### SAMMC2019 problems

SAMMC2019 Problem A: Fly Another Day Cape Town Movie Studios has won the bid to film one of the exciting action scenes for the new James Bond film. Your team has been subcontracted to model the stunt and assess its safety. The stunt involves Mr Bond driving his car (an Auston Martin DB11) along a sea-side cliff top, launching off a ramp, flying through the air between two hovering helicopters, and landing safely on a moving container ship. (Assume Q has rigged a small parachute to the back of the car to rapidly decelerate once Bond has touched down.)

Your first job is to model the trajectory of the flight and determine the required specification of the ramp and initial velocity so that the car avoids the helicopters, and lands at a suitable angle (i.e., not upside down!) on the ship. Include as much physics as you feel is relevant and state and justify any assumptions you make (such as the location of the centre of mass of the car). You must assess the safety and feasibility of the stunt in terms of various unknowns and assumptions in the model. If you think these are outside acceptable margins, suggest modifications of the scene to improve matters.

Additionally, the studio wishes to impress the producers by making the stunt even more exciting in the hope of winning future contracts. Investigate the adjustment of your model to include a backward flip (or two!) before landing safely on the container ship (for example, by using suitably designed ramp or some alternative). The studio is also open to other ideas you might have to make the scene more incredible.

As part of your final report you should include a brief letter written jointly to the producers of the film and their insurance underwriters describing how you modelled the stunt and your assessment of its safety and success. Pay particular attention to any deviations you made from the initial schematic for purposes of feasibility/safety/awesomeness and justify these. )

SAMMC2018 Problem B: The Art Gallery Security System An art gallery is holding a special exhibition of Eastern European art pieces. The exhibition will be held in a square room that is 20 meters long and 20 meters wide with entrance and exit doors each 2 meters wide as shown below. The gallery makes use of the existing walls as well as portable walls to display the artworks. For the specific exhibition, they have three portable walls of 5m each and a semi-circular portable wall with a 5m diameter available. They are also using a pedestal (50x50x100cm) to display a Fabergé egg. The current layout is given in the figure below.

Firstly, the gallery wants to make use of the minimum number of security cameras to ensure that all the artworks can be monitored. They have two cameras to their disposal: 1) a short ranged (5 m), wide angle (120 degrees) camera and 2) a long ranged (15 m), narrow angle (50 degrees) camera. Advise the gallery on how many cameras they should fix where to ensure that all the artworks are monitored. Due to the value of the Fabergé egg and the painting “Morning in the Pine Forest”, these two pieces should be monitored by at least three and two cameras, respectively.

After your initial findings, the director of the art gallery has expressed some concern over the cost of the security of the exhibit and has asked the management to analyse the security system and rearrange the portable walls to minimize the cost of the security of the exhibit.

As part of you final report you should include a brief letter to the director about your findings and make a suggestion regarding the optimal placement of the portable walls.

SAMMC2019 Problem C: Catch of the Day A large fishing company reconfigures their processing and food production plant based on the forecast of the mixture of fish brought in by various vessels. Small fish are used to make fish fingers and larger fish are used to make fillets. To make efficient use of the processing machinery at their disposal an accurate prediction of the mixture and amount of fish per vessel is crucial.

The company has approached your team to improve the accuracy of their prediction based on production data of previous years. At the moment it is assumed that the following week’s catch will be identical to the current week’s catch. This is however not accurate and does not take into account factors such as seasonality and different vessels’ fishing abilities, capacities, and docking times.

The data provided consists of the amount of fish caught each day classified according to their size, the name of the vessels that landed the fish, and some additional information. The fish are further grouped into categories small, medium, large, and fresh fish (FF). A sample of the data can be found here: https://appliedmaths.sun.ac.za/sammc/problemC2019DummyData.xlsx.

Your team should develop models for predicting the volume and size distribution of the catches arriving at the processing plant for one week in the future. As part of your final report you should include a brief letter to the board of directors describing your model and illustrate the improvement of your forecast to the current situation.

### SAMMC2018 problems

SAMMC2018 Problem A: Zip it! In a bid to counter dropping tourism following recent droughts, the city of Cape Town is considering the installation an aerial zip line from the top of Table Mountain to either the top of the nearby Lion's Head or Signal Hill, or a maybe even a "splash down" landing in to the water off of Camps Bay beach. You have been contracted develop a suitable mathematical model of the proposed zip line(s), to assess the feasibility of the project, and give recommendations on its design. (An engineer has been contracted to consider most of the construction details.)

As part of your final report you should include:
• a one-page summary for the Mayor of Cape Town

SAMMC2018 Problem B: STOP, wait, ..., GO! A busy two-way highway is reduced to a single lane via STOP-GO boards for a distance of L kilometres. Determine an optimal strategy (for example, in terms of the waiting time T, the number of queuing cars Q, or some combination of both) for when to change the direction of traffic in order to minimise delays.

As part of your final report you should include:
• a one-page summary for the provincial Minister of Transport and Public Works
Possible extensions:
• consider the case of a sequence of STOP-GOs
• consider the case of a three-way STOP-GO (e.g.,, where there is a T-junction within the closed section.)

SAMMC2018 Problem C: Home field advantage In many sports the idea of "home field advantage" describes the potential benefit a home team gets over the away team. The advantages may be physical (for example, differing climates or weather conditions, different pitch sizes, or fatigue from travel) or psychological (typically due to increased support from fans). There has been much debate recently over the effect of home field advantage in Super Rugby (particularly with the introduction of teams from Argentina and Japan). The CEO of Super Rugby has hired you to quantify the effect this has on the competition, and possibly suggest some remedies (such as introducing a bonus point for an away win or draw).

As part of your final report you should include:
• a one-page lay summary to the CEO of Super Rugby explaining your findings.
Historical Super Rugby results and standings can be found here.

### SUMMC2017 problems

Problem A - Shooting Hoops: When a basketball player is fouled in the act of shooting a basket they are awarded a "free throw" from approximately 2 metres from the hoop. Different players have different strategies for free throws. Some use a high loopy shot whereas others prefer a flat shallow shot.

By constructing a suitable model of the flight of the ball given some initial uncertainty around the initial angle and velocity of the player's shot, determine the optimal angle for a successful free throw.

As part of your report you should include a half page non-technical summary addressed to a fictional NBA player and his/her coach explaining your findings.

Problem B - Water, Water, Everywhere: The Faculty of Engineering is fortunate enough to have access to an underground (non-potable) water supply which it uses to irrigate its lawn. However, the sprinkler system it uses is expensive to buy and prone to breaking down.

In an effort to reduce costs, the Dean has asked you to redesign where the sprinklers are positioned and configured. The specification of sprinkler system they will be using can be below. (You may assume that all sprinklers supply the same amount of water to each location in their arcs.)

The department specifies that at least 95% of the grass should be watered, that an effort should be made to minimise water loss by avoiding watering the pathways, and that it is important that no areas of the grass are over-watered.

As part of your report you should include a non-technical half-page description to be sent to the building manager explaining your solution and any further necessary assumptions or considerations you had to make.

Extension: Investigate how your model changes if the assumption above is not valid, for example if a low water-pressure means that each sprinkler may only cover at most a fixed area which is smaller than pi*(5m)^2.

Problem C - Bayesian Gradebook: At Stellenbosch University, as with many other educational systems, students of a particular course are typically given many assessments (homework problems, projects, exams, etc) that are marked with individual scores. These scores are then averaged (often with weights) to arrive at a final mark. It is not always a very scientific process.

Prof Driscoll at the University of Delaware has recently suggested that one way of injecting more clarity (if not "fairness") is to appeal to Bayesian statistics: http://www.chebfun.org/examples/stats/BayesianGradebook.html.

Your challenge in this problem is to apply this idea to some real-world data. In particular, we will supply you with real data from a single cohort of Engineering students containing their marks for each assessment opportunity (tutorial tests and semester tests) from the three Applied Maths courses they take in first and second year.

What you do with this data is up to you, but some suggestions of things you might explore are:

• How do the original marks compared with the Bayesian approach?
• How important is the prior distribution?
• Should the prior for the second course be reset, or continue from the previous course?
• Are student grades independent? Can this be incorporated?
• What are the benefits / downsides to the Bayesian approach?

As part of your final report you should include a short (half page) non-technical description of Prof Driscoll's idea and of your findings addressed to the Dean of Engineering stating whether and why/why not you would support implementing such a scheme in the Faculty of Engineering.