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2.) (5) A new machine is purchased by business for S200,000. For tax reasons, we assume the machine will lose S40,000 in value each year:(a) Ift is the number of ye...

Question

2.) (5) A new machine is purchased by business for S200,000. For tax reasons, we assume the machine will lose S40,000 in value each year:(a) Ift is the number of years since the machine was purchased, and if V (t) represents the value of the machine after years, WTile an equation in the form V (t) = mt +b for the value of the machine(b) Determinc the domain and rnge for the function V(t).

2.) (5) A new machine is purchased by business for S200,000. For tax reasons, we assume the machine will lose S40,000 in value each year: (a) Ift is the number of years since the machine was purchased, and if V (t) represents the value of the machine after years, WTile an equation in the form V (t) = mt +b for the value of the machine (b) Determinc the domain and rnge for the function V(t).



Answers

A new machine was purchased by National Textile for $\$ 120,000$. For income tax purposes, the machine is depreciated linearly over 10 years; that is, the book value of the machine decreases at a constant rate, so that at the end of 10 years the book value is zero. a. Express the book value of the machine $V$ as a function of the age, in years, of the machine $n$ b. Sketch the graph of the function in part (a). c. Find the book value of the machine at the end of the sixth year. d. Find the rate at which the machine is being depreciated each year.

Okay, so we want to know the original amounts or value of the machine. Well, that's when we have Year zero. That's a T is equal to zero. So the T would be able to 5000. So this is actually what middle here on out for Part B. He wants to know the value of the machine in five years. So that's 12 years you could divide. So it's all for B of or evaluated up. Five. Let's plug in replaced this tea with five. Okay, so that gives us 2973.2 and Alfred Part C. We want to know the value. In 10 years, the V A volatile 10 were replacing, or t value with the 10 now, and that gives us 1767.77

So here it says The starting value is 120,000. So that's going to be 1 X0 when it's brand new. So 120,000 If I plugged that in here, I get big old 1.20,000. Right? And so as for the end value it depreciates by cost Over a period of 10 years. So really the slope is just gonna be negative $12,000 per year. So that's the slope. So my function is gonna be Vfx equals negative 12,000 X plus 120,000. Part B asks for the domain, that's just when it's bought, which is brand new at X equals zero And it goes for 10 years. So that's the domain I have part C. Asks me to craft this out. Starting with the Y intercept Which is at 120,000. And it depreciates in cost, wow. Or in value goes down like this. This right here is zero 120,000. The X intercept is just pen. So that's 120,000 Divided by 12,000. So that's the x intercept. That's at 10. This represents years on the horizontal axis on the vertical axis. This represents the book value. Party asks What the book value is after four years. So I need to find bu four -12000 times four Plus 120,000. This gives me 72,000. So that's the book value after four years. Part kiosks when the book value is going to be 72,000. Well we just calculated that in the previous part. So that's gonna be after four years.

So here we have a company that bought a machine for $130,000 they're going to appreciate its value over 10 years using the straight line method. So Part A is asking us to express the value of this machine as a function up. It's H X. So we know that b of zero is going to be there in 20,000. Because if we just bought it, if no years that passed, then the value is gonna remain the same. So then the we're gonna actually they were gonna actually try to find how much it is appreciating by year. So if we have 120,000 over 10 that gives us 12,000 so it's appreciating by 12,000 per year. Okay, so then its function, then it's going to be what's us? Our slope is that 12,000 says it's going down. It's gonna be a negative 12,000 next, Plus our y intercept this right here. It plus 100 and 20,000. Okay. Part breathing is asking What is the implied domain domain is how we move in our excesses or excesses our years and so we're only gonna go from 10 or started from 0 to 10. Okay, Now, Part C is asking us to graft this. So you're gonna graph it and we're gonna go by 12. Right here. This is gonna be in thousands and statuses. And then that last one is there 100 and 20. And this right here. We're just gonna go by ones. Well, until we get to 10. Okay, so we start. Uh, Cyril. Great. And we have a value of 120,000 and so every year is gonna go down by 12. Right? So we're just making a straight line all the way to 10 years. It's not gonna be worth as much. Right. Okay, so that is their line now, Part D is asking, after four years, what is gonna be the value of this machine. So for years, So then this is gonna be a negative 48 1000 waas. 120,000. That gives us that we have 72. Oh, sorry. 700 thorny 1000. Okay. Part. Um, eat. Then asked us, um, after how many years they were gonna have a value of 700 20,000 Well, we just sold for that up here, and we saw that it was after four years, so that would be after four years, and so those would be your answers.

Here we have a company that purchases a new machine for which the rate of depreciation can be modeled by. Uh This equation D. V. D. T. Equals 10,000 times t minus six. From 0 to 5. He is the value of the machine. Uh Maybe in dollars after t years. What we want to do is set up and evaluated definite integral. That gives the total loss of value of the machine for the first three years. Now if you notice DV. DT from 0 to 5, if you plug in for values of T. It's going to give you a negative number. That that makes sense. That means that the value is decreasing. So if we integrate from 0 to 3 we will get the total amount of value units dollars that we have lost and that's what it's asking for. It's not asking for the value of the machine. Uh To figure out the actual value of the machine. We would need to know if the initial value was of the machine. So all we need to do is set it up so we'll say the loss of value loss of value and that loss is already interpreting the sign that's going to end up being negative. The loss of value. Okay? Um I'm not going to put equals here because I want to be careful if I put equals um then my my negative number um you know double negative means positive. So I'm just gonna set this up here as the integral um Of 1000 or 10,000. It is t minus six D. T. And over the first three years that's from 03 Now if this is a calculated question you could just plug and chug and go for it. Um I'm gonna assume that it's not. So uh to do this I'm gonna go ahead and pull the 10,000 out front just to make it easier and we'll put a colon appear sir. Loss of value colon uh t minus six with respect to t. So now it's just a simple little integral. The 10,000 goes along for the ride and t becomes one half t squared minus 60. We evaluate that from 03 and now I plug it in, I get 10 0. Still going along for the ride, plug in the top and you get three squared is nine times a half is nine halves minus 18 minus plugging a zero and you get zero. And so now we get 10,000 times nine. Have some minus 18. We need a common denominator. So that's nine minus two times 18 is 36 all over to. And what we can do is simplify this 10 divided by two. If you hunt, that's going to be five. And then nine minus 36 that's gonna be negative 20 seven. So multiply that out if you want. Or you can pick up your handy dandy calculator 5000 times negative 27. And we're gonna end up with a negative 135000 And presumably that's dollars. So the loss of value was not negative. We didn't lose negative amount, we lost positive amount. So that's why I didn't want to say the lost equals. Because if lost equals negative it's a double negative. So now we'll answer the question. So the machine okay ah devalued um from t. Equals 02 T. Equals three years comma 135 um comma 000 value units whatever they are, dollars, pesos, pounds whatever units were using. Okay now again if you had set this up in the very beginning and it was a calculated question then you could just go ahead and say, and we could verify here, Math nine, the interval from 0 to 3 of 10,000 currencies. X minus six. Close with respect to X and you get the same man's today, So All right.


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