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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autmn Examinations 2008/
Module Title: Technological Mathematics 311
Module Code: MATH
School: Building & Civil Engineering
Programme Title: Bachelor of Engineering in Civil Engineering – Award Programme Code: CCIVL_7_Y
External Examiner(s): Dr. P. Robinson Internal Examiner(s): Mr. T. O Leary Dr. V. Morari
Instructions: Select any four questions. These questions carry equal marks.
Duration: 2 Hours
Sitting: Autumn 2009
Requirements for this examination:
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) Select any two of parts (i), (ii) and (iii):
(i) A light beam of span 6m is simply supported at its endpoints. At the points x=2m and x=4m there are loads both equal to 36kN. Express the Bending Moment M in terms of step functions. Solve the differential equation EI (^) dxd y 2 M
2
to find the deflection y at any point on the beam. (10 marks)
(ii) Both ends of a light beam of span 6m are embedded in walls. Between the points x=0m and x=2m there is a U.D.L. of 24kNm-1^. Express the Bending Moment M in terms of step functions. Solve the differential equation EI (^) dxd y 2 M
2 = y(0)=y(5)= to find the deflection y at any point on the beam. (10 marks)
(iii) The deflection y at any point on a beam of span 8m is found by solving the differential equation EI (^) dxd y 2 30 [x 4] Rx
2 =− − + where R is a constant. Solve this differential equation where y=0 at x=1 and at x=6. At x=1 the slope of y is zero. (10 marks)
(b) By using Euler’s Method or the Three Term Taylor Method with a step h=0.1 estimate the value of y at x=0.2 where dy (^) x 2 y 2 dx =^ +^ y(0)= Note:
2 k+1 k k k y =y +hy + h y ′ (^) 2! ′′ (^) (5 marks)
- (a) Samples of fifty items were taken at random from the output of a machine and for each sample the number of defective items were counted and are recorded No. of defectives 0 1 2 3 4 > No. of batches 51 25 18 5 1 0
Show that on average 1.6% of items are defective. By using the Binomial Distribution and by using the Poisson Distribution calculate the probability that a random sample of 100 items contains at most two defective items? (11 marks)
(b) The weights of bricks are assumed to be Normally distributed with a mean value of 1.02 kg and with a standard deviation of 0.006kg. (i) Calculate the percentage of bricks that weigh more than 1.033kg. (ii) If 99.8% of bricks have a weight less than some critical weight W find the value of W. (5 marks) (c) To monitor the presence of a certain chemical in a compound a quality control chart is used. Samples of size four were taken and the mass of chemical present in these samples was measured and are recorded below:
Sample 1 2 3 4 5 Mean 40.54 40.51 40.49 40.47 40. Range 0.25 0.30 0.35 0.30 0.
For the data above set up a control chart for sample means. Plot the chart and comment on whether or not that the process is under control. (9 marks)
5 (a) In a computer laboratory on average each student sends three files to a printer during a single hour. By using the Poisson Distribution calculate the probability that a student will send three or more files to the printer during any ten minute period. (6 marks)
(b) The weights of bricks are assumed to be Normally distributed with a mean value of 3.10kg and with a standard deviation of 0.007kg cont//…
(i) A sample of nine bricks is taken. Calculate the probability that the sample mean will lie between 3.095kg and 3.108kg. (ii) If 2.5% are sample means for samples of size ten are greater than some critical value W find the value of W. (6 marks)
(c) (i) Five determinations were made about the density of a particular material 4.09 4.09 4.10 4.10 4. Find the sample mean and sample variance s 2. Find 99% confidence limits for the actual value of the density. (ii) The lengths of blocks produced were assumed to be normally distributed with a mean value of 450.0mm and with a standard deviation of 1.4mm. A new type of mould was introduced and a sample of 100 blocks gave a mean length of 450.3mm. At the 0. level of significance test that the mean length has not changed and the mean has increased. (13 marks)
BINOMIAL DISTRIBUTION P(r)= (^)
r
N (^) prqN-r (^) where p+q=
POISSON DISTRIBUTION P(r)= λ^ r!e
r −λ …where λ is the mean number of occurrences P(r+1)= (^) r+1λ P(r)
NORMAL DISTRIBUTION z= x-μ σ z=σx-μ/ n
CONFIDENCE I NTERVALS Mean x 2 (x-x )^2 n x^22 x= (^) n Sample variance s = (^) n-1 = (^) n-1 ^ n -(x)
α/2 α/2,n- x±z σ^ x±t s n n
H YPOTHESIS TESTING z= (^) σx-μ/ n t= s/^ x-μ n
