MTH518 Assignment 1
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This is an individual assignment, so it requires individual submission. The date of submission
is Week 5 Wednesday, 7th August 2019, to be submitted before 5pm, either in
lecture or office. Any submission later than this time will incur a penalty of 50% of the
marked score per day unless valid reason (with evidence) is provided.
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The total marks for the assignment is 90 and contributes 2.5% towards the students coursework.
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The mode of submission is either by handing in a solution sheet with complete working or
emailing the scanned assignment to the lecturer/tutors email.
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All workings to the solutions should be provided in the solution sheet which is submitted.
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All solutions should be written in blue or black ink (no pencil or red ink allowed) and all
writing should be easily legible.
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Plagiarism is strictly prohibited and the student will be penalized accordingly. The lecturer
reserves the right to question any student about any solution in the assignment and ask the
student to reproduce any portion of the solution at any moment.
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Exercise.
1.
(Road Network) The figure below shows a network of one-way streets in central Suva. On a
particular day the traffic flow was measured at each of the intersections. The numbers in the
figure represent the average numbers of vehicles per minute through the four intersections.
(a)
Set up and solve a system of linear equations to find x1 , x2 , x3 and x4 .
(b)
Street AD is temporarily closed due to an accident. What will be the average traffic flow
in the other streets?
(c)
What are the minimum and maximum possible flows on each street?
(d)
If all the directions are reversed, how would the solutions change?
(e)
To reduce congestion Suva City Council builds a bypass road joining intersections A and
C. Solve the new system assuming a traffic flow of x5 on the new road in the direction
of D.
(10 + 5 + 5 + 5 + 10 marks)
2.
Find the value(s) of the constant k such that the system of linear equations
x+2 y =
1
kx−2ky =k + 2
has
(a)
No solution.
(b)
An infinite number of solutions.
(c)
Exactly one solution.
(5 + 5 + 5 marks)
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3.
(Temperature Distribution) We devise a simple model for estimating the temperature
distribution on a thin square plate, where the edge are held fixed at certain temperatures. A
grid is constructed on the plate and then the temperatures are estimated at the grid points.
We utilise the “mean value principle” which states that the temperature of a given point in a
body at equilibrium approximately equals the average temperature of nearby points.
Use a system of linear equations to approximate the interior temperatures T1 , T2 , T3 and T4 .
(10 marks)
4.
(Thermal Conductivity of Water) A group of students are required to conduct an experiment to find the thermal conductivity of water. The table below shows the data obtained, where x is the temperature of water in ◦ F and y is the conductivity of water in
BTU/(hr · ft ·◦ F).
x
y
(a)
32
50
100
150
212
0.337 0.345 0.365 0.380 0.395
Use least squares linear regression to find the best-fit line (regression line).
Plot the data points and regression line on the same Cartesian plane.
(c) Find the value of y at room temperature 66◦ F.
(b)
(10 + 5 marks)
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5.
(Beam Deflection) Forces w1 , w2 , and w3 (in pounds) act on a simply supported elastic
beam, resulting in deflections d1 , d2 , and d3 (in inches) in the beam, as shown below.
For our scenario, the measured deflections d and the 3 × 3 flexibility matrix F for the beam
is given below.




0.585
0.008 0.004 0.003
d = 0.640 ,
F = 0.004 0.006 0.004
0.835
0.003 0.004 0.008
Find the stiffness matrix F−1 .
(b) Use the matrix equation d = Fw to find the force matrix w.
(a)
(10 + 5 marks)
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