NCAAPMT Calculus Challenge Challenge #3 Due: October 26, 2011

Transcription

1 NCAAPMT Calculu Challenge Challenge #3 Due: October 6, 011 A Model of Traffic Flow Everyone ha at ome time been on a multi-lane highway and encountered road contruction that required the traffic to occupy only one lane each way Naturally, the Department of Tranportation would like to maximize the flow of traffic tough thi tretch of the highway What peed limit hould be et for uch a tretch of road to enure the greatet traffic flow while alo maintaining afety? When developing a mathematical model of a real-world ituation, it i uually neceary to make ome implifying aumption In thi model, we aume that all the car are the ame length and that the car follow each other at a ditance d (ee Figure 1) We know from experience that the fater we drive, the more ditance we hould leave between our car and the car in front of u Therefore, we want our model to reflect the fact that the following ditance d depend upon the aigned peed limit Figure 1: Diagram of the flow of car on the highway Car are flowing uniformly down the road, each traveling at peed and leaving a ditance of d between them and the car in front One imple model of traffic would be the equation F () d We know that the following ditance d, depend on the peed of the car Tee different rule of thumb are commonly ued to determine a afe following ditance You might want to check your tate Driver Manual to ee what i recommended The fater you are going the greater the ditance mut be between you and the car in front of you to give you time and ditance to afely top if the car in front of you top uddenly We firt note that while highway peed limit are typically given in mile per hour in the US, car length are not generally etimated in mile Therefore, it will be convenient to build into our model a unit converion of from mile per hour to feet per econd Since there are 580 feet in a mile and 3600 econd in an hour, the converion factor i

2 580 peed (in feet per econd) = 3600 feet mile econd hour peed (in mile per hour), or 1467 Building thi converion into our model in equation () give u the following form of the model: 1467 F, (3) d in which F i the rate of traffic flow in car per econd, i the length of the car in feet, d i the following ditance in feet, and i the peed limit in mile per hour Rule 1: Follow car length for every 10 It i eay to tranlate thi rule into an equation: d 10 If we ue thi rule of thumb, our model of traffic flow become: F 5 10 We need to find the critical value for thi model So fp (4) For Model 1, we have 7335 F 5 df There are no realitic d critical value for thi function A increae, the function approache F aymptotically With thi model, the maximum flow depend on, and decreae a increae Rule : Follow tee econd behind the car in front We aw in equation (3) that a car traveling at mile per hour i traveling at 1467 feet per econd So in tee econd, uch a car would travel a ditance of d feet If we ue thi rule of thumb, our model of traffic flow become: F (5) For Model, 1467 F, o 4401

3 So df d For thi model, a well, there are no realitic critical value A increae without bound, the function approache F aymptotically With thi model, the maximum flow i fixed at 0333 car/ec, or 0 car/minute Rule 3: The data below decribe the ditance needed to top at variou peed We can ue technique of data analyi to determine appropriate topping ditance Table 1 from a tate driver' handbook give ome approximate figure Speed () Thinking Ditance (ft) Braking Ditance (ft) Table1: Speed and topping ditance from driver handbook Notice that the topping ditance i broken into two part, the ditance in feet T that the car travel while the driver i reacting and putting hi foot on the brake, and the ditance in feet B that the car travel a the brake low the car to a top Thi mean that d T B The traffic flow function can be written more pecifically a 1467 F, (6) T B where T and B are function of to be determined from the data in Table 1 Clearly, T We can find function B by uing data analyi Since the data are non-linear and hould pa tough (0, 0), a power model eem appropriate If we linearize the data with a log-log re-expreion, we ee that the re-expreed data i ln B 996 ln or B 005 (See Figure ) nicely linear with equation Figure : A log-log tranformation linearize the braking ditance-peed data

4 Student hould recognize that the data in Table 1 are not really data at all, but value created from thee function Actual data would not behave o nicely (Some manual do give actual data, but the functional model hould be approximately equivalent) 1467 A model baed on Rule 3 i F 005 Now, df d There i one critical value when 0 Thi critical value i Here, the 005 maximum flow depend on, and increae a increae Since df change from poitive to d negative a we cro 0, we know we have a relative maximum value here The New Jerey Turnpike ha lane that are retricted for large truck travel only Uing Model 3, hould the peed limit be et higher or lower for thee lane? Explain your anwer Solution: If the vehicle are longer and the braking ditance tay the ame (of coure they don t tay the ame), then the optimal peed hould increae, a hown in the graph of 0 below 3 Select the model that bet decribe traffic flow and which ha a maximum flow Since thi model i dependent on the length of the vehicle, the maximum traffic flow will alo be dependent on the length Graph traffic flow v peed and identify the ordered pair that repreent maximum traffic flow for each of the following vehicle: 1467 A model baed on Rule 3 i F and we know that 0 i the peed 005 in mile per hour which maximize the flow of traffic in car per minute a) Mini Cooper length of 148 inche b) Toyota Corolla length of 1783 inche c) Hummer H length of 1899 inche d) An 18-wheeler between 70 and 80 feet Vehicle ength (, F) for the maximum Mini Cooper 148 inche mi 1543,0577 cooper ec Toyota Corolla 1783 inche 175 mi Corolla,0539 ec

5 3 The web ite Hummer H 1899 inche 1779 Hummer,058 ec Eighteen-wheeler 70 to 80 feet 3873 truck,0301 The length of 75 feet wa ued for the eighteen wheeler give the tatement that The length of time to top an eighteen wheeler i 40% greater than that of an automobile Apply thi information to find the ordered pair that repreent the peed for maximum traffic flow for an eighteen-wheeler ec 4 Our experience ugget that driver do not leave a much pace between car a they hould et p be the fraction of the required topping ditance that the driver actually leave between car Aume 01 p 1 Find the optimal peed and it correponding traffic flow that maximize traffic flow in term of the parameter and p Which ha a larger effect of the optimal peed, or p? To model thi ituation, we can ue the function 1467 F with p T B T B 005 So, df p p p d p 005 p 005 The only reaonable critical value i located at p p A p increae, the optimal peed decreae, which i what we would expect to ee Alo, a increae (without affecting braking ditance), the optimal peed increae Since p, we ee that d dp p while d Since p i mall while i large, the d change in due to mall change in p will be more ignificant than the change in due to mall change in

6 5 Some might argue that the driver would never drive o cloe to the car in front of them that they would not have ufficient time to react Thee diver might (unconciouly, of coure) leave all of the reaction ditance but only a fraction of the braking ditance Adjut your model by multiplying braking ditance by p, where 01 p 1 to find the optimal peed and it correponding traffic flow for the Corolla for p 01, p 05, and p 1 In thi cae, 1467 F T pb How different from the previou olution i the optimal velocity if thi model i ued? If 1467 F, then F 1467 T pb 005 p So, p df p p p d p The numerator in thi derivative i the ame a before, o the olution will be a well, with 447 p The flow rate of car will differ, but the elected peed will be the ame for both model involving the parameter p The Toyota Corolla ha 1783 inche or 1486 feet p Speed () Maximum Flow car/ec or 13 car/min car/ec or 65 car/min car/ec or 5 car/min car/ec or 47 car/min