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The headway \(h\) is the average time between vehicles. On a highway carrying an average of 500 vehicles per hour, the probability \(P\) that the headway is at least \(t\) seconds is given \(^{45}\) by $$ P=0.87^{t} . $$ a. What is the limiting value of \(P\) ? Explain what this means in practical terms. b. The headway \(h\) can be calculated as the quotient of the spacing \(f\), in feet, which is the average distance between vehicles, and the average speed \(v\), in feet per second, of traffic. Thus the probability that spacing is at least \(f\) feet is the same as the probability that the headway is at least \(f / v\) seconds. Use function composition to find a formula for the probability \(Q\) that the spacing is at least \(f\) feet. Note: Your formula will involve both \(f\) and \(v\). c. If the average speed is 88 feet per second \((60\) miles per hour), what is the probability that the spacing between two vehicles is at least 40 feet?

Step by step solution

01

Analyze the Limiting Value of P

We are given the formula for the probability that the headway is at least \( t \) seconds: \( P = 0.87^{t} \). To find the limiting value of \( P \) as \( t \) approaches infinity, we consider the behavior of \( 0.87^{t} \) for very large values of \( t \). As \( t \) becomes very large, \( 0.87^{t} \) approaches 0 because 0.87 is a number between 0 and 1, and raising it to increasingly higher powers results in a value approaching zero.

02

Interpret the Limiting P in Practical Terms

The limiting value of \( P \) being 0 means that as you wait for increasingly longer times \( t \), the probability of a headway being at least \( t \) seconds goes to 0. Practically, this implies that it's highly unlikely for vehicles to be separated by very long intervals of time as the average flow is high.

03

Derive Formula for Probability Q Using Function Composition

The headway \( h \) is given by \( h = \frac{f}{v} \), where \( f \) is the spacing in feet and \( v \) is the speed in feet per second. The probability that the headway is at least \( \frac{f}{v} \) seconds is equivalent to the probability that the spacing is at least \( f \) feet. Thus, substitute \( t = \frac{f}{v} \) into the equation \( P = 0.87^{t} \) to get \( Q = 0.87^{\frac{f}{v}} \). This is the formula for the probability that the spacing is at least \( f \) feet.

04

Calculate Probability for Given Spacing and Speed

We are given \( v = 88 \) feet per second and need to find \( Q \) when \( f = 40 \) feet. Substitute these values into the derived formula \( Q = 0.87^{\frac{f}{v}} \). This gives \( Q = 0.87^{\frac{40}{88}} \). Simplify \( \frac{40}{88} = \frac{5}{11} \). Calculate \( Q = 0.87^{\frac{5}{11}} \), which is approximately 0.955, meaning there is approximately a 95.5% chance that the spacing is at least 40 feet.

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Key Concepts

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

Headway Calculation

Headway, a key concept in transportation, is defined as the average time interval between vehicles crossing a specific point on a road. It provides insight into traffic flow and congestion. Calculating headway is crucial for traffic management, road safety, and infrastructure development.

The headway can be determined using the formula:

• Headway ( \( h \) ) = Spacing ( \( f \) ) / Speed ( \( v \) ).

In this equation:

• \( f \) is the average distance between vehicles, measured in feet.
• \( v \) is the average speed of traffic, calculated in feet per second.

For instance, if the spacing between vehicles is 40 feet and the vehicles are moving at an average speed of 88 feet per second, the headway is approximately 0.45 seconds. This equation is particularly useful in scenarios requiring precise traffic modeling and control measures.

Function Composition

Function composition is a mathematical operation where one function is applied to the result of another. This concept is essential when dealing with probabilistic traffic models, as it allows us to consider multiple influencing factors simultaneously.

In our traffic flow scenario, we need to determine the probability (\( Q \) ) that the spacing between vehicles is at least a certain distance. We start with the given probability formula for headway:

• Probability that headway is at least \( t \) seconds: \( P = 0.87^{t} \).

By substituting the headway formula:

• \( t = \frac{f}{v} \), where \( f \) is the spacing and \( v \) is the speed.

We derive the new function to find the probability that the spacing is at least \( f \) feet:

• \( Q = 0.87^{\frac{f}{v}} \).

Here, function composition allows us to analyze complex systems by layering functions, enabling a comprehensive understanding of intertwined variables.

Limiting Value of Functions

When discussing functions in mathematics, understanding their limiting behavior is critical. A limiting value is what a function approaches as its input grows extremely large or small. This concept is pivotal in analyzing probabilities in dynamic scenarios like traffic.

For the given mathematical model in traffic flow:

• \( P = 0.87^{t} \).

As \( t \) tends to infinity, the value \( 0.87^{t} \) steadily approaches 0. This is because, when a number less than one (\( 0.87 \)) is raised to an increasingly larger power, its value becomes closer to zero.

Practically, this translates to a decreasing likelihood of observing longer gaps between vehicles. High traffic flow limits these lengthy intervals, making them increasingly improbable as headway expands. Comprehending these limits aids in predicting traffic patterns and formulating effective regulations aimed at maintaining smooth and safe road conditions.