The table lists the practical stopping distances D (in feet)...

The table lists the practical stopping distances $D$ (in feet) for cars at speeds $S$ (in miles per hour) on level surfaces, as used by the American Association of State Highway and Transportation Officials. (TABLE CANNOT COPY) (a) Plot the data. (b) Determine whether stopping distance is a linear function of speed. (c) Discuss the practical implications of these data for safely driving a car.

The table lists the practical stopping distances $D$ (in feet) for cars at speeds $S$ (in miles per hour) on level surfaces, as used by the American Association of State Highway and Transportation Officials.
(TABLE CANNOT COPY)
(a) Plot the data.
(b) Determine whether stopping distance is a linear function of speed.
(c) Discuss the practical implications of these data for safely driving a car.

Key Concepts

Practical Implications for Safety

The practical implications for safety arise from understanding how the mathematical relationship between speed and stopping distance affects real-life driving. Recognizing that higher speeds lead to disproportionately longer stopping distances can inform appropriate speed limits, safe following distances, and overall driving behavior to enhance road safety and reduce the likelihood of accidents.

Data Modeling and Analysis

Data modeling and analysis involve evaluating whether a mathematical model, such as a linear function, adequately describes the relationship observed in the data. This includes looking at the pattern of the plotted points and potentially calculating a line of best fit or performing regression analysis to quantify how well the model fits, thereby indicating the validity of the chosen model.

Linear Functions

A linear function is characterized by a constant rate of change, meaning that as one variable changes, the other changes at a steady, unvarying rate. In determining if a relationship is linear, one examines whether the plotted data closely fits a straight line, indicating that increases in the input variable result in constant proportional increases in the output.

Data Visualization

Data visualization involves representing data graphically to reveal underlying patterns and trends. Plotting the collected data points on a coordinate plane—typically with the independent variable on the x-axis and the dependent variable on the y-axis—allows one to visually assess the relationship between the variables and to identify any outliers or deviations from expected patterns.

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