Coefficients to find the general solution of
y ''^ − 6 y '+ 5 y =− 10 x^2 − 6 x + 32 + e^2^ x (9 marks)
Q6. (a) Find the Laplace transform of each of the following functions.
(i) (^ ) (^6 )
f t = t^3 − t^4
(ii)
f ( t )= 7 cos 5 t − t^2 e −^3 t
. (5 marks)
(b) Find the inverse Laplace transform of
(i) 4 1
s
s F s (ii) s s
F s 3
= (5 marks)
(c) Use Laplace transforms to solve the differential equation
x ''^ ( t )+ 3 x '( t )+ 2 x ( t )= 10 , x ( 0 )= 0 , x '( 0 )= 2
(10 marks)
Q7. (a) The average number of vehicles arriving at a particular junction is 18 per hour. Assuming the vehicle arrivals form a Poisson distribution, calculate the probability that (i) two or more vehicles arrive in any 10 minute period; (ii) one or more vehicles arrive in any single minute. (5 marks)
(b) A sugar refinery has a processing plant. The amount of raw sugar that can be processed each day has an exponential distribution with a mean of 4.5 tonnes. Find the probability that the plant can process more than 4.5 tonnes in a day. How much raw sugar should be stocked each day so that the chance of the processing plant running out of product is only 0.09? (5 marks)
(c) In the manufacture of a certain chemical product, three measurements are made each week of the percentage acid content. Data for 8 weeks are available as shown below:
Week 1 2 3 4 5 6 7 8 Readings 5.74 6.21 5.28 5.51 5.19 5.91 5.27 5. 5.54 5.18 5.68 5.81 6.21 6.21 6.22 5. 5.97 5.50 5.00 5.75 4.71 4.71 5.85 5.
Set up a control chart for the sample means. Plot the chart and comment on the process. (10 marks)
- Sample size n A 0 '. 025 A 0 '. - 2 1.229 1. - 3 0.668 1. - 4 0.476 0. - 5 0.377 0. - 6 0.316 0. - 7 0.274 0. - 8 0.244 0. - 9 0.220 0.
- 10 0.202 0.
- 11 0.186 0.
- 12 0.174 0.
