The amounts (in billions of dollars) the U.S. federal...

The amounts (in billions of dollars) the U.S. federal government spent on research and development for defense from 2010 through 2014 can be approximated by the model $$y=0.5079 t^{2}-8.168 t+95.08$$ where $t$ represents the year, with $t=0$ corresponding to $2010 .$ (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 2010 to $2014 .$ Interpret your answer in the context of the problem.

The amounts (in billions of dollars) the U.S. federal government spent on research and development for defense from 2010 through 2014 can be approximated by the model
$$y=0.5079 t^{2}-8.168 t+95.08$$
where $t$ represents the year, with $t=0$ corresponding to $2010 .$ (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 2010 to $2014 .$ Interpret your answer in the context of the problem.

Key Concepts

Quadratic Function

A quadratic function is a polynomial function of degree two, typically expressed in the form y = ax² + bx + c. It is used to model situations where the rate of change itself changes at a constant rate. In the context of the problem, the quadratic function represents the U.S. federal government spending on defense research and development over a series of years.

Graphical Representation

Graphing the function using a graphing utility allows us to visualize the behavior of the quadratic model over time, including its curvature, vertex, and general trend. This visual representation helps in understanding how the spending changes from year to year and may indicate periods of increase or decrease in expenditure.

Average Rate of Change

The average rate of change of a function over a given interval is calculated by finding the difference in the function's values at the endpoints of the interval and dividing by the difference in the corresponding input values. This provides a measure similar to the slope of the secant line connecting two points on the curve, indicating the overall rate at which the dependent variable changes with respect to the independent variable.

Interpretation in Context

In the context of this spending model, the average rate of change represents the average annual change in defense research and development expenditure over the specified time interval. It offers insight into whether spending is increasing or decreasing over time and by how much on average each year, providing a practical understanding of fiscal trends within the defense sector.

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