2024 Assignment (MA1608)
General information
Your mark for this assignment will contribute 50% to your inal mark for the MA1608 modular block. The assignment consists of 3 problems. In your answers, you may use any material that we have covered in lectures, but you should always give your own account.
The total number of marks available is 100, including 15 marks for the quality of the text, typesetting and the general presentation. There are no restrictions to the length of your report, but try to be concise and avoid including unnecessary details. When in doubt, please ask. You need to reference the non-trivial information you use (which book, website or class notes, who did you ask etc.). What is important is to be candid about where your information comes from, not to let the reader believe you did something you did not.
Note that the values of the various parameters in the assignment are determined
individually for you on the basis of your student number. In the assignment,
the last two digits of your student number are called W and Z.
For example, if your student number is 1987650, then the last two digits are 50, and in this case W = 5 and Z = 0. You must use the numbers corresponding to your student ID.
The assignment must be typewritten, using an equation editor for mathe- matics. Although the report must be submitted via WISEFLOW online as a main single PDF document, it should be prepared in a form suitable for printing. In particular, it must be optimised for A4 sized paper; the minimum font size is 12pt.
If you need to submit any other iles, they have to be uploaded as a WISEFLOW attachment in the form of a SINGLE zipped ile. For example, if you do some calculations by using Matlab, or want to submit some appendices with a scan of your calculations, then the corresponding iles have to be uploaded as a single zip ile.
The DEADLINE to submit your report via WISEFLOW is
14:00 (London time) on Friday, 10th May 2024.
Please note that the University enforces a very strict Policy for Coursework Submission, as detailed here:
https://students.brunel.ac.uk/documents/Policies/Coursework-Submission-Policy.pdf
Please remember that this is an individual assignment. Group discussions are healthy and encouraged; however, the work you submit must be your own and MUST NOT BE prepared in collaboration with others. Please familiarise yourself with the university’s guidelines to students on the use of AI, see
https://students.brunel.ac.uk/study/using-artiicial-intelligence-in-your-studies
Misconduct in assessment is taken very seriously by the University. You are expected to abide by Senate Regulation 6 - Student Conduct (Academic and Non-Academic), see
https://www.brunel.ac.uk/about/administration/governance-and-university-committees/senate-regulations
Advice on understanding what plagiarism and collusion are and how they can be avoided can be found here: https://www.brunel.ac.uk/life/library/SubjectSupport/Plagiarism
Assignment Problems
Problem 1 [20 marks for modelling and calculation + 5 marks for presentation]
Suppose that you are planning to participate in a triathlon competition that is organized on a lake and consists of cycling, swimming and running. The lake has the form of a semicircle of diameter 6 km.
. The irst stage (cycling) starts at the corner F of the lake, and you should cycle around the lake to the transition area T that is closer to K than to F. The distance from T to the nearest point on the straight shore is (1.8 + 0.1 · Z) km, where Z is the last digit of your student number.
. The second stage is swimming from T to the transition area at any point C on the straight shore of the lake provided the distance that you need to run at the inal stage is at least 1 km.
. The inal stage is running from the transition area C to the inish point F.
Suppose that you can cycle at (30 + W) km per hour, swim at (1 + W/5) km per hour and run at (8.8 + Z/5) km per hour, where W and Z are the last two digits of your student number. The total time of the race includes cycling time, swimming time, running time, as well as the transition times from cycling to swimming (1.5 minutes) and from swimming to running (1 minute).
(a) Find the location of point C that minimizes the total time of the race.
(b) Find the minimum total time of your race. The inal answer should be given in the form hours:minutes:seconds.
Problem 2 [35 marks for modelling and calculation + 5 marks for presentation]
Suppose that a drug can be either administered to a patient by an injection or be fed continuously into their bloodstream by intravenous infusion. Assume an instantaneous rise in concentration whenever a drug is administered. It is known that the rate of decrease of the concentration of the drug in the bloodstream at any time is proportional to the current concentration. It is also known that the highest safe level of the drug concentration is ch and the lowest efective level is cL .
Assume that
cL = (4 + W/2) mg/litre, ch = (20 + W) mg/litre. (1)
where W is the second to last digit of your student number.
Consider the following two treatments of the same patient.
(a) In the irst treatment, the irst dose corresponding to (15 + W/2) mg/litre rise in concentration is administered to a patient by injection. It was measured that the concentration of the drug in the patient’s bloodstream reduces by (23 + Z)% of its initial value 2 hours after the initial injection was given. Here W and Z are the last two digits of your student number.
i. Find the concentration as a function of time t before the second injection.
ii. Find the maximum time when the second injection is to be given to prevent the concentration decreasing below cL .
iii. Suppose that the dose of the second injection is the same as the irst dose. How long does it take from the treatment start for the concentration to decrease to cL again?
(b) In the second treatment, several weeks after the irst, the same drug is fed to the patient. The rate of change of drug concentration is now determined by the continuous infusion of the drug at the rate Q(t) and by the discharge of the drug from the body proportionally to the current concentration, as in the irst treatment (a). Suppose that the drug in the second treatment is provided during 2N hours as follows:
. during the irst N hours, Q(t) = Q0 , where Q0 is a constant;
. during the next N hours, Q decreases from Q0 to 2/1Q0 (assume a linear depen- dence of Q on time t);
. there are no other injections after 2N hours,
where N = 8 + Z, and Z is the last digit of your student number.
Suppose that after the irst N hours, the concentration is 0.9 · c hmg/litre, where ch is given in (1).
i. Write a linear diferential equation and the corresponding conditions that can be used for modelling variation of the concentration with time.
ii. Solve your ODE for the irst N hours, and ind the value of Q0 .
iii. Solve your ODE for the second N hours. Find the concentration at the end of this period.
iv. Determine the maximum of the concentration during these 2N hours and check
whether it ever exceeds ch, the highest safe level of the drug concentration.
Problem 3 [30 marks for modelling and calculation + 5 marks for presentation]
Consider the diferential equation
as a model for a ish population, where
while W and Z are the last two digits of your student number.
Suppose that tis time measured in months, P = P (t) is the size of the population measured in units (one unit is equal to ten thousand ish), and hP is a harvesting rate proportional to the population P , where h is a constant.
(a) For what value of P is the unharvested population (h = 0) growing most rapidly?
(b) i. Determine the value(s) of h for which equation (2) has exactly one equilibrium solution.
ii. Find the nonnegative equilibrium solution(s) of the diferential equation (2) for the value(s) of h that you found in part (b)i and determine whether they are stable, unstable or semistable.
(c) Suppose that the present size of the ish population is Pc = (Z + 6)2 units, where Z is the last digit of your student number. Determine the values of h, for which the population is increasing.
(d) Suppose that the present size of the ish population is P0 = 0.8α units, where α is deined in (3), and there is no harvesting, that is, h = 0. Depending on what is relevant, determine
. how long does it take for the population to increase by 10 %,
or
. how long does it take for the population to become extinct.
In parts (a)-(c), you need to analyze the diferential equation (2) without solving it. You need to solve the equation in part (d) only.