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Automobile traffic passes a point P on a road of width w feet with an average rate of R vehicles per second. Although the arrival of automobiles is irregular, traffic engineers have found that the average waiting time T until there is a gap in traffic of at least 1 seconds is approximately T = teRt seconds. A pedestrian walking at a speed of 3.3 ft/s requires t = s to cross the road. Therefore, the average time the pedestrian will have to wait before crossing is f(w, R)= () ew R/3.3 s What is the pedestrian's average waiting time if w = 24 ft and R = 0.2 vehicle per second?

Automobile traffic passes a point P on a road of width w feet with an average rate of R vehicles per second. Although the arrival of automobiles is irregular, traffic engineers have found that the average waiting time T until there is a gap in traffic of at least 1 seconds is approximately T = teRt seconds. A pedestrian walking at a speed of 3.3 ft/s requires t = s to cross the road. Therefore, the average time the pedestrian will have to wait before crossing is f(w, R)= () ew R/3.3 s What is the pedestrian's average waiting time if w = 24 ft and R = 0.2 vehicle per second?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 44E
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Question
**Pedestrian Waiting Times to Cross a Road with Automobile Traffic** 1. **Problem Statement:** An automobile traffic passes a point \( P \) on a road of width \( w \) feet with an average rate of \( R \) vehicles per second. Although the arrival of automobiles is irregular, traffic engineers have found that the average waiting time \( T \) until there is a gap in traffic of at least \( t \) seconds is approximately \( T = t e^{tR} \) seconds. A pedestrian walking at a speed of 3.3 ft/s requires \( t = \frac{w}{3.3} \) seconds to cross the road. Therefore, the average time the pedestrian will have to wait before crossing is \( f(w, R) = \left(\frac{w}{3.3}\right) e^{\frac{w}{3.3}R3.3} \). 2. **Pedestrian's Average Waiting Time:** What is the pedestrian’s average waiting time if \( w = 24 \) ft and \( R = 0.2 \) vehicles per second? (Use decimal notation. Give your answer to two decimal places.) \( t = 31.15 \) 3. **Linear Approximation for Increase in Waiting Time:** Use the Linear Approximation to estimate the increase in waiting time if \( w \) is increased to 26 ft. (Use decimal notation. Give your answer to two decimal places.) \( \Delta f = 6.37 \) 4. **Estimate for Waiting Time with Increased Width and Decreased Rate:** Estimate the waiting time if the width is increased to 26 ft and \( R \) decreases to 0.18. (Use decimal notation. Give your answer to two decimal places.) \( t = 32.48 \) *Note: The provided value is marked incorrect, suggesting there might be an error in calculation or method.* 5. **Rate of Increase in Waiting Time per Foot Increase in Width:** What is the rate of increase \( \Delta \) in waiting time per 1-ft increase in width when \( w = 27 \) ft and \( R = 0.2 \) vehicles per second? (Use decimal notation. Give your answer to two decimal places.) \( \Delta = 4
Transcribed Image Text:**Pedestrian Waiting Times to Cross a Road with Automobile Traffic** 1. **Problem Statement:** An automobile traffic passes a point \( P \) on a road of width \( w \) feet with an average rate of \( R \) vehicles per second. Although the arrival of automobiles is irregular, traffic engineers have found that the average waiting time \( T \) until there is a gap in traffic of at least \( t \) seconds is approximately \( T = t e^{tR} \) seconds. A pedestrian walking at a speed of 3.3 ft/s requires \( t = \frac{w}{3.3} \) seconds to cross the road. Therefore, the average time the pedestrian will have to wait before crossing is \( f(w, R) = \left(\frac{w}{3.3}\right) e^{\frac{w}{3.3}R3.3} \). 2. **Pedestrian's Average Waiting Time:** What is the pedestrian’s average waiting time if \( w = 24 \) ft and \( R = 0.2 \) vehicles per second? (Use decimal notation. Give your answer to two decimal places.) \( t = 31.15 \) 3. **Linear Approximation for Increase in Waiting Time:** Use the Linear Approximation to estimate the increase in waiting time if \( w \) is increased to 26 ft. (Use decimal notation. Give your answer to two decimal places.) \( \Delta f = 6.37 \) 4. **Estimate for Waiting Time with Increased Width and Decreased Rate:** Estimate the waiting time if the width is increased to 26 ft and \( R \) decreases to 0.18. (Use decimal notation. Give your answer to two decimal places.) \( t = 32.48 \) *Note: The provided value is marked incorrect, suggesting there might be an error in calculation or method.* 5. **Rate of Increase in Waiting Time per Foot Increase in Width:** What is the rate of increase \( \Delta \) in waiting time per 1-ft increase in width when \( w = 27 \) ft and \( R = 0.2 \) vehicles per second? (Use decimal notation. Give your answer to two decimal places.) \( \Delta = 4
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