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CORK INSTITUTE OF TECHNOLOGY INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 1 Examinations 2009/
Module Title: Mathematics for Engineering 402
Module Code: STAT
School: Science
Programme Title: Bachelor of Engineering (Honours) in Mechanical Engineering - Award Bachelor of Engineering (Honours) in Biomedical Engineering - Award
Programme Code: EMECH_8_Y EBIOM_8_Y
External Examiner(s): Mr. J. Reilly Internal Examiner(s): Mr. D. O’Hare
Instructions: Answer any three questions. All questions carry equal marks.
Duration: 2 HOURS
Sitting: Winter 2009
Requirements for this examination: Statistical tables by Murdoch and Barnes.
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct examination paper. If in doubt please contact an Invigilator.
- (a) A company manufactures three models of a certain product. Each model has to go through three different operations: moulding, assembly, and finishing. The table below gives all the relevant information.
Model Moulding (hrs/unit)
Assembly (hrs/unit)
Finishing (hrs/unit)
Profit (€/unit) 1 3 6 10 160 2 2 3 8 120 3 4 6 12 240 Capacity/wk 48 hr 72 hr 160 hr Formulate the problem of finding the product-mix which will maximise weekly profit as an LP problem. Set up the initial Simplex table, perform one iteration of the Simplex method, and comment. (13 marks) (b) (i) Use either the dual simplex procedure or the two-phase method to find the solution to the following problem, if it exists: 1 2 1 2 1 2 1 2
Minimise 2 5 subject to 2x 4 3 6 , 0.
z x x x x x x x
(ii) Verify the solution obtained in part (i) by solving the problem graphically. (iii) Write down the dual of the problem in part (i), and deduce its solution from the final table above. Give two illustrations of the complementary slackness theorem as it applies in this example. (20 marks)
- Consider the following linear programming problem, along with the associated optimal table below (with some entries missing). 1 2 3 1 2 3 1 2 3 1 2 3
maximise 3 7 5 subject to 100 2 3 200 , , 0.
z x x x x x x x x x x x x
- (a) In an experiment aimed at improving the tool life of carbide cutting inserts, stop delay and feed rate were seen as factors potentially affecting tool life. Two stop delay settings and three feed rates were considered. The following data were gathered, with the response variable being the number of tubes cut before a failure occurred. As may be seen from the data set, two responses were recorded for each combination of stop delay and feed rate Feed rate Stop delay 1 2 3 High 125 135 194 129 130 183 Low 136 169 162 141 155 207
The sum of squares for interaction is 562.17. Produce the full analysis of variance table and state your conclusions clearly. (15 marks) (b) A study is conducted on the effect of temperature, time in process, and rate of temperature rise on the amount of dye (in mg) left in the residue bath in a dyeing process. The experiment was run at two levels of temperature (120 0 C and 140^0 C), two levels of time in the process (40 min, 55 min) and two rates of temperature rise (R 1 and R 2 ). Two readings were taken at each combination of factor levels. The resulting set of data is as follows:
Temperature 1200 140 0 40 min 55 min 40 min 55 min Rate R 1 19.9, 18.6 17.4, 16.8 25.0, 22.8 19.5, 18. R 2 14.5, 16.1 16.3, 14.6 27.7, 18.0 28.3, 26. (i) Estimate the error variance. (ii) Calculate an estimate of the temperature-time interaction effect. (iii) Test the significance of the effect estimate obtained in part (ii) using either a t- test or an F-test. (iv) Draw the temperature-time interaction plot, and comment. (18 marks)
- (a) A function f ( ) x is of period 2π and is defined over a cycle by
0 ( ) (^) - 2
x if x f x (^) x if x
π π π π π
−^ ≤^ ≤
Find a Fourier Series for this function.
( ) (^ )^ (^ )
( ) (^ )^ (^ )
2
2
Note: ( - ) sin - (^ -^ ) cos^ sin
( - ) cos (^ -^ )sin^ cos
x nx dx x^ nx^ nx n n x nx dx x^ nx^ nx n n
(13 marks)
(b) A uniform rod is aligned along the x-axis between the points x=0 and x=L. Both ends are maintained at a temperature of 20^0 C. The temperature u(x,t) at any point at any instant is found by solving the partial differential equation
2
2 x
k u t
u ∂
The initial temperature distribution is given by u(x,0)=f(x). By using a substitution
v(x,t)=u(x,t)-
solve this partial differential equation. In particular find the solution where (^) f ( ) x 20 4 x = + (^) L
Note:
2 sin cos 2 2 sin x n^ x^ dx Lx^ n^ x^ L^ n^ x L n L n L
∫ ^ ^ ^ ^ ^
2 cos sin 2 2 cos x n^ x^ dx Lx^ n^ x^ L^ n^ x L n L n L
∫ ^ ^ ^ ^ ^ (20 marks)
