The management of a company is considering three possible...

The management of a company is considering three possible models for predicting the considering three possible models for predicting the company's profits from 2003 through 2008. Model I gives the expected annual profits if the current trends continue. Models II and III give the expected annual profits for various combinations of increased labor and energy costs. In each model, p is the profit (in billions of dollars) and $t=0$ corresponds to 2003. $\begin{array}{ll}{\text { Model I: }} & {p=0.03 t^{2}-0.01 t+3.39} \\ {\text { Model II: }} & {p=0.08 t+3.36} \\ {\text { Model III: }} & {p=-0.07 t^{2}+0.05 t+3.38}\end{array}$ (a) Use a graphing utility to graph all three models in the same viewing window. (b) For which models are profits increasing during the interval from 2003 through $2008 ?$ (c) Which model is the most optimistic? Which is the most pessimistic? Which model would you choose? Explain.

The management of a company is considering three possible models for predicting the considering three possible models for predicting the company's profits from 2003 through 2008. Model I gives the expected annual profits if the current trends continue. Models II and III give the expected annual profits for various combinations of increased labor and energy costs. In each model, p is the profit (in billions of dollars) and $t=0$ corresponds to 2003.
$\begin{array}{ll}{\text { Model I: }} & {p=0.03 t^{2}-0.01 t+3.39} \\ {\text { Model II: }} & {p=0.08 t+3.36} \\ {\text { Model III: }} & {p=-0.07 t^{2}+0.05 t+3.38}\end{array}$
(a) Use a graphing utility to graph all three models in the same viewing window.
(b) For which models are profits increasing during the interval from 2003 through $2008 ?$
(c) Which model is the most optimistic? Which is the most pessimistic? Which model would you choose? Explain.

Key Concepts

Mathematical Modeling

This concept involves creating algebraic representations to predict real-world phenomena. In many business problems, functions are set up to model trends such as profits over time so that different scenarios or variables (such as costs or labor) can be incorporated and analyzed to make informed decisions.

Graphing Functions

Graphing functions using technology provides a visual representation of mathematical models. It allows for the comparison of different models simultaneously, making it easier to observe trends, intersections, and the general behavior of each model over a defined interval.

Polynomial Functions

Polynomial functions, including both linear and quadratic forms, are essential in modeling various scenarios. Quadratic functions can model phenomena with acceleration or deceleration (non-linear change), while linear functions model constant rates of change. Understanding their characteristics is important for interpreting their growth or decline trends.

Analyzing Increasing Behavior

Determining whether a function is increasing over an interval involves examining its rate of change. This can be done graphically or analytically (using derivatives in calculus) to understand if the modeled quantity (like profit) consistently grows over a given period.

Comparative Model Analysis

This concept involves assessing multiple predictive models to determine which scenario is most favorable or unfavorable. By comparing the outputs of different models, one can identify which assumptions yield more optimistic or pessimistic projections, thereby informing strategic business decisions.

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