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9.1
Analyzing and Using Mathematical Models
P
urpose
The role this topic plays in quantitative reasoning
Mathematical models are used to analyze real-life situations. These models help us to learn more
about the situations, do what-if modeling, and make better decisions by understanding the set of
relationships between input and output data. Amathematical model can be deterministic or probabilistic.
In deterministic models, there is a unique set of output values for each given set of input values, and
rerunning the model will not produce different results, as long as identical input values are used. For a
probabilistic mathematical model, the output values will exhibit variation when iterated with fixed input
values. In many different disciplines, we can discover and formulate a precise relationship between a
set of variables, such as with the combined gas law that relates the temperature, pressure, and volume
of an ideal gas. We can test this mathematical model by changing the input variables (volume and
temperature of a given gas), to see how the output variable (pressure) changes. The output variable
should be consistent, except for slight experimental errors, for a given volume and temperature inputs.
Therefore, the combined gas law is deterministic.
An example of a mathematical model that interests most people is one that calculates the amount of your
paycheck. In this situation, a basic model might equate your monthly income with the number of hours
worked in a month, multiplied by your hourly wage. The purpose of studying a model such as this one is
to understand the mathematical relationship between the input variables: hours of work and hourly salary,
and the output variable: income. Ideal mathematical models use the fewest number of variables and the
simplest set of relationships to explain most, if not all, of the behavior of a phenomenon. In the previous
example, our initial model of monthly income only calculated base pay and is thus an accurate model only
if we do not work overtime. The model can be evolved or advanced to address overtime hours. Overtime
workers would need to have the model expanded to include the overtime rate (1.5 times the hourly
rate) and overtime hours as input variables. Further expansions of this model could also account for
adjustments for holiday hours (2 times the hourly rate), taxes, and contributions to an IRA, for example.
L
earning Goals
What you should learn while completing this activity
1. Determine the components of a mathematical model, including input and outpur variables
2. Understand and explain the relationships within a mathematical model
3. Determine the strength, limitations, and validity of a given model
4. Apply a generalized mathematical model to a specific situation by selecting the relevant inputs
and outputs, experimenting with possible input values, and interpreting the variation of the output
results.
D
iscovery
Finding out for yourself
Learn how to win a jellybean-counting contest. Read the blog in the link available on the companion
website before completing this activity.
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Formulate a basic mathematical model for estimating the number of jellybeans in a cylindrical jar
based on the dimensions of the candy and the container.