Integrations with traffic

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In summary, the problem involves an intersection in Thomasville, Oregon where cars turn left at a rate of L(t) = \sqrt{t}*sin^2(t/3) cars per hour over the time interval 0 < t < 18 hours. The graph of y = L(t) is shown above. The question is whether a traffic signal is needed, based on the condition that in any two-hour interval, the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. However, it is not feasible to solve for the exact value of x using integration and a TI-83 calculator. It is better to focus on understanding the problem
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http://img85.imageshack.us/img85/9919/calculusgraphda5.jpg [Broken]

An intersection in Thomasville, Oregon, cars turn left at the rate L(t) = [tex]\sqrt{t}*sin^2(t/3)[/tex] cars per hour over the time interval 0 < t < 18 hours. The graph of y = L(t) is shown above.Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.So... basically, I read that last part, and my mind was drawing a blank. Could someone give me a little push as to what I'm supposed to think of when trying to figure out this problem?
 
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  • #2
Basically you want to find if there is any 2 hours in which the number of cars turning left is 200,000/500 = 400.

L(t) gives the number of cars per unit of time, what will integrating with respect to time give?
 
  • #3
this is from the AP calc exam 2006 where you need to use a TI-83.
don't bother integrating.
 
  • #4
So your saying no one can solve for x:

[tex]\frac{2}{3}x^{\frac{3}{2}} \sin^2 \frac{x}{3} - \frac{2}{3}(x-2)^{\frac{3}{2}} \sin^2 \frac{x-2}{3} [/tex] >[tex] 400[/tex]?
 
  • #5
Gib Z said:
So your saying no one can solve for x:

[tex]\frac{2}{3}x^{\frac{3}{2}} \sin^2 \frac{x}{3} - \frac{2}{3}(x-2)^{\frac{3}{2}} \sin^2 \frac{x-2}{3} [/tex] >[tex] 400[/tex]?

I wouldn't want to try to solve that question by hand (nor do I think I could!) .
what I am saying is that there isn't time enough to do the integration by hand, plug in the messy numerical limits, and solve.

There are always 2-3 questions out of the 6 on the AP exam that are far more challenging than the rest. This is one of them because you need to apply a great deal of knowledge to know how to set up the traffic function, determine the limits, and evaluate the definite integral with the TI-83.

it's better to struggle with the questions they have rather than see how they found the solution.
you won't learn anything. period.
 
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1. What is traffic integration?

Traffic integration is the process of combining data from various sources, such as traffic cameras, sensors, and GPS systems, to provide a comprehensive understanding of traffic patterns and conditions in a given area.

2. How does traffic integration work?

Traffic integration involves collecting data from different sources and using algorithms and software to analyze and combine the data. This allows for a more accurate and real-time view of traffic conditions.

3. What are the benefits of traffic integration?

There are several benefits of traffic integration, including improved traffic management, more accurate travel time predictions, and better decision-making for drivers and transportation agencies. It can also help reduce congestion and improve overall traffic flow.

4. What types of data are used in traffic integration?

Data used in traffic integration can include information from traffic cameras, GPS systems, sensors, weather data, and historical traffic patterns. It can also incorporate data from social media and other sources to provide a more complete picture.

5. How can traffic integration be used in transportation planning?

Traffic integration can be a valuable tool in transportation planning. It can provide insights into traffic patterns and help identify areas for improvement, such as the need for new roads or public transportation options. It can also aid in predicting future traffic trends and inform decision-making for infrastructure projects.

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