QUESTION 1 [18 MARKS]
(a) What is the usefulness of simulation in Mathematical modelling? Hence discuss Monte
Carlo methods of simulation.
(3 Marks)
(b) Describe the Monte Carlo procedure for plotting the area between the quarter of a
circle below
and the following ellipse.
fl 2 +&] 2 =3
Then sketch the region of focus.
(10 Marks)
(c) Study the following MATLAB code and hence correct and edit it for plotting the region
described in (a) using 42,000 random numbers when the minor and major radi of the
ellipse are respectively 1 and 2, while the circle radius is 5; and computing the value of
the region.
(S Marks)
r=5; a=2; b=1; N=100; A=0; for i=1:N
p=r*abs(rand(1,2)); x=p(1);y=p(2); q1=((x/a).2)+((y/b).2);
if (q1>=1.00 & q2<=r)
A=A+1; plot(x,y,'*'); hold on;
q2=sqr((x.42)+(y.42));
QUESTION 2 [32 MARKS]
(a) Consider a small-scale engineering firm that produces two farming implements: hoes and
shovels and realizes a net unit profit of NS125 per hoe and NS$140 per shovel. Assume
that the firm has up to 250 square metres of iron sheet and 200 metres of wood length
to devote to a farming project plus a signed contract of supplying 10 hoes and 15 shovels
to a family farm during the period of the project. Moreover, it requires 2 square metres
of iron and 0.65 metre of wood to fabricate and 3 square metres of iron and 0.85 metre
of wood to produce a shovel. Formulate and solve the model for maximising the firm’s
profits for hoes and shovels.
(8 Marks)
(b) Define post-optimality analysis for linear optimisation problems and hence discuss the
analysis for change in the firm’s profits on hoes, showing all expressions to support your
conclusion.
(10 Marks)
(c) Consider the following production profit maximisation model:
Maximise f (x1, x2) = 25x, + 30x,
subject to
20x, + 30x, < 690
2.4)
Sx, + 4x, < 120
x, 24
x, 22
Page 2 of4