The estimated revenues r (in millions of dollars) from sales...

The estimated revenues $r$ (in millions of dollars) from sales of in-vehicle technologies in the United States from 2003 through 2008 can be approximated by the model $r=157.30 t^{2}-397.4 t+6114, \quad 3 \leq t \leq 8$ where $t$ represents the year, with $t=3$ corresponding to 2003. (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 2003 through 2008 . Interpret your answer in the context of the problem.

The estimated revenues $r$ (in millions of dollars) from sales of in-vehicle technologies in the United States from 2003 through 2008 can be approximated by the model $r=157.30 t^{2}-397.4 t+6114, \quad 3 \leq t \leq 8$ where $t$ represents the year, with $t=3$ corresponding to 2003.
(a) Use a graphing utility to graph the model.
(b) Find the average rate of change of the model from 2003 through 2008 . Interpret your answer in the context of the problem.

Key Concepts

Interpretation of the Average Rate of Change

In the context of a real-world application, interpreting the average rate of change means converting the computed numerical value into a practical understanding. For instance, understanding whether a positive or negative rate indicates revenue growth or decline over time, and quantifying that change in relatable terms such as dollars per year or millions of dollars per year.

Average Rate of Change

The average rate of change is computed as the change in the function’s value divided by the change in the independent variable over an interval. It provides a simplified description of how a quantity (like revenue) increases or decreases on average across a specified period.

Graphing Utilities

Graphing utilities are technological tools that assist in visualizing mathematical functions. They allow users to input polynomial models and generate graphs, helping to identify features like maxima, minima, and overall trends, which are essential for analyzing the behavior of a function over its domain.

Domain Restriction and Variable Interpretation

This concept involves defining the range of input values (the domain) over which the model is applicable and ensuring that each variable is clearly interpreted in context. For example, time may be represented with a specific scale or transformation to align with the real-world scenario being modeled.

Quadratic Function Model

A quadratic function model is used to represent relationships where the rate of change itself changes over time. These models, expressed as a second-degree polynomial, are commonly used in fields like economics to forecast trends, such as revenue growth or decline, because they capture acceleration or deceleration in the data.

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