"Time headway" in traffic flow is the elapsed time between the time that one car finishes passing a fixed point and the instant
that the next car begins to pass that point. Let X = the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow (sec). Suppose that in a particular traffic environment, the distribution of time headway has the following form.f(x) = k/(x^7) ; x > 1 0 ; x<1a) Determine the value of K for which f(x) is a legitimate pdfb) Obtain the cumulative distribution functionF(x) = [ Blank ] ; x > 1 0 ; x < 1c) Use the cdf from (b) to determine the probability that headway exceeds 2 sex. (Round your answer to three decimal places.)Use the cdf from (b) to determine the probability that headway is between 2 and 3 sec (round your answer to three decimal places)d) Obtain the mean value of headway and the standard deviation of headway. (Roun to nearest 3 decimal places)mean [ blank ]standard deviation [ blank ]e) what is the probability that headway is within 1 standard deviation of the mean value
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Answer:
Around 5.5 square meters
Step-by-step explanation:
You can start by finding the area of the segment. Since the rest of the circle that is not in the segment is 240 degrees, the segment is 120 degrees or a third of the circle. You can therefore find the area of that segment with the formula square meters. Now, you need to find the area of the triangle inside the sector. This is more difficult than last time, because it is not a 90 degree angle. However, you can solve this by dividing this triangle into two 30-60-90 triangles, which you know how to find the ratio of sides for. In a 30-60-90 triangle, the hypotenuse is twice the length of the smallest leg, and the larger leg is
times larger than the smaller leg. In this case, these dimensions are a base of
for the smaller leg and
for the larger leg, or the base. Using the triangle area formula and multiplying by 2 (because remember, we divided the big triangle in half), you get
square meters. Subtracting this from the area of the segment, you get about 5.5 square meters. Hope this helps!