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[10 marks] It costs S10,000 to buy backhoe attachment for tractor tnat asts maximum of five years, and the estimated maintenance cost in each year of ownership is s...

Question

[10 marks] It costs S10,000 to buy backhoe attachment for tractor tnat asts maximum of five years, and the estimated maintenance cost in each year of ownership is shown in the table below. The table Iso shows the trade-in value for the old backhoe if it is replaced after that many years of service You need to buy backhoe right now: What is the least expensive policy for buying and replacing backhoes over the next six years? Use Dynamic Programming methods to solve this proble em.vearMaintenance

[10 marks] It costs S10,000 to buy backhoe attachment for tractor tnat asts maximum of five years, and the estimated maintenance cost in each year of ownership is shown in the table below. The table Iso shows the trade-in value for the old backhoe if it is replaced after that many years of service You need to buy backhoe right now: What is the least expensive policy for buying and replacing backhoes over the next six years? Use Dynamic Programming methods to solve this proble em. vear Maintenance (S) 00O Trade-in value (S_ 00O OOO 5,000 OOO 3,000 500 00O



Answers

Use a calculator to help solve each problem. Depreciation A lawn tractor, costing $C$ dollars when new, depreciates $20 \%$ of the previous year's value each year. How much is the lawn tractor worth after 5 years?

We know about this tractor At time zero had a value of $142,000. And after 10 years, I believe that it's going to depreciate to 67,000. And we want to find a linear model that looks like this and we'll have a slope times T plus B. So let's find that slope. So we want to find the difference in the dependent values over the difference in the time and this is in dollars and this is in years. So that 67,000 -342000, it has dropped down in value, $75,000, divided by 10 means it's dropped $7500 um per year. And in fact, we're supposed to talk about what the meaning of the slope is. Uh, the sloping this many dollars per one year means that for every one year increase we end up as the year goes up by one, the value of the truck goes down goes down $7500. That's that meaning of that slope. You look at the units of the second coordinate, divided by the first coordinate. So it's dollars per year that it drops. So let's continue. Soviet T. Is equal to negative 7500 times T plus B. And we actually know the Y intercept because the beginning point is that 142,000. So our model looks like this plus 142,000. Now, the last thing that we want to find is what will be the value after six years. And so we're just gonna plug in six in place of that T. And let's see what we get times six plus 142,000 And the value will be $97,000. That's what the depreciated value would be predicted from that straight line depreciation model.

Que machine that costs in the onset. $60,000 then at the end of five years, it's worth nothing. So we're gonna find a formula for this which would follow y equals MX plus B. So we have the b value because that is the wine intercept. And you always have an X value of zero for why intercept? So now we need the slope, which we can do with slope formula change and why over change in X. So we'll get 60,000 divided by a negative five, which gives us a negative 12,000. So that will be your formula. So then, if you want to find some different values so at the end of one year, we're gonna use our formula, Why equals we had negative 12,000 one year plus 60 1000. Uh, okay, so you can calculate that. So that means the machine is worth 48 1000 and then the accumulated depreciation you would just take your 60,000 minus for 48,000 so depreciated by 12,000. And of course you could see it. That also corresponds to the slope. So then, after two years, do the same thing. Anger is going to appreciate another 12,000 so we could think 12,000 times two. 24,000. And then you could just take the 60,000 minus for 24,000 feet, like we'll calculate that and then that would give you 36,000 and then we'll do your three same thing. Depreciation would be 36,000 and then 24,000 would be how much it's worth. And then four years depreciation will be 48,000. So then it's worth 12,000 and then you see, by the time you get to five years, depreciation is the full 60,000.

Okay, here. Initial cost of the machine is 5200. So or the value of why intercepts, see, is it called toe 5200. Now equation off line is given by why is it called? Um X plus C Right now. Here, salvage value is 1100 after eight years. So when X is equal to eight, then why is it will do 1100. Now substitute the value off X Y and see in this equation and find the value off them. Now subtract 5200. Both sides of your eight m is he called toe 1100 minus 5200. And that is equal to negative 4100. Now divide by eight board side. So we're slope of the line is negative. 512.5. Now substitute the well you off em and see in our equation off line. Therefore, our equation off line is why is equal to negative 512.5 X plus 5200. Okay, now we need to find the book value after 01234 and seven, and eight years after. So first of all substitute X is equal to zero in this equation. Okay, zero multiplied by any value that is equal to zero and zero plus 5200. That is a call Do 5200. So book value after lawyers is 52,000. Sorry, $5200. Now substitute T is equal to one. Sorry X is equal to one here already a ballistics noti, you hear negative 512.5 multiplied by one that is equal to negative. 512.5 plus 5200 that is equal to $4687.5 Modesty or book value after one year. Right now substitute x they called to So when x When X is equal to two, then the value of wise for the $175. So this is Theo horrible. While you After two years now substitute X is equal to three. So when X is equal to three, then why is equal to 3600 and $62.5. So this is Thea our book when you after three year now substitute X is equal to four. So when X is a call to four, then the value of y is 3100 and $50. So this is your book value after for your okay. Now substitute X. Is it called Do seven. So here. Why is it called Negative? 512.5. Multiplied by five. Plus 50. 200. So when X is equal to five, then why is equal to 20? $637.5. So this is your book while you after why you're now substitute X is equal to okay. Seven year. So here. Negative. 512.5 multiplied by seven. Class 52,000. And that is equal to 1612.5. That is to your book value. After seven years. Now, substitute X is equal. Do eight. Sorry. Here. Eight, not seven. So when X is they cultivate, then why is it called zero? So solid value after eight years is zero. All right. Thank you.

So since the new machine costs $60,000 that means at time zero it has a value of 60,000. And then, since the lifetime is five years, that means that at year five, AT T equals five. The value of the machine is zero as no solid value at this point. Now, to find the slope of in order to find the formula for the value we need to first find Slope. So that's going to be 60,000 minus zero over his year on minus five. That will be 60,000 over a negative five. So M is going to equal a negative 12,000. So negative 12 thousands are slope and then I want to use the 0.5 Actually, look, to use 0 60,000 as the point that we inputs, that'll be 60,000 equals negative 12,000 time zero plus B so that the value of 60,000 which tells us that our value formula will be why equals actually will do well call V for value so V equals negative 12,000 um T plus 60,000 and then for part B or being asked is too um, following. We have to complete the depreciation schedule. So, um, in order to complete the depreciation schedule, we first need to look at what the schedule is showing at the end of year zero. We know that nothing has been depreciated, so the value is 60,000. At the end of your 1 12,000 was depreciated. So the value is 48,000. At the end of your 234 and five, obviously 12,000 will be depreciated each year, and the accumulated depreciation will keep increasing by 12,000. So in that third column it will go. 0 12,000 24,000 36,000 48,000 60,000 and then the value of it is going to constantly decrease by 12,000. So we'll start with 60,000. Then you're one will go. 48,000, 36,000, 24,000, 12,000 and then obviously, by year five, we will have $0. So that's what the depreciation schedule look like.


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