Exam Instructions

The questions are of equal value.

5% of your mark for this assessment is comprised of your presentation and explanation of your work.

Mandatory Items are marked **.

To ensure academic integrity, submissions missing mandatory items will have no marks recorded for the assessment.

i)  You are to complete the test under standard exam conditions, with hand- written solutions.

•  Make an electronic copy by, for instance, photographing, scanning, or using a tablet to handwrite the solutions, and saving as image or PDF ile(s).

•  It is suggested that you clearly number all the pages of your solution.

ii)  Record a video presentation explaining your solutions.

** You must appear at some point in the video and introduce yourself.

•  Video and audio in your video submission must be sucient to deliver the information, but presentation marks will not be awarded based on the video quality.

•  The video can be split across multiple parts in separate iles, but you must speak in every part.

•  The video should include:

–  Your handwritten work.

–   A voiceover/video explaining the work you are presenting indi- cating the steps and reasoning.

iii)  ** Create a ile that includes a photograph of your student  ID  card with the signed, handwritten originality statement:

“I declare that this submission is entirely my own original work.”

iv)  Submission:

•  Upload your handwritten solutions as pdf or separate image ile(s).

•  Upload your originality statement ile**.

•  Upload your video ile(s), including your introduction**.

•  You can delete and/or reload iles until the deadline.

•  Make sure you submit all your answers.

Dimensions of Some Physical Entities in the MLT System

Start a new page clearly marked Question 1

1.   Consider a two tank system as shown below. Tank T1  initially contains 100L of water in which 90g of salt are dissolved.  Tank T2  initially contains 60L of pure water.  Liquid is pumped through the system as indicated, and the concentrations in each tank are kept uniform by stirring.

i)  Find the volumes of liquid in Tank T1   and Tank T2 , V1 (t) and V2 (t), respectively, as a function of time, t, in minutes.

ii)  Find the concentrations of salt y1 (t) and y2 (t) in Tank T1  and Tank T2 , respectively, where t is time in minutes.

iii)  What happens to the volume of liquid in each tank, and concentrations of salt in each tank in the long term?

Start a new page clearly marked Question 2

2.   In mining operations it is important to predict the size of the crater result- ing from an explosion in the soil.  Assume that the craters are geometrically similar.

i)  Consider the case where the crater size depends on three variables: the radius of the crater, r , the density of the soil, ρ, and the mass of the explosive, W .

Find a dimensionless product for this system and thus propose a model for the volume of the crater created by the explosion.

ii)  If the amount of explosive is very large, it has been proposed that gravity plays a key role in the explosion process.

Consider the case where the crater size depends on four variables: crater radius, r , density of the soil, ρ, gravity, g , and the energy of the explosive, E .

a)  Find a dimensionless product for this system and thus propose a model for the volume of the crater created by the explosion.

b)  Two identical explosions are set of – one on the Moon and one on Earth.

Stating any assumptions you make, predict the volume of a crater on Earth, compared to that on the Moon, which has 17% of Earth’s gravity.

Start a new page clearly marked Question 3

3.   In some experiments on competition between two species of hydra, it was found that coexistence was only possible if a fraction of the population of each species was removed at regular intervals.  A model for the system with this experimental manipulation is given by

dN1dt = T1N1K1  (K1 — m1 — N1 — aN2 )

dN2dt = T2N2K2  (K2 — m2 — bN1 — N2 )

i)  Explain the variables and parameters of the model.

ii)  Consider the system with K1  = 100, K2  = 90, a = 1.2, b = 0.8, m1  = 2, and m2  = 10.

a)  Show that for these parameters stable coexistence occurs.

b)  Sketch a phase plane portrait showing the nullclines,  equilibrium points and the direction of motion on or across the nullclines as time advances for these parameters.

iii)  Show that if the parameters in part (ii) are changed so that m1  = m2  = 0, that stable coexistence does not occur. Why?