mathbfm-in-detroit-hertz-rent-a-car-has-a-fleet-of-about-2000-cars-the-pattern-of-rental-and-return-

Question

$[\mathbf{M}]$ In Detroit, Hertz Rent $A$ Car has a fleet of about 2000 cars. The pattern of rental and return locations is given by the fractions in the table below. On a typical day, about how many cars will be rented or ready to rent from the downtown location?
$$
\left[\begin{array}{ccc}{.90} & {.01} & {.09} \ {.01} & {.90} & {.01} \ {.09} & {.09} & {.90}\end{array}ight]
$$

Michael Jacobsen · Accepted Answer

in this example were considering a stochastic Matrix P that will describe the migration for rental cars that come from the airport downtown and a metro airport area. So, for example, if we consider this entry, that means the car came from the Metro Air area and there is a 1% chance it ends up in the downtown area. So what we want to do next is to determine the steady state vector Q of P. And to do that, we have to first solve the matrix equation that p Times X equals X. If we subtract X Trible sites and factor it out, were then solving the equation p minus the identity matrix Times X equals the zero vector. So let&#x27;s write down what the augmented matrix will be for this system of equations. I first noticed that when we subtract the identity matrix frumpy will just be subtracting one from each one of these entries, and that will give us a negative 10.1 in each one of those positions. So are augmented. Matrix is going to now be of the form negative 0.1, then copy 0.1 point 09 Calling to is 0.1 Then, from this subtraction, we have a negative 0.1 again and copy 0.9 0.9 0.1 And then we have a negative 0.1 once again. And once again these entries here came from P minus I. Now we&#x27;re augmenting with a zero vector. It&#x27;s all placed, though, that here and now we need to reduce this system. This is a very tedious system to do by hand, and so it&#x27;s heavily encouraged to use a calculator or a computer algebra system in a calculator. I found the road reduce echelon form is 10 negative. 0.919 zero Row two is 01 then negative point 192 zero and then the third row contains Onley zeros. This tells us then that x one is equal to positive. 0.919 x three x two Next is positive point 192 x three and x three is a free variable. Saw right x three equals x three. This tells us then that the solution X to this matrix equation is the following. I&#x27;m going to pick a one in place of x three, then that will produce a 0.192 and a point 919 for the entries of X. The next thing we need to do to find the steady state vector is take the entries of X 0.919 and add them together. So at 0.192 to the first century plus one, and this is 10.2111 altogether now, with this value 0.2111 we can assert that the steady state vector Q is now equal to 1/2 0.111 scaling that Vector X. So multiply each entry of X by the reciprocal of 2.111 And after rounding to three decimal places, we obtained point for 35 in the first entry, 0.91 and then point 474 for their remaining entries. Next, let&#x27;s say what this means in particular and will focus on the second entry here, which corresponds to the downtown area. So we&#x27;ll say here that if there are, let&#x27;s say, 2000 cars in this rental car company fleet, then in the long run, there will be the 2000 cars multiplied by the 0.91 that we have highlighted here. This funny is equal to 182 cars, but that tells us that the 102 cars ends up in the downtown area, so this will be in downtown. So this will allow the rental car company to plan on how to redistribute cars in an equal way, looking at the steady state vector and where the cars will eventually end up.

Mihir Nayar · Answer

Hey, everyone, today we&#x27;re solving problem number 14. Uh um chocolate in section two of your textbook. So let&#x27;s jump right into it. In part A. You&#x27;re asked to draw time series graph, um, off the car sales for all three companies from 2007 to 2000 Tech. So, um, in times, years graphs, When you&#x27;re making them the year or any, you know, the time element always goes on the Y axes and then your number of cars and service I&#x27;ll just write cars here will always go about their variable Essentially will always go on the x axis. I mean, on the y axes, whereas the time always goes on the X axis is so again we have 07 08 So I and just 10 something like that. 07 I got 07 is representative of the year 2007 08 is representative of the year 2008 09 is representative of the year 2000 and nine and finally, 10 was the representative of the year 2010. All right, And then you look at your maximum value in terms of number of cars and service and top and thousands. And for hurts, it&#x27;s somewhere above 300 but below 350. So I&#x27;m gonna make a scale by 50 for this particular problem, Hogan. 350. I wasn&#x27;t course. And service is going to be my max for my wife. Value on the scale at least. Not like Max about you. I&#x27;m gonna go by fifties of 50. 100 1 50 200. I want too small. Go. 50 100. 150. 200. 250. 300. So I can just leave it by 50. So you have 50 and obviously this would be 100 101 150 would be here then. I don&#x27;t know the 5200. And then you have 250 here so again goes up by fifties. All right, now we&#x27;re time. It&#x27;s time to grab. Um, I recommend making key here. So right. A key for your graph, in part at least. And the reason for making he is to kind of just show what different colors if you&#x27;re using different colors when you&#x27;re drawing this final answer or different shapes. So like if I had a star on with different instead of a dot on them in like an ex The ex represent Avis story represent hurts don&#x27;t represent dollar. You could do something like that. But since I have the colors on this my port, I&#x27;m going to utilize that. So, God, for my key for a I&#x27;m going to say anything red. So ride thought or a red line is always gonna, um, indicate something about their hurts. Car company. All right, then we&#x27;ll say green. It&#x27;s a green dot or green line would be indicative of values for Avis and the line that we&#x27;re gonna make it. And then last but not least, I have blue. So blue obviously has to represent anything blue on my graph pasta represent information regarding dollars company for their cars and servers from 2007 to 2010. So now we&#x27;re just basically in a graft. The data, um for hurts. I can see that I&#x27;m going to have a raffle. It&#x27;s like this. So a little bit about 2 300 then a little bit below that. But still look 300 that dips below. I&#x27;m drinking too, that it picks back up just a little bit in 2000 gone straight lines. Probably easier. D&#x27;oh! With a pen and paper on. Then we&#x27;re moving on to Avis again. This is just graphing Tina points, guys. So in 2007 Avis, how little bit over 200,000 Parcel service that I went off in 2008 beholding about 2 52,009 It&#x27;s shot up close to 300 and then, oh, Tana, dip below Hertz value. But again, just by a little bit. Not too much. So again, connect the dots with straight lines, try a bus to make it street and the beautiful thing about her keys. Now we don&#x27;t have to tell her. We already told our reader with our key, but all the colors, record sights. Um, So with dollar above 1 50 it started out in 07 That in 08 Um, it dipped below 1 50 Just minimally, I think, 09 dipped a lot 10 just but very even space. Not much difference in 09 and 10 for a dollar. So there you have it. There&#x27;s your times. Here&#x27;s Graf again with times, Here&#x27;s graphs you connect all your points of straight lines and on the X axis is always going to be your time. All of it. Moving on to part two. Any of the questions which asks us for the first part Time to figure out what the market share linger is in terms of, um, number of car sell sold in thousands from 2070 laws and attend. So if I look at my graph, I see specifically that the red is leading the charge and the red is hurts, So hurts gonna be you&#x27;re a guy. This is all based off of the graph you just drew, as will typically be the questions with stats, they&#x27;ll follow previous part Always. So that&#x27;s something good about it. All right? Hertz is your leader. And then if you just wanted to notice different graphical trends, which they do ask about, um, I&#x27;ll just say it out loud for you guys. So Hertz had a general decrease in sales overall, um, they picked up a little bit from 09 to 10 but overall decrease, um A. This was the one that had a drastic increase, especially from 2008 to 2009. But then they had a bit of a dip so leveled off in 2010. But it&#x27;s still increased overall from 07 to 10. And then finally, dollar howto probably. Yeah, it did have the biggest decrease in cars in service. Uh, there you have it. That&#x27;s kind of the brief summer you give and then in part, see your astronaut. You are asked to now draw another grab this time a park, Um, for rental cars in service in 2006. I&#x27;m just gonna draw by two axes. Mom, again, The year 2000 and 10 is going to be on your ex axes. The different bars are gonna be representative of different companies. So I would have again, I&#x27;m using this key for both grafts. So this would apply to this and then this he would have lied about. So I&#x27;m just going to make a red mark for Hertz. I got nuts. The beauty of your key could apply to more than one draft if you are soft and more than one part and need to draw a graph into the parts of the question So we know that route is representative. Anything regarding hurts information. Green is representative of anything regarding a this information. And finally, Blue&#x27;s representative of anything. Great party dollars affirmation and the number, parts and service. And again, um, you&#x27;re why axes is going to be the same again. It&#x27;s the number of cars and service I&#x27;m not writing. Got out fully. So just like ours. You guys were aware of that. And now we are going to draw our Marga. Um, but the first thing you&#x27;d use to make a scale for a buy axes I would choose to start it. Zero kind of drops with lying to show. Get somewhere else that&#x27;s larger. This is 100 right here. Um, and from 100 onwards in our y axes, I would go up by 25 1 25 1 50 We need to get to 300. Rubber that to 25 2 50 She was 75 300 25. So I&#x27;m gonna make that bigger. It&#x27;s home of space that I just like a 1 25 12 one said to home drug 25 Do you for certainly 3 25 there. So I&#x27;m just gonna mark I&#x27;m Jake. I mean, the fifties. So 350 200 that all right, now we&#x27;re going to draw our three our army, our three traps. Where? Each individual. So again, you&#x27;re only asked about the year 2010. So for Hertz, in 2000 and 10 I had a value of 290,000. So get to 90 somewhere out there, Draw wine there to shave it in. Try your best to shit in. And I&#x27;m going to write the value on top of the bar two. All right, um, for Davis in 2000 and 10 my value. Well, I cz 270 so that&#x27;s gonna be kind of somewhere, but not exactly height, too. Not the same. Height. Never gets as by hurts, Bill, but it&#x27;s still pretty solid. 270,000 got shade that in much easier dependent papers. 270,000. And then finally, dollar, which did the worst, Uh, it had in 2010. Uh, a total of 100. 1000. We&#x27;ll be out here 18 Shape that in 108 And there you have it. That is how you saw problem number 14 from chapter one of the text. If you guys enjoyed my video, please of the light. But next to it I hope you all have a nice day. And thank you for watching.

Zach Steedman · Answer

all right. In a couple of my previous videos, I went ahead and calculated the means variances and standard deviations by hand. I&#x27;m not gonna do that because this textbook assumes that we kind of have Microsoft excel. And so I&#x27;m actually thinking being better use of our time to show you how you can broke plug this into Excel or Google sheets, which is what I use to find those three values. Okay, so the first thing we need to do is mean where you type in is your type in average, after you type in average. Oh, put equal shirts Excuse me, but equals a V. And once you do, a V average will probably pop up, select it and then highlight all the cells. So I am put it all this data into people sheets highlighted all the cells and it spit out the average. Oh, I guess I didn&#x27;t put that number similar to variants, right? Type of equal v a r. And you can see it&#x27;s the 1st 1 that comes up similar to standard deviation for standard deviation type in sd Dev. And then it should come up. And so here are the three values after and for all those you&#x27;re selecting the cells and clicking Enter. Right, Um so hear those three values That means 38 the variances 97 the standard deviation is 9.8 in variance and standard deviation are different ways to kind of say how much this data fluctuates and or how close it is to the mean. And so when we compare it to the West, the mean is exactly the same. So we can say, on average, you would pay the same amount of money per day on the West with these West Coast cities, as you would on these East Coast cities. However, the variants on the West Coast is a lot smaller, and so is the standard deviation, which is another way to measure how close we are to the mean. So if we had these numbers for the West Coast cities are imagined, they would all be a lot closer to 38. So you can see here this one gets all the way up to 58 all the way down to 30 I imagine for the West Coast cities, we&#x27;d be in the high thirties. Low forties, probably wouldn&#x27;t have any in the fifties or the twenties, we would all remain a lot closer to that mean compared to these East Coast cities where there are several different prices.

Nathalie Luna · Answer

Okay, So for this problem, we are going to define a linear function that describes the price for one day rental. So in this firm, carbon tal charges $44 and the charges an extra 28 miles per mile. So since the price of pence on the miles we&#x27;re gonna name or variable m to be hello stripping, okay. And the function itself, it&#x27;s going to be pfm. Um, still, it&#x27;s the price for the rental. Yeah, having driven and looks. Okay, so since Yep. So the function is called P for price. Um, it is it&#x27;s going to be in dollars, and it&#x27;s going to depend on the miles driven, which is so now to define lips, too. Yeah. Define the actual formula P S m. So first, we&#x27;re going to do this part. So, um yep. So it&#x27;s going to be 20 cent, 28 cents per mile, So that&#x27;s zero point 28. Since we want this to be in dollars and times. Um, that&#x27;s our variable. And these $44 which is constant. It&#x27;s going to be over here with no ml just by itself. So this is our linear function. Describing the price for the one day rental and in terms of the mouths driven

Alaina Shelton · Answer

Welcome back, friends. Let&#x27;s take a look at this problem. So in this problem, we&#x27;re talking about a car rental agency that is going to rent vehicles and win this car. Rental agency rents vehicles. It is always, ah $180 per week, plus an additional 25 cents. So 250.25 per mile, I&#x27;m gonna use em for miles. And in your budget, you have budgeted $395 for your vacation rental with this vehicle, and your job is to figure out how many miles you&#x27;re able to drive for $395. So what we&#x27;re going to do here with our linear programming model here is to first subtract 180. So, you know, without it out that it&#x27;s going to cost $180 out of your $395 budget to rent this car. So we&#x27;re gonna have your 0.25 mile or so 25 cents per mile. Um, and we know that 3 95 minus 1 80 gives us 215. So you have $215 left to budget for your mileage. So now we&#x27;re gonna divide by 2150.25 on both sides in what we end up with is miles could be 860 miles Max. So 860 miles is how many miles you will be able to drive.

Victoria Dollar · Answer

So for this problem, we&#x27;re gonna have X equals fans. Why is going to equal large trucks? I&#x27;m sorry, Small trucks. And then I&#x27;m gonna set Z t equal large trucks. And for, um, for this particular problem, it is saying that the company plans to spend $7 million on two new vehicles there on 200 new vehicles. So we know that the total of all of the&#x27;s is going to be 200. We&#x27;re also told that each commercial vein will cost 35,000. Small truck will be 30 and large truck will be 50,000. So we&#x27;re gonna have 35,000 plus 30,000. Why, plus 50,000 Z. And we&#x27;re told the total is going to be seven million. So we&#x27;re gonna come back so I can actually decrease this by three zeros. So if I do that, all have 35 x plus 30. Why plus 50 z equals 7000, and you could, I guess, go hand go through across out another. But I like to keep all of my whole numbers and then last but not least, we are told that past experience shows they&#x27;re going to twice as many vans. Thin, small trucks. So that tells us that we&#x27;re gonna have X minus two. Why is going to equal zero? Another way to think about this is that we need twice Aziz. Many small trucks. Did they end? So we&#x27;re gonna have to balance that out to equal that 20 or to have that equal. So when I set up my matrix, which is 1 35 wine 1 30. Negative, too. 1 50 0 with my solutions is being 207,000 and zero. So what I find using a calculator is that I get a I need a, um 120 vans, 60 small trucks and 20 large trucks.

Problem

Let $P$ be an $n \times n$ stochastic matrix. The…

Problem 16 Hard Difficulty

Answer

on a typical day, about $( .090909 ) ( 2000 ) = 182$ cars will be rented or available from the downtown location.

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on a typical day, about $( .090909 ) ( 2000 ) = 182$ cars will be rented or available from the downtown location.

Linear Algebra and Its Applications

Lily A.

Caleb E.

Kristen K.

Michael J.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Lily A.

Caleb E.

Kristen K.

Michael J.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications