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Numerade Educator



Problem 9 Easy Difficulty

Show that the median waiting time for a phone call to the company described in Example 4 is about 3.5 minutes.


$x=5 \ln 2 \approx 3.47$


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Video Transcript

OK, so food this question. Um, we have to take the probability events to thank you for giving an example for, um textbook. So that is FT is what you've given to you, eh? For tea was defined. It is zero for all t for all t. Your tea is lesson 10 and then it is 0.2 you to define us t over five if Ty's raven article sitio Now, the question is we have to show well, to show that the media for this, uh, what is probably build inside Duncan, if 3.5 really show median your 3.5. So what, you handed his question? Um, What does it mean? We talk about the media. If you have an interval like this, you're going from a to be now. The media is exactly between A and B, which is, you know, if you were to market his brother's every year halfway, so so the media and it should be between half to be here because this is linear the line. Um, so that's exactly why we need to do here to answer this question. Way need. We need some some real number. So some real number X number X. It's like that side that, um if you take the integral from zero to X and then you integrated f of tea, you should get 1/2 orders, you know, 0.5. So, um, these X here is what we're looking for. So let's do this integral to find out what What kind of ex valley lies exactly in the middle? Um, So fft you get right. So when she's lessons hero this in thefifties So we don't need to worry about that. The only to worry about at that half of t when she is greater than all of course, is your Oh, so that means this injury will becomes, um, you know, condoms from zero to x, and then you have 0.2 year to the minus a tea for five teeth, and he's has to be equals to 0.5. So one thing you will notice that Ah minus t over five. The zoo closes minus one of t and 1150.2. This is equivalent to minus your point to t so I can replace. This is just that it will. It will make you want to go Easier to computing. One step. Uh, you're not come from what is You could keep one of the five. This is over. I'm gonna do this. So is minus and then years Europe going to tea right now. Uh, this one here did appoint Tuesdays a constant so you can push that outside of the integral. And then you just integrated you to the mind of little cling to a T. So during this, we can't with you. No 0.2. Now, if I intubate this part here, I will get E to the minus. Theater went to tea divided by finest, beautiful to t 0.2. And then I have to value these forms, you know, tracks. And of course, this is the coast right inside, which is here upon five. Clearly, we can see that you know you're going to get going to cancer. You got one here, you cancel this. You get minus one. So you divide the left hand side with minus one. So whatever you get from here, you get minus one. So let me create myself room here now. So minus sign would come along. So you get a minus side, and then you plugging X the upper limit of integration. So you get minus E to the minus, you know, going to exit and then then minus minus. And then you plug in X equals zero. I mean, t it was you for the low limit. So you get minus minus E to the minus zero. So use mine a zero on any Sequels, your 00.5. So use my view will be what so here you got, uh, So the two minor science you will cancel, so you get minus e 80 minus, you know, going to X plus one. It was a tough one. Fire. Now you take, uh, you bring these bone fired his way, and then you take the one with the minus side. You take it to the right hand side. So doing theirs, we obtain. So you get one miners. Get up. 15 isn't closed to e to the minus. You're a 0.2 eggs. Which would imply that you know, 45 is opposed to eat mine of beautiful into Rex Now to solve for X In this equation, we take the natural logarithms you goes and then you have 0.5. Music was too long. The agents of minors. You four into five x so you can see the left of the right hand side here will become gets minus 0.5 x right way. No solving for athlete off 10 you divide both sides minus 0.2. So x will be loan over 0.5 divided by minus two. You plug you plug this interview calculator you're looking actually sit close to 314 and then 657 which is roughly 3.5. And that's what the question was asking. Huge asking you for you around this after you use rifles to 3.5. So the median is X except for 3.5.