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MMO701S - MATHEMATICS MODELLING 1 - 2ND OPP - JULY 2023 |
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n Am I B I A u n IVE Rs I TY
OF SCIEnc E Ano TECHn OLOGY
FACULTYOF HEALTH,NATURAL RESOURCESAND APPLIEDSCIENCES
SCHOOLOF NATURALAND APPLIEDSCIENCES
DEPARTMENTOF MATHEMATICS, STATISTICSAND ACTUARIALSCIENCE
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSAM
LEVEL: 7
COURSE CODE: MMO701S
COURSE NAME: MATHEMATICAL MODELLING 1
SESSION: JULY 2023
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 120 (to be converted to 100)
SUPPLEMENTARY/SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINERS
PROF.S. A. REJU
MODERATOR: PROF.0. D. MAKINDE
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 [25 MARKS]
(a) Discuss mathematical modelling and its process with appropriate illustrated diagram.
[9]
(b) State the method of Conjecture in Mathematical modelling and employ the method
to show that the solution of the dynamical system
is given by
(1.1)
(1.2)
for some c (which depends on the initial condition).
[16)
QUESTION 2 [35 MARKS]
(a) Consider an annuity where a savings account pays a monthly interest of 1.2% on the
amount present and the investor is allowed to withdraw a fixed amount of N$1200
monthly until the account is depleted. What is the solution of the dynamical system
model for the annuity problem and how much of the initial investment will be needed
to deplete the annuity in 22 years? State all appropriate theorems used in your
solution.
[24)
(b) Given the following experimental data from a spring-mass system:
Mass
50
100 150 200
250
Elongation 1.200 1.650 2.000 3.150 4.200
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. [11)
QUESTION 3 [30 MARKS]
(a) Construct natural cubic splines that pass through the following data points.
[15)
(b) Consider the following table of data:
X
y
153..55 I 2.3
I 4.5 I 6.5 I 7.0
3.2
6.2
4.5
7.5
(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.
[SJ
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(ii) If the largest absolute deviations for the Chebyshev's criterion and that of the Least
Squares criterion are given respectively by Cmax and dmax, define them and then compute
their values including their least bound D to express their relationship for the above data
and the model line.
[10)
QUESTION 4 [30 MARKS]
(a) Suppose a certain drug is effective in treating a disease if the concentration remains
above 120 mg/L. The initial concentration is 645 mg/L. It is known from laboratory
experiments that the drug decays at the rate of 25% of the amount present each hour.
(i) Formulate a model representing the concentration at each hour.
[2]
(ii) Build a table of values (answer correct to 2 decimal places) and determine when the
concentration reaches 120 mg/L.
[12)
(b) Consider the following table showing the experimental data of the growth of a
micro-organism.
n0
Yn 8.2
!J.yn 8.7
1
2
15.3 29.2
11.7 16.3
3
45.5
23.9
4
71.1
52
5
120.1
55.5
6
174.6
85.6
where n is the time in days and Yn is the observed organism biomass.
(i) Construct a linear model for the above organism growth 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 665. Hence formulate a nonlinear dynamical system model for the organism using
your constant obtained from an appropriate ratio similar to the example given in class,
= for n 3 in the above data.
[16)
END OF QUESTIONPAPER
TOTAL MARKS= 120
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