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The speed of vehicles on a highway with speed limit 100 km/h are normally distributed with mean 112 km/h and standard deviation 8 km/h.

(a) What is the probability that a randomly chosen vehicle is traveling at a legal speed?

(b) If police are instructed to ticket motorists driving 125 km/h or more, what percentage of motorists are targeted?

A. $\approx 6.68 \%$

B. $\approx 5.21 \%$

Applications of Integration

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Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

So if you were saying a driveway, uh, or I'm loving you Like, uh, well, I wrote them, um, Carson driving there on the there are some sleep experience. So if we assume that I'm that car's, uh, have their speed, they drive around Allah, Speed on a speed. There has a normal distribution off our off the cars. Tribal on this. Very weak. All the cars, uh, following normal distribution. So the normal distribution here with normal distribution has deviation. Um, mhm human meters. For our it has mean I mean, um, one so kilometers for our. So you have the following situations. If the spilling it is 100 speed, lead this 100 kilometers per hour. Um, so how many? How many cars off away? The cars that go through this driveway? Uh, Dr, uh, legal speed. So this number can be computed by making well, the probability that a car this question is the same as What is the problem? Is that a car as probably the velocity of all the velocity has to be positive. You can zero and, uh, 100. That is our speed limit. Where this is the velocity of the car. so these probably they can be computed as they develop off. But normal distribution Well, the the way of a normal distribution is of the form need to the minus X minus The explains the mean square over two times this other. The deviation is sigma. It can be noted like that over to squared off to by Mana that times deviation. So this is the normal district. They intensity the density function off these normal distribution distribution. So we would be to integrate that density from zero after 100. You have to integrate from 0 200 ft, minus X minus one. Well, not a square divided by two times for 64 the 64 is going toe 80 square that is able to the standard to the deviation squared being divided by squared off to Bye. I'm Dana. I'm saying that is the alleviation the X. So this internal is, um is about 0.0, six from 68 So these is relative toe one. So if you want to make this in numbers off percentage, well, one is 100 percent. So that this number six point, uh, point point or six 68 corresponds to point those six six mhm times. 100 that is, um, all that is gonna be the total percentage in that moves. So serious. So exam E 6% she 6.68%. So that is the the number off cars that gold, legal speed. And then, um yeah. So I have that until this Proclivities 100 on a second. It was really would want to consider if we are take a team. We take it to the cars because that go beyond, let's go faster than than 1 25. Yeah. Okay. How many? What percentage of cars will be ticket? Many tickets. Yes. So this is not reckon we will be, is it? If you want to compute the probability off, our velocity lies between 1 25 on the wall infinity. Just, uh, it is the same as the probability is that X is bigger than one 25. On the level that is the we are integrating the same function, but from 1 25 after infinity. So we need to the minus x minus 1 12 square. Divided by two times 64 over a squared off to five grams eight the X on. Then we do this integral. We should get something that is about point 05 21 Also, in terms of percentages multiplying by what 100% The pain there multiplied by 100 chopped in the percentage this corresponds toe 5% 55.21 percent of the off The cars will be ticket cars with ticket 5% yeah.

University of Colorado at Boulder

Applications of Integration