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

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

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.

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.

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.

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.

Transcript

00:02 So we're given the function model that represents the amount in millions of dollars the department of energy spent for research and development from 2005 to 2010.

00:12 The model is given there by y equals.

00:16 T represents the year, so a t of 5 is the year 2005.

00:23 And we're only looking at the time from 2005 to 2010.

00:27 So if t equals 5, it's 2005.

00:29 If t equals 10, it's 2010.

00:31 And we're looking at all the years between 2005 and 2010.

00:34 So the first question says to graph it.

00:36 So we're just going to go to our, we're going to use desmos.

00:41 We're going to go to desmos here and we're going to enter the graph 56 .77 x squared.

00:50 So desmos won't take anything when you're graphing like this.

00:54 You want to always use x no matter what your variable is.

01:01 If you're just looking at the graph, you want to use x for your independent variable.