# South African Mathematical Modeling Contest (SAMMC)

## Registration

Those even slightly interested in entering the competition in 2018 should pre-register (with no obligation) here.

## Contest outline and instructions

Outline: The main aim of SAMMC ("Sam-see") is to provide South African undergraduate students in science and engineering disciplines some exposure to applied mathematics problems more relevant to real-world applications than they might otherwise encounter in the classroom. It is a chance to challenge their brain and develop problem-solving skills, gain experience in working in a team, and possibly win some prizes!

The format of the contest is loosely based on the international COMAP MCM competition, held every January/February: http://www.comap.com/undergraduate/contests. A secondary aim of SAMMC is to gain experience in solving MCM-type problems and to help select teams for the international competition.

More details can be found in the here and a more precise formulation of the contest rules 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.