The amounts y (in millions of dollars the U.S. Department of...

The amounts $y$ (in millions of dollars the U.S. Department of Energy spent for research and development from 2005 through 2010 can be approximated by the model $$y=56.77 t^{2}-366.8 t+8916, \quad 5 \leq t \leq 10$$ where $t$ represents the year, with $t=5$ corresponding to $2005 .$ (Source: American Association for the Advancement of Science) (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 2005 to $2010 .$ Interpret your answer in the context of the problem.

The amounts $y$ (in millions of dollars the U.S. Department of Energy spent for research and development from 2005 through 2010 can be approximated by the model
$$y=56.77 t^{2}-366.8 t+8916, \quad 5 \leq t \leq 10$$
where $t$ represents the year, with $t=5$ corresponding to $2005 .$ (Source: American Association for the Advancement of Science)
(a) Use a graphing utility to graph the model.
(b) Find the average rate of change of the model from 2005 to $2010 .$ Interpret your answer in the context of the problem.

Key Concepts

Graphing a Function

Graphing a function involves plotting its behavior over a specified interval on a coordinate system to visualize the relationship between the variables. This process helps in understanding the nature of the function, locating key points such as intercepts and minima or maxima, and observing overall trends and behavior over the interval of interest.

Quadratic Models

Quadratic models are mathematical representations where the function includes a squared term, leading to a parabolic shape when graphed. These models are frequently used to describe situations where the rate of change itself changes over time and are valuable in capturing accelerating or decelerating trends in various contexts.

Average Rate of Change

The average rate of change of a function between two points is calculated as the difference in the function's values divided by the difference in the corresponding inputs. It represents the overall change or slope of the function over that interval, serving as a summary of how rapidly or slowly the dependent variable changes relative to the independent variable.

Interpretation in Context

Interpreting results in context involves relating the mathematical findings back to the real-world situation. This means explaining what the calculated average rate of change signifies within the specific scenario, considering the units of measurement and the practical implications of the values obtained.

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