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2011 MCM Problems
PROBLEM A: Snowboard Course
Determine the shape of a
snowboard course (currently known as a “halfpipe”)
to maximize the production of “vertical
air” by a skilled snowboarder.
Tailor the shape to optimize other
possible requirements, such as maximum
twist in the air.
What tradeoffs may be required to
develop a “practical” course?
PROBLEM B: Repeater
Coordination
The VHF radio
spectrum involves line-of-sight transmission and
reception. This
limitation can be
overcome by “repeaters,” which pick up weak
signals, amplify
them, and retransmit
them on a different frequency. Thus, using a
repeater,
low-power users (such as
mobile stations) can communicate with one another
in situations where direct user-to-user
contact would not be possible. However,
repeaters can interfere with one
another unless they are far enough apart or
transmit on sufficiently separated
frequencies.
In addition to
geographical separat
ion, the
“continuous tone
-coded squelch
system” (CTCSS), sometimes nicknamed
“private line” (PL), technology can be
used to mitigate interference problems.
This system associates to each
repeater
a separate subaudible tone that is transmitted by
all users who wish to
communicate
through that repeater. The repeater responds only
to received
signals with its specific
PL tone. With this system, two nearby repeaters
can
share the same frequency pair (for
receive and transmit); so more repeaters
(and hence more users) can be
accommodated in a particular area.
For a circular flat area of radius 40
miles radius, determine the minimum
number of repeaters necessary to
accommodate 1,000 simultaneous users.
Assume that the spectrum available is
145 to 148 MHz, the transmitter
frequency in a repeater is either 600
kHz above or 600 kHz below the receiver
frequency, and there are 54 different
PL tones available.
How
does your solution change if there are 10,000
users?
Discuss the case
where there might be defects in line-of-sight
propagation
caused by mountainous
areas.
2012 MCM Problems
PROBLEM A: The Leaves of a
Tree
weight of the leaves (or for
that matter any other parts of the tree)? How
might
one classify leaves? Build a
mathematical model to describe and classify
leaves.
Consider and answer the
following:
? Why do leaves have the
various shapes that they have?
? Do the shapes “minimize”
overlapping individual shadows that are
cast
, so as
to maximize
exposure? Does the distribution of leaves within
the “volume” of
the tree and its
branches effect the shape?
? Speaking of profiles, is leaf shape
(general characteristics) related to tree
profile/branching structure?
? How would yo
u
estimate the leaf mass of a tree? Is there a
correlation
between the leaf mass and
the size characteristics of the tree (height,
mass,
volume defined by the profile)?
In addition to your one page summary
sheet prepare a one page letter to an
editor of a scientific journal
outlining your key findings.
PROBLEM B: Camping along
the Big Long River
Visitors
to the Big Long River (225 miles) can enjoy scenic
views and exciting
white water rapids.
The river is inaccessible to hikers, so the only
way to enjoy
it is to take a river trip
that requires several days of camping. River trips
all start
at First Launch and exit the
river at Final Exit, 225 miles downstream.
Passengers take either oar- powered
rubber rafts, which travel on average 4
mph or motorized boats, which travel on
average 8 mph. The trips range from 6
to 18 nights of camping on the river,
start to finish.. The government agency
responsible for managing this river
wants every trip to enjoy a wilderness
experience, with minimal contact with
other groups of boats on the river.
Currently,
X
trips travel down the Big Long River each year
during a six month
period (the rest of
the year it is too cold for river trips). There
are
Y
camp sites
on the Big Long River, distributed
fairly uniformly throughout the river corridor.
Given the rise in popularity of river
rafting, the park managers have been asked
to allow more trips to travel down the
river. They want to determine how they
might schedule an optimal mix of trips,
of varying duration (measured in nights
on the river) and propulsion (motor or
oar) that will utilize the campsites in the
best way possible. In other words, how
many more boat trips could be added to
the Big Long River’s rafting season?
The river managers have hired you to
advise them on ways in which to develop
the best schedule and on ways in
which
to determine the carrying capacity of the river,
remembering that no two
sets of campers
can occupy the same site at the same time. In
addition to your
one page summary
sheet, prepare a one page memo to the managers of
the
river describing your key findings.
2013 MCM Problems
PROBLEM A: The Ultimate
Brownie Pan
When baking in a rectangular pan heat
is concentrated in the 4 corners and the
product gets overcooked at the corners
(and to a lesser extent at the edges). In
a round pan the heat is distributed
evenly over the entire outer edge and the
product is not overcooked at the edges.
However, since most ovens are
rectangular in shape using round pans
is not efficient with respect to using the
space in an oven. Develop a model to
show the distribution of heat across the
outer edge of a pan for pans of
different shapes - rectangular to circular and
other shapes in between.
Assume
1. A width to length
ratio of
W/
L for the oven
which is rectangular in shape.
2. Each
pan must have an area of
A
.
3. Initially two racks in the oven,
evenly spaced.
Develop a
model that can be used to select the best type of
pan (shape) under
the following
conditions:
1. Maximize number of pans
that can fit in the oven (N)
2.
Maximize even distribution of heat (H) for the pan
3. Optimize a combination of conditions
(1) and (2) where weights p and (1-
p
)
are assigned
to illustrate how the results vary with different
values
of
W/L
and
p
.
In
addition to your MCM formatted solution, prepare a
one to two page
advertising sheet for
the new Brownie Gourmet Magazine highlighting your
design and results.