In this PullOut, students explore
strategies that could make the
difference between whether or not a
business is successful. The basic
mathematical idea in the activities is
optimization. Students develop several
mathematical models that can be used to
optimize the performance of a small
manufacturing business. These models
include linear, quadratic, and rational
functions.
In Activity 1, students fit a linear model
to sales and price data. They use this
model to create a quadratic revenue
model and then determine the price that
maximizes revenue. However,
maximizing revenue does not necessarily
lead to a successful business; in order for
a business to be successful, it must take in
more money than it pays out. Realizing
that profit, the difference between
revenues and costs, may be a better
measure of success, students develop a
quadratic profit model and then
determine the price that maximizes profit.
The profit model in Activity 1 is based on the
assumption that the raw materials needed to
produce each monthâ€™s finished products
were ordered monthly. In Activity 2, students
revise their profit model by relaxing this
assumption. The business is allowed to
purchase sufficient raw materials to last
multiple months. Under this assumption,
students must figure in the cost of losing
interest on the money spent on large
quantities of raw materials. The goal in this
activity is to minimize costs. Students create
an average cost model, which turns out to be
a rational function, and maximize profits by
minimizing costs.
In Activity 3, conclusions from profit models
similar to those developed in Activity 1 are
revisited in a setting in which the business is
considering expansion. Students determine
maximum profits given three possibilitiesâ€”
the company employs one shift, the
company allows the workers to work
overtime, and the company expands to
two shifts.
