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Problem 85 Hard Difficulty

A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate $ f = f(t) $, where $ t $ is the time measured in months since its last overhaul. Because a fixed cost $ A $ is incurred each time the machine is overhauled, the company wants to determine the optimal time $ T $ (in months) between overhauls.

(a) Explain why $ \displaystyle \int^t_0 f(s) ds $ represents the loss in value of the machine over the period of time $ t $ since the last overhaul.
(b) Let $ C = C(t) $ be given by $$ C(t) = \frac{1}{t} \Biggl[ A + \int^t_0 f(s) ds \Biggr] $$
What does $ C $ represent and why would the company want to minimize $ C $?
(c) Show that $ C $ has a minimum value at the numbers $ t = T $ where $ C(T) = f(T) $.

Answer

a. $[0, t]$
b. $\frac{A+F(t)}{t}$
c. $C(t)$

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spotlight. One of them probably didn't want. We're told that the manufacturing company owns a major piece of equipment that appreciates at the continuous rate at which is a function of T. Where T. Is the time make it in the months since its last overhaul? We're told that a fixed cost A. Is encourage each time the machine is overhaul. And the company wants to determine the optimal time, big T. And months between overhauls part A. We have to explain why the integral from zero to little T. Of F. Of S. D. S. Represents the loss and value of the machine over the period of time T since the last overhead bills good things. Well, well, let big F F. T. B. Value of the equipment at the time. Little teeth. We know from what we're given the greatest depreciation. Little FFT. It will be clear. Oh, oh yeah. You know what? I rewatched this within the last six years and now it's all come because the derivative is the rate of change in function. It follows that the derivative of big F. F. T. Is equal to little Fft. It's the high dollar in the integral of both sides. And applying the fundamental form a calculus. It therefore follows that Big F. F. T. Is equal to the integral from zero to T. Of little F. Of S. B. S. You got to get a gay guy to watch your girlfriend. This tells us that the integral gives us the total loss of value at time. Little teeth. You fuck grown me. Yeah. No boys or girls. I need you. I need you to watch my girlfriend and part. We're told that see which is a function of T. Is given by the equation. You see it equals one over T. So good Times A. plus the integral from 0 to of little F. Of S. Mhm. I don't know. They start dating. Sorry. Whereas what C. Represents and why the company would want to minimize the seat. Yeah. A friend who is well from party, we already know that this integral represents the total depreciation in the value of the machine since the last overhaul. We're also told that A. Is the cost encouraged by machines overhaul and add the two. We get the total costs related to the maintenance of machine. So this expression is the total cost. Yeah. Therefore the function C. Of T. Is the total cost over the times people last overhaul? Oh geez I don't like doing it for this video. You're going to add observance. Every system ever. Japanese imports a full like arcade, their house. Anything from. I don't overhaul the machine. It costs the company in terms of the depreciation of the machine. If you overhaul there's also it's possible to do it so minimizing C. T. And we were it was the first time like his Children like, oh my God, this rule is just the best thing that ever happened. My dad just gave it to one of his friends company dad redid some bar and his payment they gave spends less overall. And um it was like a country weather proofing would that was like thinner than it usually looks, and the guy was like and it spends less. This means that the company and my dad, she's been like, no, this is one creepy profit. Just complain. Which is the goal of the company generally is to make as much profit. And a guy who made him reduce just some fucking pussy, like finally in part C, whereas to show that C. Has a minimum value at the numbers, little T. Equals big T where Cf Beatty is little. So this Hitchcock would get Marnie. Yeah, we haven't seen it. I wouldn't love pacman machine. Wait, listen. But he's an amateur zoology. I haven't seen it. Well, at the time they team well, if we want to find the minimum value of C will want to take a derivative inside of equal to zero. To find the critical numbers from part B. We are to determine that they see was the average expenditure per month. We also determined the company want to minimize C. A. T. So let's take the derivative of C. F. T. C. Prime of T. You know. Uh Using the product rule, this is the derivative of one over T, which is negative one over she squared times that integral expression A. Plus the integral from zero, the little T. Uh F S D S plus one of the T times the derivative of the second factor is a conference of served as simply zero. And by the fundamental theorem of calculus, the derivative of this integral is simply FFT. Of course I do. Now let's factor out of one over T. We can write this as one of her tee times and she's holding. Huh? That's right. See -1 of the two times A. Plus the integral from here. The key of F. F. S. Plus, that's a team. And as I said before, we want to find when this expression is equal to zero. One of the treatment that would be equal to zero. This implies that the other factors must be equal to zero. So it follows that she steals from his safe and then the traps are negative. One of the T. Times A. Plus the integral from zero to T. Of FMS. He changed her like an animal. Like he's like he's like, she's like zero shares. You know, he's reading books about the female criminal mind and that's fucking awesome. It's actually really in fact this looks very similar part of this to the equation for C. And C. Yeah. Sean Connery talks about so I'll solve for F. F. T. F. T. Equals one over T. Times A. Plus the integral from zero to T. Uh F. S. D. S. Which this is the same as CFT. I wasn't gay. Take it back. You're hurting Hillary, you're not allowed to say that ever see list Elizabeth Warren's and uh somebody asked her about like now you see the two time of T. Equals zero only to the time T. We'll call this time the T. His dick in his mouth at least anyway. And therefore it follows that C. As a minimum value. Oh yeah I want to see him in the chest and fuck nothing. There is no. He's fucking american cycle at the number T. Equals dignity. Our definition. He's shaky. Was the optimal time between overhauls. Where as you saw by solving our differential equation, effort didn. T equals C. A big issue on toys. In other words, our costs are monthly cost of the minimum value where the monthly cost is equal to the great as appreciation Liz, What do you certainly? Mhm. Anyway, time for some maize? He's just got a whole year.