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MMO701S - MATHEMATICAL MODELLING 1 - 2ND OPP - JULY 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: MMO/701S
COURSE NAME: MATHEMATICAL MODELLING 1
SESSION: JULY 2020
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 120 (to be converted to 100)
SUPPLEMENTARY/SECOND OPPORTUNITY
EXAMINERS
PROF. S. A. REJU
MOpDERATOR: | 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)
Discuss mathematical modelling and its process with appropriate illustrated diagram.
[7.5]
(a)
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 N$1000
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 20 years?
[22.5]
QUESTION 2 [30 MARKS]
(a)
Discuss the method of Conjecture in Mathematical modelling.
(b)
Employing the Conjecture method, show that the solution of the
dynamical
[9]
system
Ansi = TA, +b, r#1
(2.1)
Is given by
dy =r*c+—1-br
(2.2)
for some C (which depends on the initial condition).
[8]
(c) Given the following experimental data from a spring-mass system:
Mass
Elongation
50
100
1.200 1.650
150
2.000
200
3.150
250
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. [13]
QUESTION 3 [30 MARKS]
(a)
Consider a drug that is effective in treating a disease if the concentration remains
above 100 mg/L. The initial concentration is 440 mg/L. Laboratory experiments show
that the drug decays at the rate of 18% of the amount present each hour.
(i) Formulate a model representing the concentration at each hour.
(ii) Build a table of values and determine when the concentration reaches 100 mg/L.
[13.5]
(b)
Consider the following table showing the experimental data of the growth of a micro
organism
n
0
Vn
8.2
Ayn
8.7
1
2
15.3
29.2
11.7
16.3
3
4
5
6
45.5
Tiel
120.1
174.6
23.9
52
55:5
85.6
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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 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.
[16.5]
QUESTION 4 [30 MARKS]
(a)
Construct natural cubic splines that pass through the following data points.
Xj 0;1/]2
Vj
0
5
8
[15]
(b)
Consider the following table of data:
xX
1
2.3
3.5
4.5
6.5
7.0
y
a0
3.2
5.5
6.2
4.5
7.3
(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.
[5.5]
(ii) If the largest absolute deviations for the Chebyshev’s criterion and that of the Least
Squares criterion are given respectively by Cg, and ding, define them and then compute
their values including their least bound D to express their relationship for the above data
and the model line.
[9.5]
END OF QUESTION PAPER
TOTAL MARKS = 120
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