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Instudying the value of secondhand laptops, regression produces the precliction formula of y = 395.174 9.4682 where! is the predicted laptop price, and xis the lapt...

Question

Instudying the value of secondhand laptops, regression produces the precliction formula of y = 395.174 9.4682 where! is the predicted laptop price, and xis the laptop age in months. Which of the following correctly interprets the slope and the intercept estimate? Ex- plain and show your work:(A) 395.174 is the preclicted price of a laptop that is brand new_ B) 395.174 is the predlicted laptop age in months when its price is 0. For every OHIC month increase in laptop $ age, its predicted price in

Instudying the value of secondhand laptops, regression produces the precliction formula of y = 395.174 9.4682 where ! is the predicted laptop price, and xis the laptop age in months. Which of the following correctly interprets the slope and the intercept estimate? Ex- plain and show your work: (A) 395.174 is the preclicted price of a laptop that is brand new_ B) 395.174 is the predlicted laptop age in months when its price is 0. For every OHIC month increase in laptop $ age, its predicted price increases by 9.468 . (D) For every OHC MOH MCTOASC M A laptop' $ age; its predlicted pice decreases by 9.468 , E) For every one month inerease in a laptop $ age; its preclicted price decreases by 9.468 .



Answers

DEPRECIATION A laptop computer that costs $\$ 1150$ new has a book value of $\$ 550$ after 2 years.
(a) Find the linear model $V=m t+b .$
(b) Find the exponential model $V=a e^{k t}$
(c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years?
(d) Find the book values of the computer after 1 year and after 3 years using each model.
(e) Explain the advantages and disadvantages of using each model to a buyer and a seller.

All right for this problem are ordered. Pairs are t comma V, where t is the number of years or the age of the computer and V is the value. And we know that it went that when the computer was new, it was $1150. And we know that after two years, its value was $550 and the first thing we want to do is find the linear model V equals M T plus B. So we're going to find the slope em. So we have y tu minus y 15 50 minus 11. 50 over x two minus x one to minus zero. And that gives us negative 300. We already know the Y intercept. We have the 0.0 11 50. So we have the model V equals negative 300 t plus 11 50 for Part B. We're going to find the exponential model, which is V equals a times E to the K T. So what we need to do is find the value of A and find the value of K, and they would be the initial amount or the initial value of the computer, so that would be 11 50. We still need to find the value. Okay, so what we'll do is we'll use our second order pair. We'll use 550 for V and 11 or excuse me to for tea, and this will allow us to sell for K. So the next step is to divide both sides by 11 50 and 550. Divided by 11 50 reduces to 11 23rd. Then we take the natural log of both sides, and then we divide both sides by two. So we have our value of K. We can substitute it into our model, so that gives us the value is 11 50 times he to the natural log of 11 23rd over two times T. That's the exponential model. So the next step in the process is to use a graphing utility and view the two models. So here's my graphing calculator and have gone into y equals and a typed the equations in there. So we have the linear model as why one and we have the exponential model is why, too, and then for a window I went with 0 to 10 on the X axis. Those would be the years the age of the computer. And then I went with 0 to 1200 on the Y axis. That would be the value of the computer. So let's take a look at those graphs. We have the lying, and then we have the exponential curve in red. The point of intersection is going to be time to. That's where they have the same value, but before time to so between time zero. And to notice that the red curve is lower. That means the exponential curve is lower. And that means that the exponential curve shows greater depreciation in the 1st 2 years because it shows a lower value. Okay for Part D, let's use the table. So I've gone into table set. Second window is table set. I changed my settings to independent Ask that way I can type in my own X values, and then we go into table, which is second graf. And if we want to know the value of the computer after one year, we can type in a one press enter, and the calculator will generate those y values for us. So for the linear model, the value would be $850 for the exponential model, it would be $795.30. Similarly, if we want to find the values according to the models for three years, the linear model shows the value of 2 50 the exponential model shows a value of $380.36. So in terms of advantages and disadvantages, if you were the seller, you would want to make it look like the computer was worth more, so you would want to use the linear model before two years, and the exponential model after those would be advantageous to you. However, if you're the buyer, you want to make it look like the value of the computer is lower, so you can pay less. So you would want to use the exponential model before two years on the linear model. After two years

Question 35 tells you that a laptop computer costs $1150 new, so At T equals zero, and it costs $550 at two years. So the first thing you want you to do is find the linear model V equals M T plus B. So for starters, you can use these two points zero 1150 and to 5. 50. Um, And to do so, you can use the point slope formula. So why to minus y one cools m Which is your slope x two minus x one, or in this case, getting em? You can say why to minus y one over x two minus X one, which is equal to 5. 50 minus 1150 divided by tu minus zero. So you're slope. Is it negative 300. Okay. And using the equation y equals MX plus B and plugging in the 0.0 1500 you can see that your baby is going to be equal toe 1150. Therefore, equation is B equals negative 300 t plus 1150 for part B. They want you to find the exponential model V equals ae to the K T. Ah, starting off with our 0.2 equals zero V equals 1150. You can plug that in to find a two u zero each at the zero is always gonna be one. So our A is equal to 1150. Well, again, our next point t equals two b equals 5 50. You have 5 50 equals 1150 e to the K times two. From there, you can take Ln of 5 50 divided by 1150 equals Ln of e to the two k that gets rid of the East, you have Ln 5 50 over 1150 equals two k. So K is negative. 0.36 87994716 in full. But you can shorten that to be equals 1150 e. It's the negative point 369 t to get your expression moving on to part C. They want you to use a graphing utility, um, to graph the two models in the same viewing window and see which model depreciates faster in the first two years. Eso to do so. I'm gonna use a T I 84 and your white equals said why one equal to negative 300 X +1150 y two equals 1150 e to the negative. I put the full decimal amount just to get the most accurate answer in your window. You can set your X men equal to zero X max equal to five. You're Wyman should be zero and your y max, your maximum value going ever going to get its 15 11550 So I just said it to 1200. Ah, What you should see is first, you're why one value decreasing linearly through. I won and you're white too. First dips low and then kind of comes out like that. Um, and right here is about your two year point, and you can see from there that V equals 1150 e to the minus 11500.36 90 depreciates faster in the first two years. Yeah, but moving on to part D, they want you to find the book values. After one years and three years, um, using both models. So for T equals one. Yeah. First model negative 300 multiplied by one plus 1150 b equals $850 for your second function. Well known 50 e to the minus 500.3 69 times one you get V equals 795 $0.14. T equals three. Yeah, plugging in again minus 300 times three plus 1150 equals to 50. And for our second function on 150 e to the minus 1500.369 times three you get V equals 380 $0.13. Ah, for the last part party, they want you to explain the advantages and disadvantages of using each model to the buyer and seller. So you can see from that graph again. That's your first function. Your second function. The advantages of the linear model to the cellar is after that two year mark, you're gonna be able or excuse me to the buyer. They're gonna be getting it at a great price. So it's basically at this market's going about 3.8 years. This model is telling the seller that the value is zero. So, uh, fire gets a good price. But the seller maybe undervaluing their book. And after 3.8 years, they don't make any money for your exponential model. Your seller will always make a profit because this function is never going to reach zero. And the buyer may be paying more than it's worth because it's never going to breach that. That same zero value is a linear model. So and there you that is your answer for a question 35.

So we have a situation in which a computer at time zero was worth $575 and then two years later it's worth $275. So 5 75 to 2. 75 in two years and we want to start out. We want to write one model as linear, and we want to write one model as exponential and exponential with the base E. And they show that is X could also do. It is T. So I have a couple of things to do here. First of all, let's do the linear model, and we know that that's the Y intercept. We also know that we can figure out what the rate of change is. So let's find the slope. The slope is the difference in the Wise, which is to 75 minus 5 75 over the difference in the exes, which is two years minus zero years. And so this is dropping $300 in two years. So it's dropping $150 per year is what a linear model would predict and, like, get this citizen move up. So let's write our model our model is negative 1 50 times the input variable of time and years plus $575. So there is linear. Now let's find our exponential model. Well, we know little A is 5 75 and then we want to plug in a time. So we know what this value is. That is going to be that 5 75 automatically here, because is our initial amount. And the only thing we have to find is R. K. So we'll plug in a point. We know that it will be worth 275 once we have a time of two years. So we divide both sides by the 5. 75 and we have e toothy two k is equal to this to 75/5 75 and then we can take the natural aga both sides a couple different ways. You can saw that exponential, but I'm gonna move that equal sign there and right, Ellen right there. And then we could do our log of a power. In fact, I'm not even well that well equal sign even more. We'll move away over there. I'm gonna do log of a power and I will get to K times, Elena V. And now that's moved down. But Elena B is one. So we deceptive divide both sides by two. And we'll have what that little K isn't that quick. Hit that on my calculator. So I have Ln of 2 75 divided by 5 75 calls my parentheses and then divide that by two. And I find that that K value is negative. 0.3688 and it's negative because we're decaying. So our model for exponential is that 5 75 times e to the negative 0.3688 times X or T whichever variable you want to use. Then we're asked to compare and grab those on our calculator. So I would set up my window, you know, your decaying. So let's look at a time frame. Really? We're looking at, like, the first two years, so I'm gonna have my windows set up to go from zero to, but I'm gonna go up to five years. What the heck? So 0 to 5 scaling by ones, and then I'm gonna have my why minimum be at zero and my highest value out of school up to 600 and I'll scare by 100. So again, this is going to go up to 600 we can put both of our models in. We've got our negative 1 50 x plus 5 75 and then we have our 5 75 times e to the power of and we have negative 750.3688 x, and we can let the calculator graph those. And when we dio, we naturally end up getting after all this right, We end up getting that linear model. Does that and the exponential model put in red. When I look at it on my screen, it dips below a little bit. Not a lot. But then at two, that's a common intersection point. Because we know two and 2 75 is a common point for both those as is this zero and 5 75 and then this graph levels off. So I believe part C or part part V as you look at the graph, and we can see that during that first year, our first two years that it appears as though the, uh, the exponential model seems to decay faster and drops down faster. Now, if we look at a little table, we can see what happens in those first years. So here's our linear model and here is our exponential model and at time, zero. We know they're both 5 75. We know that, and so I could go to table set and I can start my table at zero. Now I'll just go up by one. So let's just look at a few years. And now we set up table and we have after one year, the linear is worth 24 25. But the exponential is birth 397.65 And then, after two years to 75 we know that their goal is gonna be the same and then notice what happens. Linear keeps dropping until we get thio time. Three were at 1 25 time, four were negative. So apparently you have to pay somebody to take the computer and versus here for part, 390.1 831.52 so we can see that although the exponential does decay faster, then later on, it decays more slowly and linear drops out between three and four years to have absolutely no value with the computer, which is probably the case anyway, so hopefully that's helpful.

Leaving you to solve a problem. And that if here, a parties leader the position model. So a off peak was 11 by 80 plus 5 50 minus 1150 divided by tu minus zero. In the deep, we get a off T equals 1140 minus 300. These exponential depreciation model. So a f t equals a. Not here is to beauty we just 11 For Israel, it is to beauty. 5. 15. It was 1150 It is to be into they get be because and then 55 is the road they wanted by one month, 50 they wouldn't, but they got because minus 0.3688 pay off deep equals 11 by zero. Here is the mind a zero point 3688 solving using equation for decay because equals, you know, here is to when a 0.693 indu do you do it by behalf? The graphic utility record and this steep and he purchased exponential shows. Exponential shows faster depression in first to is, the linear model has same depreciation value throughout the life period. off left off. Also, the book value off the car is depreciating every year, so it helps the by it too artificially different. The price off car and seller will get lower priced, and it's really good. Exponential decay helps to spread the decay, and it is compounded continuously, so it represents the really value off the car. Helping the buyer and seller toe get the rial value tax benefit force a lot. It's slowing exponential deposition because the appreciation about is it smaller as time goes on, that kind of question Thank you.


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