Solve each problem. Selected values of the stopping distance...

Solve each problem. Selected values of the stopping distance $y,$ in feet, of a car traveling $x$ miles per hour are given in the table. $$\begin{array}{|c|c|}\hline\hline\begin{array}{c}\text { Speed } \\\text { (in mph) }\end{array} & \begin{array}{c}\text { Stopping Distance } \\\text { (in feet) }\end{array} \\\hline 20 & 46 \\\hline 30 & 87 \\\hline 40 & 140 \\\hline 50 & 240 \\\hline 60 & 282 \\\hline 70 & 371\end{array}$$ (a) Plot the data. (b) The quadratic function $$f(x)=0.056057 x^{2}+1.06657 x$$ is one model that has been used to approximate stopping distances. Find $f(45)$ to the nearest foot, and interpret this result. (c) How well does $f$ model the car's stopping distance?

Solve each problem.
Selected values of the stopping distance $y,$ in feet, of a car traveling $x$ miles per hour are given in the table.
$$\begin{array}{|c|c|}\hline\hline\begin{array}{c}\text { Speed } \\\text { (in mph) }\end{array} & \begin{array}{c}\text { Stopping Distance } \\\text { (in feet) }\end{array} \\\hline 20 & 46 \\\hline 30 & 87 \\\hline 40 & 140 \\\hline 50 & 240 \\\hline 60 & 282 \\\hline 70 & 371\end{array}$$
(a) Plot the data.
(b) The quadratic function $$f(x)=0.056057 x^{2}+1.06657 x$$ is one model that has been used to approximate stopping distances. Find $f(45)$ to the nearest foot, and interpret this result.
(c) How well does $f$ model the car's stopping distance?

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Key Concepts

Quadratic Functions in Modeling

Quadratic functions are often used in modeling scenarios where the relationship between variables is not linear but instead exhibits a curved trend. They capture accelerating or decelerating rates of change, making them particularly useful for representing phenomena like physical distances or growth patterns.

Model Validation and Accuracy Assessment

Model validation involves comparing the predictions generated by a mathematical model against actual observed data. This assessment ensures that the model not only fits the existing data well but also provides reliable estimates for new or untested values, highlighting its practical effectiveness and any potential limitations.

Data Visualization

Data visualization involves graphically representing data, such as through scatter plots, to observe patterns, trends, and anomalies. It helps in understanding the relationship between variables by providing a clear, intuitive picture of how one variable may change in relation to another.

Function Evaluation and Prediction

Function evaluation entails substituting specific values into a function to obtain predictions based on a mathematical model. This process is crucial for making informed estimates and interpreting the practical significance of the model's output in real-world contexts.

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