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MMO702S - MATHEMATICAL MODELLING 2 - 1ST OPP - NOVEMBER 2023 |
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nAmlBIA UnlVERSITY
OF SCIEnCEAno TECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoolof NaturalandApplied
Sciences
Department of Mathematics,
Statistics and Actuarial Science
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
NAMIBIA
T: +264612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION : BACHELOR of SCIENCEIN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BSAM
LEVEL:7
COURSE:MATHEMATICAL MODELLING 2
COURSECODE: MMO702S
DATE: NOVEMBER 2023
SESSION: 1
DURATION: 3 HOURS
MARKS: 225 (To be converted to 100%)
EXAMINER:
MODERATOR:
FIRST OPPORTUNITY:.QUESTION PAPER
Prof Sunday A. Reju
Prof O/uwole D. Makinde
INSTRUCTIONS
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the
examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
6. Use of COMMA is NOT ALLOWED for a DECIMAL POINT.
PERMISSIBLE MATERIALS
1. Non-Programmable Calculator
ATTACHMENTS
NONE
This paper consists of 3 pages including this front page.
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QUESTION 1 (97 MARKS]
(a) Discuss the role of Simulation Modelling as an extension of conventional Scientific
Method with an appropriate diagram to reflect Mathematical modelling as a simulation
technique.
(7 Marks)
(bl Define the Linear Congruential Generator (LCG), and using a seed 3, multiplier 17,
increment 2 and modulus 80, obtain the sequence of fifteen pseudo-random numbers
using the LCG.
(33 Marks)
Define cycling property and state if it occurs in the generated sequence, indicating when it
occurs and the first two cycled pseudo-random numbers.
(3 Marks)
(c) A customised LAN Email-to-Fax application delivers a block of textual data every 10
microseconds (µs). A conversion application checks each data block for conversion errors
and corrects the errors, if necessary, before spontaneous conversion. It takes 1 µs to
determine whether the block has any errors. If the block has one error, it takes 5µs to
correct it and if it has more than 1 error it takes 20µs to correct the error. Blocks are
queued when the converter falls behind. Assume that the converter is initially empty and
that the number of errors in the first 15 blocks are: 1, 0, 3, 1, 0, 4, 0, 1, 0, 3, 1, 2, 0, 2, l.
Construct a data conversion simulation table for the queueing model, showing arrival times,
number of errors, waiting, conversion (service) and departure times.
(45 Marks)
(d) From your simulation table in (c), determine the following performance measures (correct
to 2 decimal places for non-integer numbers):
(12 Marks, 2 Marks each)
(i) Average number of data blocks in the system.
(iii) Maximum data conversion time.
(v) Decoder utilisation time.
(ii) Average block waiting time.
(iv)Decoder busy duration.
(vi) Decoder idle time.
QUESTION 2 [32 MARKS]
(a) AGRIMAN Windhoek produces two farming implements: hoes and shovels and realises a
net unit profit of N$115.50 per hoe and N$120.65 per shovel. Assume that the firm has
up to 130 square metres of iron sheet and 120 metres of wood length to devote to a small
farming project plus a signed contract of supplying 10 hoes and 15 shovels to a Rehoboth
farm during the period of the project. In addition, assume that it requires 2.5 square
metres of iron and 1.65 metre of wood to fabricate a hoe and 1.2 square metres of iron
and 1.85 metre of wood to produce a shovel. Formulate and solve the model for
maximising the firm's profits during the project, stating also how many of each of the
implements will the firm produce for the project apart from the contract. (15 Marks)
(b) (i) Define post-optimality analysis for linear optimisation problems.
(5 Marks)
(ii) Discuss the analysis for change in the firm's profits on hoes, showing all
expressions to support your conclusion.
(12 Marks)
Course Name (Course Code)
1'1 Opportunity November 2023
2
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QUESTION 3 [54 MARKS]
(a) Discuss the Method of Substitution for solving nonlinear optimisation problems.
(2 Marks)
Hence u·sethe method to solve the following problems:
( i)
= Minimise f (x 11 x 2 ) 4xf + Sxi
= subject to 2x 1 + 3x 2 6
(13 Marks)
( ii)
= Maximise z(xi, x 2) 4x 1 - 0.lxf + Sx2 - 0.2xi
= subject to x 1 + 2x 2 40
(14 Marks)
{b)
(i) Solve the problem in (a)(i) above using the Lagrange Multiplier's method.
(10.S Marks)
(ii) State the theorem for necessary and sufficient conditions for optimality for nonlir.ear
optimisation problems and specifically the Karush-Kuhn-Tucker (KKT) conditions within
the theorem.
(14.5 Marks)
QUESTION 4 [42 MARKS]
Suppose a large lake that was formed by building a dam over a river holds initially 100 million
gallons of water. Because a nearby agricultural field was sprayed with a pesticide, the water
has become polluted. The concentration of the pesticide has been measured and is equal to
35ppm (parts per million), or 35 x 10-6.
The river continues to flow into the lake at a rate of 300 gal/min. The river is only slightly
po!luted with a pesticide and has a concentration of 5 ppm. The flow of water over the dam
can be controlled and is set at 400 gal/min. Assume that no additional spraying causes the
lake to become even more polluted.
Forr.--,ulatethe pollution model and hence determine how long will it be before the lake water
reaches an acceptabie level of pesticide concentratior. equal to 15 ppm.
END OF EXAMINATION
TOTAL MARKS:225 (CONVERTTO 100%)
Course Name (Course Code)
l"Opportunity November 2023
3