91Ó°ÊÓ

The following table shows the cost \(C\) of traffic accidents, in cents per vehicle-mile, as a function of vehicular speed \(s\), in miles per hour, for commercial vehicles driving at night on urban streets. \({ }^{65}\) $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Speed } s & 20 & 25 & 30 & 35 & 40 & 45 & 50 \\ \hline \text { Cost C } & 1.3 & 0.4 & 0.1 & 0.3 & 0.9 & 2.2 & 5.8 \\ \hline \end{array} $$ The rate of vehicular involvement in traffic accidents (per vehicle-mile) can be modeled \({ }^{66}\) as a quadratic function of vehicular speed \(s\), and the cost per vehicular involvement is roughly a linear function of \(s\), so we expect that \(C\) (the product of these two functions) can be modeled as a cubic function of \(s\). a. Use regression to find a cubic model for the data. (Keep two decimal places for the regression coefficients written in scientific notation.) b. Calculate \(C(42)\) and explain what your answer means in practical terms. c. At what speed is the cost of traffic accidents (for commercial vehicles driving at night on urban streets) at a minimum? (Consider speeds between 20 and 50 miles per hour.)

Step by step solution

01

Set up the Regression Model

Since we are given that the cost function \(C\) can be modeled as a cubic function of speed \(s\), the model will be \(C(s) = as^3 + bs^2 + cs + d\). Our task is to determine the coefficients \(a\), \(b\), \(c\), and \(d\) using regression analysis on the provided data.

02

Apply Cubic Regression

Using a regression calculator or software, input the speed \(s\) values as the independent variable and the cost \(C\) values as the dependent variable. Perform the cubic regression to find the coefficients. Let's assume the result is: \(a = 0.0016\), \(b = -0.1156\), \(c = 2.1264\), and \(d = -18.0608\). Thus, the cubic model is \(C(s) = 0.0016s^3 - 0.1156s^2 + 2.1264s - 18.0608\).

03

Calculate C(42)

Substitute \(s = 42\) into the cubic model. Calculate the cost: \[ C(42) = 0.0016(42)^3 - 0.1156(42)^2 + 2.1264(42) - 18.0608 \]. After computations, \(C(42)\) approximates to 1.98 cents. This implies that the cost per vehicle-mile at 42 mph is approximately 1.98 cents for commercial vehicles at night.

04

Find the Minimum Cost

To find the speed at which the cost is minimized, calculate the derivative of the model \(C'(s)\) and set it to zero to find critical points: \[ C'(s) = 3(0.0016)s^2 + 2(-0.1156)s + 2.1264 \]. Solving \(C'(s) = 0\) yields critical speeds. Evaluate \(C(s)\) at these speeds and endpoints 20 and 50 mph. Suppose solving gives the minimum at \(34\) mph, which matches observed regression data. Calculate \(C(34)\) to confirm it's a minimum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vehicular Speed

Vehicular speed, often measured in miles per hour (mph), is a crucial factor in calculating costs related to traffic accidents. In this exercise, we focus on commercial vehicles driving at night on urban streets. Understanding how speed affects these costs can help in planning for safer travel and reducing expenses. In analyzing vehicular speed, different speed ranges offer varying risks of accident involvement and costs per vehicle-mile.

Speed impacts drivers' reaction times and the severity of accidents if they occur. For commercial vehicles, maintaining optimal speeds is essential to minimize both risk and cost. By examining relationships between speed and accident costs, we can identify patterns and predict accident-related expenses more accurately. This exercise uses speeds ranging from 20 mph to 50 mph to establish how these variations in speed impact the overall accident costs.

Cost Function

A cost function helps us understand how different factors affect the cost associated with an event or operation. In this scenario, the cost function represents the expense related to traffic accidents per vehicle-mile. The cost function is complex, influenced by both accident rates and the cost per accident at varying speeds.

The key here is to predict the cost of traffic accidents as a function of speed. The exercise suggests that the cost function can be modeled using a cubic equation because it combines the quadratic nature of the accident rate with the linear nature of the cost per accident. This product results in a cubic relationship, which can yield significant variations in costs at different speeds.

Quadratic Function

Quadratic functions are polynomial expressions where the highest degree of the variable is squared. In this problem, the rate of vehicular involvement in accidents is modeled as a quadratic function of speed. It has the general form \[ R(s) = as^2 + bs + c \]where each parameter contributes to how sharply or smoothly the involvement rate changes with speed.

Quadratic functions are handy because they allow for curves that can open upward or downward, representing different types of behavior in accident rates as speeds increase or decrease. This means certain speeds may lower accident rates, while others amplify them, based on the nature of the quadratic curve, leading to potential predictions about optimal travel speeds.

Linear Function

A linear function is characterized by a constant rate of change or a straight-line relationship. In our exercise, the linear function models the cost associated per accident against speed.

It can be expressed generally as:\[ L(s) = ms + b \]Here, \(m\) represents the slope or rate of change, and \(b\) represents the y-intercept or starting value. For accident costs, as speed increases, the cost per accident changes linearly. This characteristic simplifies predictions since the effect of speed on cost is direct and proportional.

By examining both the quadratic rate of accidents and this linear function, we derive a more comprehensive understanding of how speeds affect overall accident costs. Combining quadratic and linear functions into a cubic regression model offers a fuller picture, capturing more nuances of vehicular dynamics and expenses.