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PAPER CODE NO.
MATH 122
THE UNIVERSITY
of LIVERPOOL
SUMMER 2005 EXAMINATIONS
Bachelor of Science : Year 1 Bachelor of Science : Year 2 Master of Mathematics : Year 1 Master of Mathematics : Year 2 Master of Physics : Year 1
DYNAMIC MODELLING
TIME ALLOWED : Two Hours and a Half
INSTRUCTIONS TO CANDIDATES
Candidates should answer the WHOLE of Section A and THREE questions from Section B. Section A carries 55% of the available marks. Take g = 9. 81 ms−^2. Give numerical answers to 3 significant figures. You may use
dx
dv v dt
dx dx
dv dt
dv = =.
Page 1 of 6 Continued
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SECTION A
1. Newton’s law of cooling is expressed mathematically as
k ( T TS )
dt
dT = − − ,
where T is the temperature of the cooling body. What is? By defining , determine the differential equation for x. Write down the solution of this equation.
T S
x = ( T − TS )
A cake baked at is removed from the oven and placed on the kitchen sideboard to cool. If the kitchen temperature is a constant , how long does it take for the cake to cool to? [Assume ]
800 C
200 C
30 0 C k = 2 × 10 −^4 s−^1 [6 marks]
2. My credit card company has offered me a loan of £4000 to be repaid over a period of 6 years at rate of 8% per annum. I wish to pay off the loan by paying a fixed amount c each month. Show that the annual rate of interest is equivalent to a monthly rate of 0.643%. Write down the equation for , the balance owed at the end of month
um + 1 ( m + 1 )in terms of and c. If where N is the equilibrium solution of this equation, write down a second equation for in terms of. If after 6 years (72 months) calculate the size of my monthly repayments c and the total amount I will pay during the loan period.
u m x (^) m = um − N
x m x 0 u 72 = 0
[7 marks]
3. Consider two one-way roads that cross at right angles. There are traffic lights at the intersection. Cars arrive at the rate of 3 per minute at one set of lights, which are green for a time mins. At the other set, cars arrive at a rate of 6 per minute and the green light lasts for mins. Both lights are red for one minute at the same time to allow pedestrians to cross. Traffic passes a green light at the rate of 10 cars per minute.
T 1
T 2
Show that to avoid congestion 7 T 1 (^) ≥ 3 T 2 + 3 and 2 T 2 (^) − 3 T 1 ≥ 3. Shade the region in the plane for the times satisfied by these inequalities. If the maximum time the lights can be green is 90 seconds will congestion build up?
T 1 − T 2
[7 marks]
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8. Let and represent the levels of two animal populations governed by
the following coupled differential equations
x^ ( ) t y ( ) t
( ) ( x )
dt
dy y dt
dx = − 2000 = 21000 − 2
Initially x = 1040 and y = 1940. Obtain and solve the differential equation for y in terms of x. From your results draw a phase diagram for this situation, indicating which way around the curve the point ( x , y )moves. [7 marks]
SECTION B
9. Explain what is meant by a stochastic process. [2 marks] Consider a two-state stochastic system. At time t , let P ( A , t )be the probability that the system is in state A and P ( B , t )the probability that it is in state B. Let be the probability per unit time the system goes from state B to state A and let be the probability per unit time the system goes from state A to state B. Write down an equation for
W^ (^ B → A ) W ( A → B ) P ( A , t + δ t ), the
probability the system is in state A at time t + δ t , and show that
P ( At ) P ( B t ) W ( B A ) P ( At ) W ( A B
dt
d
[5 marks] Once a week Mike likes to spend an evening either at the cinema or meeting friends in the pub. If he visits the cinema one week the probability he will do so again next week is 0.4. In contrast if he goes to the pub one week the probability he goes to the cinema the next week is 0.7. If P ( cin , t )and are the probabilities Mike goes to the cinema and pub respectively on week t , show that
P ( pub , t )
+. Pcint =. dt
dP cint .
Solve this equation if P ( cin , 0 ) = 0. 5. In the long term, how many times is Mike likely to go to the cinema during one year (52 weeks)? [8 marks]
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10. In London at the time of the Black Death in 1348 the city was infested with
rats. The rat population x ( t ) × 104 per unit area at time t was thought to grow at
a rate
xy dt
dx = ,
where is the human population at that time. The rats carry the bubonic plague virus which is lethal to humans. As a result the rate of change of the human population was estimated to satisfy
y ( ) t × 104
y ( x )
dt
dy = 2 4 −.
Find the equation for dy dx and integrate it, assuming that at t = 0 , x = 2 and y = 1. [7 marks] Sketch the graph of y against x , indicating the direction that x and y change with time. Describe what happens to the two populations. Calculate the maximum human population before the plague virus takes hold. [8 marks]
11. The rate at which people join a queue is per minute where is
the number of people in the queue at time t. If no one is leaving the queue, what would be the final number in the queue?
10 + 9 n − n^2 n ( ) t
[2 marks] However, people are leaving the queue at a rate of 28 per minute. Show that
n ( ) t satisfies the differential equation
= − ( n 2 − 9 n + 18 )
dt
dn .
What are the possible equilibrium values of n ( t )and which ones are stable?
[6 marks]
Solve the differential equation to find n ( t )given there are four people in the
queue at time t = 0. [Hint: use partial fractions.] [7 marks]
