MCM: The Mathematical Contest in Modeling
ICM: The Interdisciplinary Contest in Modeling
2001 Mathematical Contest in Modeling
Problem A: Choosing
a Bicycle Wheel
Problem B: Escaping a Hurricane's
Wrath (An Ill Wind...)
Problem A: Choosing a Bicycle Wheel
Cyclists have different types of wheels
they can use on their bicycles. The two basic types of
wheels are those constructed using wire spokes and those
constructed of a solid disk (see Figure 1) The spoked
wheels are lighter, but the solid wheels are more aerodynamic.
A solid wheel is never used on the front for a road race
but can be used on the rear of the bike.
Professional cyclists look at a racecourse
and make an educated guess as to what kind of wheels should
be used. The decision is based on the number and steepness
of the hills, the weather, wind speed, the competition,
and other considerations. The director sportif of your
favorite team would like to have a better system in place
and has asked your team for information to help determine
what kind of wheel should be used for a given course.
Figure 1: A solid wheel
is shown on the left and a spoked wheel is shown on the
The director sportif needs specific information
to help make a decision and has asked your team to accomplish
the tasks listed below. For each of the tasks assume that
the same spoked wheel will always be used on the front
but there is a choice of wheels for the rear.
- Task 1. Provide a table giving the wind speed
at which the power required for a solid rear wheel
is less than for a spoked rear wheel. The table should
include the wind speeds for different road grades
starting from zero percent to ten percent in one percent
increments. (Road grade is defined to be the ratio
of the total rise of a hill divided by the length
of the road. If the hill is viewed as a triangle,
the grade is the sine of the angle at the bottom of
the hill.) A rider starts at the bottom of the hill
at a speed of 45 kph, and the deceleration of the
rider is proportional to the road grade. A rider will
lose about 8 kph for a five percent grade over 100
- Task 2. Provide an example of how the table
could be used for a specific time trial course.
- Task 3. Determine if the table is an adequate
means for deciding on the wheel configuration and
offer other suggestions as to how to make this decision.
Problem B: Escaping a Hurricane's Wrath
(An Ill Wind...)
Evacuating the coast of South Carolina ahead
of the predicted landfall of Hurricane Floyd in 1999 led
to a monumental traffic jam. Traffic slowed to a standstill
on Interstate I-26, which is the principal route going
inland from Charleston to the relatively safe haven of
Columbia in the center of the state. What is normally
an easy two-hour drive took up to 18 hours to complete.
Many cars simply ran out of gas along the way. Fortunately,
Floyd turned north and spared the state this time, but
the public outcry is forcing state officials to find ways
to avoid a repeat of this traffic nightmare.
The principal proposal put forth to deal
with this problem is the reversal of traffic on I-26,
so that both sides, including the coastal-bound lanes,
have traffic headed inland from Charleston to Columbia.
Plans to carry this out have been prepared (and posted
on the Web) by the South Carolina Emergency Preparedness
Division. Traffic reversal on principal roads leading
inland from Myrtle Beach and Hilton Head is also planned.
A simplified map of South Carolina is shown.
Charleston has approximately 500,000 people, Myrtle Beach
has about 200,000 people, and another 250,000 people are
spread out along the rest of the coastal strip. (More
accurate data, if sought, are widely available.)
The interstates have two lanes of traffic
in each direction except in the metropolitan areas where
they have three. Columbia, another metro area of around
500,000 people, does not have sufficient hotel space to
accommodate the evacuees (including some coming from farther
north by other routes), so some traffic continues outbound
on I-26 towards Spartanburg; on I-77 north to Charlotte;
and on I-20 east to Atlanta. In 1999, traffic leaving
Columbia going northwest was moving only very slowly.
Construct a model for the problem to investigate what
strategies may reduce the congestion observed in 1999.
Here are the questions that need to be addressed:
- Under what conditions does the plan for turning
the two coastal-bound lanes of I-26 into two lanes
of Columbia-bound traffic, essentially turning the
entire I-26 into one-way traffic, significantly improve
evacuation traffic flow?
- In 1999, the simultaneous evacuation of the state's
entire coastal region was ordered. Would the evacuation
traffic flow improve under an alternative strategy
that staggers the evacuation, perhaps county-by-county
over some time period consistent with the pattern
of how hurricanes affect the coast?
- Several smaller highways besides I-26 extend inland
from the coast. Under what conditions would it improve
evacuation flow to turn around traffic on these?
- What effect would it have on evacuation flow to
establish more temporary shelters in Columbia, to
reduce the traffic leaving Columbia?
- In 1999, many families leaving the coast brought
along their boats, campers, and motor homes. Many
drove all of their cars. Under what conditions should
there be restrictions on vehicle types or numbers
of vehicles brought in order to guarantee timely evacuation?
- It has been suggested that in 1999 some of the coastal
residents of Georgia and Florida, who were fleeing
the earlier predicted landfalls of Hurricane Floyd
to the south, came up I-95 and compounded the traffic
problems. How big an impact can they have on the evacuation
Clearly identify what measures of performance are used
to compare strategies. Required: Prepare a short newspaper
article, not to exceed two pages, explaining the results
and conclusions of your study to the public.
Clearly identify what measures of performance are used
to compare strategies.
Required: Prepare a short newspaper article,
not to exceed two pages, explaining the results and
conclusions of your study to the public.