Instruction: Submit your answers and codes to SEPARATE links on iSpace. Late or
incorrect submissions will not be graded. Each question carries the same weight.
1. (Mixing tanks) Two very large tanks A and B are each partially lled with 100 gallons of
brine. Initially, 100 pounds of salt is dissolved in the solution in tank A and 50 pounds
of salt is dissolved in the solution in tank B. The system is closed in that the well-stirred
liquid is pumped only between the tanks, as shown in Figure 1.
Figure 1: Container within a container.
(a) Use the information given in the gure to construct a mathematical model for the
number of pounds of salt x1(t) and x2(t) at time t in tanks A and B, respectively.
(b) Find a relationship between the variables x1(t) and x2(t) that holds at time t.
Explain why this relationship makes intuitive sense. Use this relationship to help
nd the amount of salt in tank B at t = 30 min.
2. (Newton’s Law of cooling/warming) As shown in Figure 2, a small metal bar is placed
inside container A, and container A then is placed within a much larger container B. As
the metal bar cools, the ambient temperature TA(t) of the medium within container A
changes according to Newton’s law of cooling. As container A cools, the temperature of
the medium inside container B does not change signi cantly and can be considered to be
a constant TB. Construct a mathematical model for the temperatures T(t) and TA(t),
where T(t) is the temperature of the metal bar inside container A. Find a solution of
the system subject to the initial conditions T(0) = T0, TA(0) = T1. (Assume here that
the temperature, T(t), of the metal bar does not a ect the temperature, TA(t), of the
medium in container A)
Figure 2: Container within a container.
3. (Electoral system / Voting theory) An electoral system is the system that determines how
elections and referendums take place and how their results are arrived at. Di erent rules
of voting may sometimes lead to di erent winner(s) even if the preferences of constituents
remain the same.
Consider the following voting rules (assume only a single winner)
1. Plurality method: A single round of election is held. The candidate who polls the
most among their counterparts is elected.
2. Single runo method: A preliminary round of election is held and two candidates
who receive the most number of votes advance to the nal round. The nal round
is then based on the plurality method. The winner of the nal round is chosen as
the elect.
3. Instant runo method: Multiple rounds of election are held. In each round, the
candidate who receive the least number of votes is eliminated. When there are only
two candidates remaining, use the plurality method to determine the elect.
4. Coombs method: Multiple rounds of election are held. In each round, the con-
stituents are required to rank every candidate in terms of their preferences. The
candidate who received the most number of votes for the last place is eliminated.
When there are only two candidates remaining, use the plurality method to determine
the elect.
5. Borda count: A single round of election is held. The constituents are required
to rank every candidate in terms of their preferences. For each vote casted, the
candidate in the last place gets 1 mark, the candidate in the second last place gets
2 marks, ..., the candidate in the rst place gets n marks (n is the total number of
candidates). The candidate who gets the highest mark is the elect.
(a) Consider the following voting results from 30 students on their favourite sports team.
There are 4 teams and each student ranks them from the rst (most like) and the
fourth (least like). For example, 11 students vote B as the rst, D as the second, C
as the third and A as the fourth. Assume that the voters’ preferences do not change
if multiple rounds of voting are necessary. For each voting rule above, determine the
winner.
Votes 11 10 9
First B C A
Second D D D
Third C A C
Fourth A B B
(b) Investigate the 2002 French presidential election. Which rule was used? Write a
note to discuss the e ect if 1% of all voters for Jean-Marie Le Pen instead voted
for Jacques Chirac in the rst round (other votes remain unchanged - see the tables
below).
You will need to get additional information on your own to draw the conclusion.
Table 1: Real result ( rst round)
Candidate % Rank
Jacques Chirac 19.88% 1
Jean-Marie Le Pen 16.86% 2
Lionel Jospin 16.18% 3
... ...
Table 2: Hypothesised result ( rst round)
Candidate % Rank
Jacques Chirac 20.88% 1
Lionel Jospin 16.18% 2
Jean-Marie Le Pen 15.86% 3
... ...
4. (Secretary problem / Marriage problem) Use simulation to demonstrate the secretary/marriage
problem (refer to case study 5) with N = 10 and N = 50 candidates, respectively. For
each N, obtain the probability that the best candidate is chosen pN(k) for k = 1;2;:::N.
For each value of k, perform. su cient number of independent runs such that the 95%
con dence interval with Ur 0:01 is obtained for pN(k). Use your simulation results to
plot a gure (see Figure 3 as an example, the vertical error bar at each point represents
con dence interval) to illustrate the relationship between k and pN(k).