# Integrations with traffic

http://img85.imageshack.us/img85/9919/calculusgraphda5.jpg [Broken]

An intersection in Thomasville, Oregon, cars turn left at the rate 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.

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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Gib Z
Homework Helper
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?

this is from the AP calc exam 2006 where you need to use a TI-83.
don't bother integrating.

Gib Z
Homework Helper
So your saying no one can solve for x:

$$\frac{2}{3}x^{\frac{3}{2}} \sin^2 \frac{x}{3} - \frac{2}{3}(x-2)^{\frac{3}{2}} \sin^2 \frac{x-2}{3}$$ >$$400$$?

So your saying no one can solve for x:

$$\frac{2}{3}x^{\frac{3}{2}} \sin^2 \frac{x}{3} - \frac{2}{3}(x-2)^{\frac{3}{2}} \sin^2 \frac{x-2}{3}$$ >$$400$$?

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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