The stopping distance of an automobile is the distance...

The stopping distance of an automobile is the distance traveled during the driver's reaction time plus the distance traveled after the brakes are applied. In an experiment, researchers measured these distances (in feet) when the automobile was traveling at a speed of $x$ miles per hour on dry, level pavement, as shown in the bar graph. The distance traveled during the reaction time $R$ was $R=1.1 x$ and the braking distance $B$ was $B=0.0475 x^{2}-0.001 x+0.23$ (a) Determine the polynomial that represents the total stopping distance $T$. (b) Use the result of part (a) to estimate the total stopping distance when $x=30, x=40,$ and $x=55$ miles per hour. (c) Use the bar graph to make a statement about the total stopping distance required for increasing speeds. (GRAPH CAN'T COPY)

The stopping distance of an automobile is the distance traveled during the driver's reaction time plus the distance traveled after the brakes are applied. In an experiment, researchers measured these distances (in feet) when the automobile was traveling at a speed of $x$ miles per hour on dry, level pavement, as shown in the bar graph. The distance traveled during the reaction time $R$ was
$R=1.1 x$ and the braking distance $B$ was $B=0.0475 x^{2}-0.001 x+0.23$
(a) Determine the polynomial that represents the total stopping distance $T$.
(b) Use the result of part (a) to estimate the total stopping distance when $x=30, x=40,$ and $x=55$ miles per hour.
(c) Use the bar graph to make a statement about the total stopping distance required for increasing speeds.
(GRAPH CAN'T COPY)

Key Concepts

Nonlinear Relationships

Nonlinear relationships occur when changes in one variable do not produce proportional changes in another. Here, the braking distance includes a quadratic term, showing that as speed increases, the stopping distance increases at an accelerating rate. Understanding nonlinear behavior is crucial in scenarios where small increases in one variable can lead to disproportionately large changes in another.

Function Evaluation

Function evaluation involves substituting specific values into a function to determine the corresponding output. In the problem, after forming the total stopping distance polynomial, particular speeds are substituted into the function to estimate the stopping distances. This is a fundamental step in using mathematical models to make predictions based on given input data.

Graphical Data Interpretation

Graphical data interpretation is the skill of analyzing and extracting information from visual representations such as bar graphs. In this problem, the bar graph supports the observation that total stopping distances increase with speed. This concept underscores the importance of linking graphical trends with algebraic models to validate and visualize real-world data relationships.

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients, involving operations of addition, subtraction, and nonnegative integer exponents. In this context, both the reaction distance and braking distance are expressed as polynomials, which are then combined to represent the total stopping distance. This use of polynomial functions provides a powerful way to model and predict outcomes based on varying input values.

Adding Functions

Adding functions is the process of combining two or more functions by adding their outputs to create a new function. In the problem, the total stopping distance is found by summing the reaction distance and the braking distance, each expressed as functions of speed. This concept is central to forming comprehensive mathematical models from simpler component functions.

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