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MMO701S - MATHEMATICAL MODELLING 1 - 1ST OPP - JUNE 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 35BAM
LEVEL: 7
COURSE CODE: MMO701S
COURSE NAME: MATHEMATICAL MODELLING 1
SESSION: JUNE 2022
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 130 (to be converted to 100%)
EXAMINERS
MODERATOR:
FIRST OPPORTUNITY
PROF. S. A. REJU
PROF. O. D. MAKINDE
EXAMINATION
QUESTION
PAPER
INSTRUCTIONS
1. Attempt ALL the questions.
2. All written work must be done in blue or black ink and sketches must
be done in pencils.
3. Use of COMMA is not allowed as a DECIMAL POINT.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (including this front page)
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QUESTION 1 [30 MARKS]
(a) (i) Characterise the method of Conjecture in Mathematical modelling.
(ii) Show that the solution of the dynamical system
An41 =TAa, +b, r#1
Is given by
a, =r*ct+ =
(1.1)
(1.2)
for some C (which depends on the initial condition).
[12]
(b) Given the following experimental data from a spring-mass system:
Mass
45
90
135
180
225
270
315
Elongation | 1.30
1.65
2.20 | 3.15
4.20
5.25
6.10
Formulate two different models that estimate the proportionality of the elongation to the
mass, clearly showing how your proportionality constant is obtained for each model (correct
to 4 decimal places).
[18]
QUESTION 2 [30 MARKS]
(a)
Suppose a certain drug is effective in treating a disease if the concentration remains
above 100 mg/L. The initial concentration is 625 mg/L. It is known from laboratory
experiments that the drug decays at the rate of 20% of the amount present each hour.
(i) Formulate a model representing the concentration at each hour.
(ii) Build a table of values (answer correct to 2 decimal places) and determine when the
concentration reaches 100 mg/L.
[12.5]
(b)
Consider the following table showing the experimental data of the growth of a micro
organism
n
0
Vn
10.6
AYy
8.7
1
2
18.3
29.2
11.7
16.3
3
4
5
6
45.5
71.1
120.1
174.6
23.9
52
55.5
85.6
where 7 is the time in days and y,, is the observed organism biomass.
(i) | Formulate a linear model for the above organism and show that the model predicts an
increasing population without limit.
(ii) | Assume that contrary to your model prediction in (i), there is a maximum population
of 320. Hence formulate a nonlinear dynamical system model for the organism
growth using your constant obtained from an appropriate ratio similar to the example
given in class, forn = 3 in the above data.
[17.5]
2/Pag
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QUESTION 3 [40 MARKS]
(a) Consider the following data for bluefish harvesting (in Ib) for the years shown.
Year
Blue Fish
1940
15,000
1945
150,000
1950
250,000
1955
275,000
1960
270,000
1965
280,000
Using 1940 as the base year represented by x = 0 for numerical convenience, construct a
SINGLE TERM MODEL for the fish harvesting and hence predict the weight y (Ib) of the fish
harvested in 2020. HINT: Employ the least squares fit of the model form logy = mx + b for
your procedure, where log is to base 10.
[14]
(b) Consider the following table of data:
X
1
2.3
a5
4.5
6.5
7.0
y
3.5
3.2
5.5
6.2
4.5
72
(i)
Estimate the coefficients of the straight line y = ax + b such that the sum of the
squared deviations of the data points and the line is minimised.
(ii)
State the general normal equations arising from the use of least squares criterion
for your answer in (i) and hence obtain the normal equations from your data.
(iii)
State the MATLAB commands for obtain the parameters q and b.
(iv) If the largest absolute deviations for the Chebyshev’s criterion and that of the
Least Squares criterion are given respectively by Cyqgx and dmgx, define them and
then compute their values including their least bound D to express their
relationship for the above data and the model line.
[26]
QUESTION 4 [30 MARKS]
(a)
A sewage treatment plant processes raw sewage to produce usable fertilizer and clean
water by removing all other contaminants. The process is such that each other 15% of
remaining contaminants in a processing tank are removed.
i. What percentage of the sewage would remain after half a day?
ii. How long would it take to lower the amount of sewage by half?
iii. | How long until the level of sewage is down to 12% of the original level?
[12]
(b)
Consider an annuity where a savings account pays a monthly interest of 1% on the
amount present and the investor is allowed to withdraw a fixed amount of NS$1000
monthly until the account is depleted. What is the solution of the dynamical system
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model for the annuity problem and how much of the initial investment will be needed
to deplete the annuity in 20 years?
[18]
END OF QUESTION PAPER
TOTAL MARKS = 130
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