Problem A: The Velociraptor Problem

The Velociraptor, Velociraptor mongoliensis, was a predatory dinosaur that lived during the late Cretaceous period, approximately 75 million years ago. Paleontologists think that it was a very tenacious hunter, and may have hunted in pairs or larger packs. Unfortunately, there is no way to observe its hunting behavior in the wild as can be done with modern mammalian predators. A group of paleontologists has approached your team and asked for help in modeling the hunting behavior of the velociraptor.
They hope to compare your results with field data reported by biologists studying the behaviors of lions, tigers, and similar predatory animals.

The average adult velociraptor was 3 meters long with a hip height of 0.5
meters and an approximate mass of 45 Kg. It is estimated that the animal
could run extremely fast, at speeds of 60 km/hr., for about 15 seconds.
After the initial burst of speed, the animal needed to stop and recover
from a buildup of lactic acid in its muscles.

Suppose that Velociraptor prey on Thescelosaurus neglectus, a herbivorous
biped approximately the same size as the Velociraptor. A biomechanical
analysis of a fossilized thescelosaurus indicates that if could run at a
speed of about 50km.hr. for long periods of time.

Part1
Assuming the velociraptor is a solitary hunter, design a mathematical model
that describes a hunting strategy for a single velociraptor stalking and
chasing a single thescelosaurus as well as the evasive strategy of the
prey. Assume that the thecelosaurus can always detect the velociraptor when
in comes within 15 meters, but may detect the predator at even greater
ranges (up to 50 meters) depending upon the habitat and weather conditions.
Additionally, due to its physical structure and strength, the velociraptor
has a limited turning radius when running at full speed. This radius is
estimated to be three times the animal's hip height. On the other hand, the
thescelosaurus is extremely agile and has a turning radius of 0.5 meters.

Part 2
Assuming more realistically that the velociraptor hunted in pairs, design a
new model that describes a hunting strategy for two velociraptors stalking
and chasing a single thescelosaurus as well as the evasive strategy of the
prey. Use the other assumptions and limitations given in Part 1.

Problem B: Mix Well For Fruitful Discussions

Small group meetings for the discussion of important issues, particularly
long-rang planning, are gaining popularity. It is believed that large
groups discourage productive discussion and that a dominant personality
will usually control and direct the discussion. Thus, in corporate board
meetings the board will meet in small groups to discuss issues before
meeting as a whole. These smaller groups still run risk of control by a
dominant personality. In an attempt to reduce this danger it is common to
schedule several sessions with a different mix of people in each group.

A meeting of an Tostal Corporation will be attended by 29 Board Members of
which nine are in-horse members(i.e., corporate employees). The meeting is
to be an all-day affair with three sessions scheduled for the morning and
four for the afternoon. Each session will take 45 minutes, beginning on the
hour from 9:00 A.M. to 4:00 P.M., with lunch scheduled at noon. Each
morning session will consist of six discussion group with each discussion
group led by one of the corporation's six senior officers. None of these of
officers are board members. Thus each senior officer will lead three
different discussion groups. The sessions will consist of only four
discussion groups.

The president of the corporation wants a list of board-member assignments
to discussion group for each of seven sessions. The assignments should
achieve as much of a mix of members as much as possible. The ideal
assignment would have each board member with each other board member in a
discussion group the same number of times while minimizing common
membership of groups for the different sessions.

The assignments should also satisfy the following criteria:
1.For the morning sessions, no board member should be in the same senior
officer's discussion group twice.
2.No discussion group should contain a disproportionate number of in-house
members.

Give a list of assignments for members 1-9 and 10-29 and officers 1-6.
Indicate how well the criteria in the precious paragraphs are met. Since it
is possible that some board members will cancel at the last minute or that
some not scheduled will show up, an algorithm that the secretary could use
to adjust the assignments with an user to make assignments for future
meetings involving different levels of participation for each type of
attendee.