The police department must determine the speed limit on a...

The police department must determine the speed limit on a bridge such that the flow rate of cars is maximum per unit time. The greater the speed limit, the farther apart the cars must be in order to keep a safe stopping distance. Experimental data on the stopping distance $d$ (in meters) for various speeds $v$ (in kilometers per hour) are shown in the table. $\begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{v} & 20 & 40 & 60 & 80 & 100 \\ \hline \boldsymbol{d} & 5.1 & 13.7 & 27.2 & 44.2 & 66.4 \\\hline\end{array}$ (a) Convert the speeds $v$ in the table to the speeds $s$ in meters per second. Use the regression capabilities of a graphing utility to find a model of the form $d(s)=a s^{2}+b s+c$ for the data. (b) Consider two consecutive vehicles of average length $5.5$ meters, traveling at a safe speed on the bridge. Let $T$ be the difference between the times (in seconds) when the front bumpers of the vehicles pass a given point on the bridge. Verify that this difference in times is given by $$ T=\frac{d(s)}{s}+\frac{5.5}{s} $$ (c) Use a graphing utility to graph the function $T$ and estimate the speed $s$ that minimizes the time between vehicles. (d) Use calculus to determine the speed that minimizes $T$. What is the minimum value of $T$ ? Convert the required speed to kilometers per hour. (e) Find the optimal distance between vehicles for the posted speed limit determined in part (d).

The police department must determine the speed limit on a bridge such that the flow rate of cars is maximum per unit time. The greater the speed limit, the farther apart the cars must be in order to keep a safe stopping distance. Experimental data on the stopping distance $d$ (in meters) for various speeds $v$ (in kilometers per hour) are shown in the table.
$\begin{array}{|c|c|c|c|c|c|}
\hline \boldsymbol{v} & 20 & 40 & 60 & 80 & 100 \\
\hline \boldsymbol{d} & 5.1 & 13.7 & 27.2 & 44.2 & 66.4 \\\hline\end{array}$
(a) Convert the speeds $v$ in the table to the speeds $s$ in meters per second. Use the regression capabilities of a graphing utility to find a model of the form $d(s)=a s^{2}+b s+c$ for the data.
(b) Consider two consecutive vehicles of average length $5.5$ meters, traveling at a safe speed on the bridge. Let $T$ be the difference between the times (in seconds) when the front bumpers of the vehicles pass a given point on the bridge. Verify that this difference in times is given by
$$
T=\frac{d(s)}{s}+\frac{5.5}{s}
$$
(c) Use a graphing utility to graph the function $T$ and estimate the speed $s$ that minimizes the time between vehicles.
(d) Use calculus to determine the speed that minimizes $T$. What is the minimum value of $T$ ? Convert the required speed to kilometers per hour.
(e) Find the optimal distance between vehicles for the posted speed limit determined in part (d).

Key Concepts

Unit Conversion

This concept involves converting measurements from one unit system to another, such as converting speeds from kilometers per hour to meters per second. It is a fundamental skill in physics and engineering problems, ensuring that all quantities in a calculation are expressed in compatible units, which is necessary for accurate analysis and problem solving.

Regression Modeling

Regression modeling is the process of determining the relationship between a dependent variable and one or more independent variables using a best-fit function. In this context, quadratic regression is used to model how the stopping distance varies with speed. This allows for predictions of distance at speeds not directly measured and enables further analysis using the derived model.

Time-Distance Relationships

Time-distance relationships involve understanding how time, speed, and distance are interrelated. Here, the time gap between vehicles is calculated by dividing the distance (comprising both the safe stopping distance and the vehicle length) by the speed. This concept is crucial in formulating timing and spacing strategies in traffic flow and safety analyses.

Graphical Analysis

Graphical analysis employs visual tools like plotting functions using graphing utilities to examine behavior, such as identifying minima or maxima. In optimization problems, graphing a function provides an approximate solution and insight into how the function behaves across a range of values, helping to pinpoint optima before applying formal methods.

Calculus-based Optimization

Calculus-based optimization involves finding the extrema of a function by computing its derivative, setting the derivative equal to zero, and solving for the critical points. This method is used to determine the speed that minimizes the time interval between vehicles, ensuring an optimal flow. It is an essential technique in many fields including engineering, economics, and operations research.

Traffic Flow Optimization

Traffic flow optimization is concerned with maximizing the number of vehicles passing a point per unit time while maintaining safety standards. This involves balancing speed, vehicle spacing, and reaction time to ensure that roadways are used efficiently. The problem under discussion exemplifies how mathematical modeling and optimization techniques can be applied to real-world traffic management challenges.

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