QUESTION 1 (40 MARKS]
Let a mass-spring system of mass of 2.5kg with natural length 0.54m be stretched to a length
of 0.76m and released by a force of 20.5N while it is immersed in a fluid of damping constant
= C 42.
(a) Formulate the general model differential equation for the undamped system and find
the position of the mass at any time t if it starts from the equilibrium position and is
given a push with an initial zero velocity, stating all relevant physical laws.
(Your answers correct to 2 decimal places).
(25 Marks)
(b) Formulate the general model differential equation for the damped system, assuming
that the damping force is proportional to the velocity of the mass and acts in the
direction opposite to the motion. Then obtain only the general solution without using
the initial conditions.
(15 Marks)
QUESTION 2 (31 MARKS]
(a) Consider a local company that produces bowls and mugs and assume that that per unit
profit contribution for bowls is given by ($4 - 0.1x 1) and that per unit profit
contribution for mugs is given by ($5 - 0.2x 2 ).
Formulate and a nonlinear profit maximization problem from the above subject to just a
+ = labour constraint given by x 1 2x 2 40 hours
(15 Marks)
(b) Consider the following data
X
1.2
1.5
2.0
2.6
3.2
4.5
5.2
5.7
6.0
6.8
y
1.1
1.3
1.6
2.0
3.4 4.1
3.2
4.5
2.5 5.2
Obtain the normal equations for f(x) defined by (a) above using the above data.
(16 Marks)
QUESTION 3 (115)
(a) Define the Middle Square Method for generating pseudo-random numbers. Hence
using a seed 642246, obtain ten pseudo-random numbers by the method. (26 Marks)
Is there cycling? (YES/NO). If so, when does it occur?
(1 Mark)
(b) Consider a single server seaport freight system model where seven vessels dock at a
seaport to unload cargo according to the following time data (in minutes):
Vessels
Vessel 1 Vessel 2 Vessel 3 Vessel 4 Vessel 5 Vessel 6 Vessel 7
Random Inter-
18
55
65
185
212
40
35
Arrival Times
Cargo Unloading
55
45
60.5
75
80
70
90
Duration
Course Name (Course Code)
1" Opportunity November 2023
2