THE UNIVERSITY OF NEWCASTLE: SCHOOL OF ENGINEERING
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MECH2450 Engineering Computations 2 Assignment #1 (2018: 2024 – Write My Essay For Me | Essay Writing Service For Your Papers Online)
Numerical Analysis
Q.1 (From Chapter 1 of Numerical Analysis)
The diagram above shows two corridors of widths a and b (a b) which meet at right angles. The
longest rectangular object (desk or piano etc) of width w (w < a) that can be manoeuvred around the
corner has a length l, given by:
2 2
l mb a w 1 m 11/ m (1)
provided that the object cannot be moved vertically. The value of m can be found from the condition that
in the interval [0, (a/b)1/3 ] m is the only root of the equation:
2 0
2 2 6 2 4 3 2 2 2 2 b w m w m abm w m a w (2)
(a) Write a program using MATLAB to solve Eq. (2) by the Regula Falsi method, and hence find m and
l. As a test, for a = 50, b = 70, and w = 40, l = 87.4723. Discuss how the initial points should be
chosen.
(b) For thin objects like a ladder, w 0. When w = 0 show that the method will not work if the interval
[0, 1] is searched. Explain why this happens. Hint: solve Eq. (2) analytically for w = 0.
(30 marks)
Q.2 (From Chapter 2 of Numerical Analysis)
(a) Write a program that implements the LU factorisation algorithm 2.3, with partial pivoting, given in
the Text and in the supplementary section with MATLAB versions of the algorithms. Your code
should have separate functions for the factorization, forward and back-substitution steps. Check
your code by analysing the problem solved on page 41 of the Text:
l w
b
a
12
2
11
3 1 2
2 2 1
1 2 3
3
2
1
x
x
x
(b) (From Chapter 3 of the Numerical Analysis) Implement the least squares fitting algorithm 3.2
given in the Lecture Notes and in the supplementary section with MATLAB versions of the
algorithms. Use your LU factorisation program (from part (a) of this question) to solve the linear
equations and a separate function to evaluate the various expressions. Check your program by
analysing the problem given on pages 67-68 of the Text.
(c) As an example for an application, you are given the following x-y data representing student
attendance at lectures in a particular course:
point x (week) y (no of students at
lecture)
1 1 64.0
2 2 68.0
3 3 60.0
4 5 55.0
5 6 62.0
6 9 64.0
7 10 52.0
8 11 57.0
Use your program to find the coefficients for a least squares fit to these data using the expansion:
2 3
0 1 2 3 f x a a x a x a x ( )
Write your program in MATLAB so that it prints the value of f x( ) at each of the x data points. How
well does f x( ) fit the data?
(30 marks)
Probability
Q.3 (From Chapter 3 of Probability and Statistics)
A consulting engineer must meet a deadline for a project consisting of two independent phases:
(a) Field work – if weather conditions are favourable, the probability that the field work will be
completed on schedule is 0.95. Otherwise, the probability of on-schedule completion is reduced to
0.45. The probability of unfavourable weather is 0.65.
(b) Computations – two independent computers are available to perform the required calculations.
Each computer has a reliability of 95% (i.e. the probability of working is 0.95). If only one of the
computers is working, the probability of completing the computations on time is 0.55, whereas, if
both computers are working, this probability increases to 0.9. Furthermore, if both of computers are
not working, the engineer must perform his/her calculations using desk calculators which are 100%
reliable, but will decrease the probability of completing the computations on time to 0.35.
(i) What is the probability that the field work will be completed on schedule?
(ii) What is the probability that the computations will be completed on time?
(ii) What is the probability that the engineer will meet his/her deadline for the project?
(10 marks)
Q.4 (From Chapter 5 of Probability and Statistics)
The size (in millimetres) of a crack in a Pontiac Trans Am’s front sub-frame weld is described by a
random variable X with the following PDF:
elsewhere
x
x x
f x X
0
1/ 4, 4 6
/16, 0 4
( )
(a) Sketch the PDF and CDF.
(b) Determine the mean crack size. (c) What is the probability that a crack will be smaller than 3 mm?
(d) Determine the median crack size. (e) Suppose there are four cracks in the weld. What is the probability that only one of the four cracks is
larger than 3 mm?
(15 marks)
Q.5 (From Chapter 5 of the Probability and Statistics)
A “heavy rain” is defined as rain with a rainfall over 50 mm. Let X be the amount of rainfall in a heavy
rain. The cumulative distribution (CDF) of X in a given town is:
0 for 50
for 50 50 ( ) 1
4
x
x
x
F x X
(a) Determine the median of X;
(b) Determine the PDF.
(c) What is the expected amount of rainfall in the town in a heavy rain?
(d) Suppose a flood condition is defined as a rain with over 75mm of rainfall. What is the percentage of
heavy rains representing flood condition?
(e) Suppose the probability that the town will experience 0, 1, and 2 heavy rains in a year is 0.5, 0.4,
and 0.1, respectively. Determine the probability that the town will not experience a flood condition
in a given year. Assume the amounts of rainfalls between the rains are statistically independent.
(15 marks)
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