The braking distance required for a car to stop depends on...

The braking distance required for a car to stop depends on numerous variables such as the speed of the car, the weight of the car, reaction time of the driver, and the coefficient of friction between the tires and the road. For a certain vehicle on one stretch of highway, the braking distances $d(s)$ (in $\mathrm{ft}$ ) are given for several different speeds $s$ (in mph). $$ \begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{s} & 30 & 35 & 40 & 45 & 50 \\ \hline \boldsymbol{d} \boldsymbol{(} \boldsymbol{s}) & 109 & 134 & 162 & 191 & 223 \\ \hline \boldsymbol{s} & 55 & 60 & 65 & 70 & 75 \\ \hline \boldsymbol{d}(\boldsymbol{s}) & 256 & 291 & 328 & 368 & 409 \\ \hline \end{array} $$ a. Use regression to find a quadratic function to model the data. b. Use the model from part (a) to predict the stopping distance for the car if it is traveling 62 mph before the brakes are applied. Round to the nearest foot. c. Suppose that the car is traveling 53 mph before the brakes are applied. If a deer is standing in the road at a distance of $245 \mathrm{ft}$ from the point where the brakes are applied, would the car hit the deer?

The braking distance required for a car to stop depends on numerous variables such as the speed of the car, the weight of the car, reaction time of the driver, and the coefficient of friction between the tires and the road. For a certain vehicle on one stretch of highway, the braking distances $d(s)$ (in $\mathrm{ft}$ ) are given for several different speeds $s$ (in mph).
$$
\begin{array}{|c|c|c|c|c|c|}
\hline \boldsymbol{s} & 30 & 35 & 40 & 45 & 50 \\
\hline \boldsymbol{d} \boldsymbol{(} \boldsymbol{s}) & 109 & 134 & 162 & 191 & 223 \\
\hline \boldsymbol{s} & 55 & 60 & 65 & 70 & 75 \\
\hline \boldsymbol{d}(\boldsymbol{s}) & 256 & 291 & 328 & 368 & 409 \\
\hline
\end{array}
$$
a. Use regression to find a quadratic function to model the data.
b. Use the model from part (a) to predict the stopping distance for the car if it is traveling 62 mph before the brakes are applied. Round to the nearest foot.
c. Suppose that the car is traveling 53 mph before the brakes are applied. If a deer is standing in the road at a distance of $245 \mathrm{ft}$ from the point where the brakes are applied, would the car hit the deer?

Key Concepts

Regression Analysis

Regression analysis is a statistical method used to understand and model the relationship between a dependent variable and one or more independent variables. In contexts like estimating braking distances, regression allows us to create predictive models that capture the trends found in empirical data, providing a mathematical formula that approximates how changes in one variable can affect another.

Quadratic Function Modeling

A quadratic function is a polynomial of degree two, typically expressed in the form ax² + bx + c. This type of model is particularly useful when the relationship between variables is non-linear. When variables such as speed and braking distance do not change at a constant rate, a quadratic model can more accurately capture the curvature in the data, allowing for better predictions and understanding of the dynamics involved.

Prediction and Inference Using the Model

Once a regression model is built, it can be used to predict values of the dependent variable for given inputs of the independent variable. In practical terms, this involves substituting specific values into the quadratic equation to estimate outcomes like braking distance at a certain speed. These predictions are crucial in evaluating scenarios and making decisions, such as determining if a vehicle will come to a stop before reaching an obstacle based on the computed stopping distance.

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