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Walking in Seattle: It is common in large cities for people to travel to the center city and then walk to their final destination. In Seattle the percent P of pedestrians who walk at least D feet from parking facilities in the center city is given partially in the accompanying table.22 $$ \begin{array}{|c|c|} \hline \text { Distance } D & \begin{array}{c} \text { Percent } P \text { walking } \\ \text { at least } D \text { feet } \end{array} \\ \hline 300 & 60 \\ \hline 500 & 40 \\ \hline 1000 & 18 \\ \hline 1500 & 9 \\ \hline 2000 & 5 \\ \hline \end{array} $$ a. Make a model of \(P\) as an exponential function of \(D\). b. What percentage of pedestrians walk at least 200 feet from parking facilities? c. When models of this sort are made, it is important to remember that many times they are only rough indicators of reality. Often they apply only to parts of the data and may tell very little about extremes of the data. Explain why it is not appropriate to view this as an accurate model for very short distances walked.

Step by step solution

01

Write the problem description

We need to determine an exponential model for the percentage of pedestrians walking at least a certain distance, \( D \), from their parking facilities in Seattle. The relationship should follow the form \( P(D) = a \cdot b^D \), where \( P \) is the percentage of pedestrians and \( a \) and \( b \) are constants we must find. We will use the provided distance-percent data points from the table.

02

Establish mathematical relationships

By analyzing the percent and distance values given (300, 60), (500, 40), (1000, 18), (1500, 9), and (2000, 5), we assume an exponential decay function. We write two equations based on the form \( P(D) = a \cdot b^D \) using two data points: \( a \cdot b^{300} = 60 \) and \( a \cdot b^{500} = 40 \).

03

Solve for constants using data points

From the two equations \( a \cdot b^{300} = 60 \) and \( a \cdot b^{500} = 40 \), divide them to eliminate \( a \): \( \frac{b^{500}}{b^{300}} = \frac{40}{60} = \frac{2}{3} \). Simplifying, we find \( b^{200} = \frac{2}{3} \). Solving for \( b \), we get \( b \approx (\frac{2}{3})^{1/200} \approx 0.9965 \). Substitute \( b \) back into one of the original equations to find \( a \): \( a = \frac{60}{b^{300}} \approx 60 \cdot (0.9965)^{-300} \approx 80 \).

04

Formulate the exponential model

With \( a \approx 80 \) and \( b \approx 0.9965 \), the model becomes \( P(D) = 80 \cdot 0.9965^D \). This describes pedestrian behavior as an exponential decay function with distance \( D \).

05

Calculate the percentage for 200 feet

Using the model \( P(D) = 80 \cdot 0.9965^D \), substitute \( D = 200 \): \( P(200) = 80 \cdot 0.9965^{200} \). Calculating, \( P(200) \approx 68.2 \). Thus, approximately 68.2% of pedestrians walk at least 200 feet.

06

Discuss model limitations

The model attempts to estimate pedestrian behavior based on certain data points. However, for very short distances like 200 feet, the model may not accurately depict reality due to the potential nonlinear nature of pedestrian behaviors at low distances, as parking patterns and other urban design factors could significantly influence more localized pedestrian movement.

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Key Concepts

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

Pedestrian Behavior

In large urban settings like Seattle, people often need to walk from parking areas to reach their destinations. Studying pedestrian behavior involves understanding how far people are typically willing to walk after parking. Such behavior is crucial for urban planning and developing efficient pedestrian pathways.

We often find that fewer people are inclined to walk longer distances. This behavior can be observed and recorded in data tables, such as the one given in the exercise showing the percentage of pedestrians who walk varying distances.

• Different distances are recorded in the table, with columns indicating distance walked and the corresponding percentage of pedestrians.
• This data helps in identifying patterns in walking behavior and is significant in planning city infrastructure.
• By modeling this data, we can predict pedestrian behavior for other distances not specifically measured in the original data.

Urban designers and planners use insights from pedestrian behavior data to improve transportation systems and the overall walkability of the city.

Exponential Decay

Exponential decay is a mathematical process where quantities decrease at a rate proportional to their current value. This concept is crucial in many natural and physical phenomena, and it also applies to pedestrian behavior modeling.

When we describe the percentage of people walking a certain distance using exponential decay, the number of pedestrians decreases as the distance increases. This is captured in the mathematical model we derived:

• The formula follows the general form of exponential decay: \( P(D) = a \cdot b^D \), where \( a \) is the initial percentage of pedestrians and \( b \) is the decay factor.
• In our model, this factor \( b \) is slightly less than 1 (0.9965), indicating a slow decay as distance \( D \) increases.
• The value of \( a \) is initially 80, representing the maximum percentage of pedestrians willing to walk very short distances.

This decay factor demonstrates how quickly the pedestrian numbers drop off as distance increases, aligning with our common understanding that fewer people will walk longer distances.

Mathematical Modeling

Mathematical modeling involves creating abstract representations of real-world scenarios using mathematical equations. This helps in better understanding complex systems and predicting outcomes.

In the exercise, a mathematical model in the form of an exponential function is used to represent pedestrian behavior as they walk varying distances:

• Our model is represented by \( P(D) = 80 \cdot 0.9965^D \), which predicts the percentage of people walking at least a distance \( D \).
• The model is constructed using data points from the given table, ensuring that the predictions are grounded in observed behavior.
• This model allows us to estimate pedestrian percentages for distances not originally included in the data table.

By using mathematical models, we can analyze patterns and trends in pedestrian behavior, which are valuable for city planners to address transportation needs effectively. It's important to remember, however, that models may not capture all real-world variables and are often approximations.

Data Analysis

Data analysis is the systematic inspection, cleansing, transformation, and modeling of data to discover useful information and support decision-making. In the context of the exercise, data analysis helps us decode the relationship between distance walked and pedestrian percentage.

The key steps in data analysis here involved:

• Plotting the given data, which details the percentage of pedestrians at various distances, helps visualize the trend.
• Applying mathematical techniques to fit an exponential model, illustrating the decay in pedestrian percentages as distances increase.
• Interpreting the output, such as calculating the percentage at a specific distance like 200 feet, to extract actionable insights.

This analysis is vital not just for academic exercises but also for urban ecological studies, transportation predictions, and improving city planners' ability to design infrastructures that cater to pedestrian needs and encourage walking.