.The police department must determine a safe speed limit on a bridge so that the flow rate of cars is...

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.The police department must determine a safe speed limit on a bridge so that the flow rate of cars is at a maximum per unit time. The greater the speed limit, the farther apart the cars must be expected to be in order to allow for a safe stopping distance. The total distance needed for a car to stop, if a car in front of it stops suddenly, depends on two factors: time needed to react and the speed of the car. Experimental data on the stopping distance d (in feet), on the bridge surface, for various speeds s (in miles per hour) is given in the following table. The table also provides an estimate for reaction distance r (in feet); this is the distance the car will travel before the driver reacts. s (in mph) 5 10 20 30 40 50 60 d (in feet) 4 11 33 62 100 149 203 r (in feet) 5 10 20 30 40 50 60 The police department has also identified the lengths of the 5 most common types of vehicles that are expected to use the bridge: Model Length (in inches) Fiat 500 142.0 Ford Fiesta 153.1 Dodge Caliber 173.8 Honda Civic 176.5 Dodge Grand Caravan 202.5 The bridge will also be used occasionally by tractor-trailers with an average length of 75 feet and stopping distances that are about 40% greater than the stopping distances for an automobile. For this project, you will produce and analyze models for the traffic flow across the bridge. Problem Set Include units with all of your results. 1. Find a function of the form d(s) = as2 +bs+c that models the stopping distance in terms of speed. What should d(0) equal? What is a reasonable estimate for d′(0)? Select the constants a, b and c that best fits the data and produces the value of d(0) and d′(0) that you identified. 2. Produce a model for the flow rate of the cars crossing the bridge. Consider two consecutive cars each with vehicle length ℓ and traveling a speed s. If the driver in the second car is traveling a safe distance from the first car (i.e. allowing enough time to react and stop if the first brakes suddenly), what is the distance between the cars? How much time elapses between the moment the front bumper of the first car enters the bridge to the moment the front bumper of the second car enters the bridge? Find a formula in terms of s the provides the number of cars that enter the bridge per minute, assuming each car is the same length and there is the same (safe) distance between each car. 3. Provide a graph of traffic flow in terms of speed if all the cars are Fiat 500’s. Provide a similar graph if all the cars are Dodge Grand Caravans. How does the length of the cars seem to affect the traffic flow? 4. Assuming all the cars have vehicle length ℓ, what should be the speed limit (in miles per hour) in order to maximize traffic flow? Provide a graph of this optimal speed limit in terms of the vehicle length ℓ. How does the vehicle length affect the optimal speed limit? What should the speed limit on the bridge be to safely accommodate the 5 most common vehicle types expected to use the bridge. 5. What should be the speed limit if the stopping distance and length of tractor-trailers is also taken into account? 6. It is fairly common for drivers to leave less than the safe distance between cars while driving. Suppose that all drivers only allow k times the safe distance between each car, with 0 < k  1 a fixed number. How does the value of k affect the optimal speed limit? 7. What is unrealistic about the assumptions used in modeling the traffic flow across the bridge? What modifications would make the model more realistic? (You do not need to produce a new model; you should just identify some things that if taken into consideration might produce a more realistic model.)
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