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© 2014 Pacific Crest
419
Step
Explanation
Watch it Work!
3.
What is
important?
Choose the most important aspects,
ones which contribute the most to the
behavior, of the situation to include in
your initial model .
A simple model will include only the
cylindrical geometry of the tire
4.
What can be
ignored?
Omit characteristics that are less
significant or will make the model overly
complex.
We will ignore the walls, treads, and
any deviation from an ideal shape
5.
Establish
relationships
identify the mechanisms that relate the
important characteristics using actual
data or the phenomenology of the
situation
The volume of a thin cylindrical shell
is approximated by
V
= 2π
rht
r
= radius,
h
= height,
t
= thickness
6.
Abstract a
model from
the problem
Express a simplified version of the
problem as a set of mathematical
objects (mathematical model) using the
results of steps 4-6
The shape of the tire is modeled here
h
z
r
1
r
2
The volume is approximated by
V
= 2π
rht
where
t
=
r
2
–
r
1
,
r
=
r
2
7.
Reframe the
problem
Restate the original problem in terms of
the mathematical model
Given the actual size of our tires we
determine that
r
= 0.4m,
h
= 0.2m,
and
t
= 0.01m
8.
Find a
solution
Solve the model problem using an
algorithm, brute force, or trial and error.
We use order of operations to
simplify the formula
V
= 2π(0.4)(0.2)(0.01)
V
@
0.005m
3
9.
Test the
model
Compare the results of the model with
actual data from the situation modeled
The weight and density of a tire
is used to determine the actual
volume to compare with the model’s
estimate.
10.
Improve the
model
Return to step 3 and refine the model
with fewer or more assumptions and
repeat the remaining steps.
If necessary, we repeat the process
including the walls in our model.
9.2 Constructing a Mathematical Model