profitability: revenue gen. at rate of R'(t) = 6025-8t^2....

[mvickyru]

Hello guys I was wondering if you can help me with this problem.

Suppose that when it is t years old, a particular industrial machine generates revenue at the rate of R'(t) = 6,025 - 8t^2 dollars per year and that operating and servicing costs accumulate at the rate C'(t) = 4,681 + 13t^(2 ) dollars per year.

a) How many years pass before the profitability of the machine begins to decline?

b) Compute the net profit generated by the machine over its useful lifetime.

c) Sketch the revenue rate curve R' (t) and the cost rate curve C' (t) and shade the region whose area represents the net profit computed in part (b).

Thanks...

 

[Deleted member 4993]

Re: profitability

Hello guys I was wondering if you can help me with this problem.

Suppose that when it is t years old, a particular industrial machine generates revenue at the rate of R' (t)=6,025-8t^2 dollars per year and that operating and servicing costs accumulate at the rate C' (t)=4,681+13t^(2 )dollars per year.
a) How many years pass before the profitability of the machine begins to decline?
b) Compute the net profit generated by the machine over its useful lifetime.
Sketch the revenue rate curve R' (t) and the cost rate curve C' (t) and shade the region whose area represents the net profit computed in part (b).
Thanks...

Please show your work/thoughts - indicating exactly where you are stuck?

 

[mvickyru]

Re: profitability

hello, I do not even know where to start. This is not even on the class notes or within the chapters we are required to study. I even ask the instructor to review the problem but she did nothing...it is not fair and that is why I am coming here for help.
Thanks

 

[Deleted member 4993]

Re: profitability

This is really a simple problem - where understanding of "rate" and "accumulation" in terms of integration/differentiation is required.

So study the problem carefully and make sure you understand the terms used.

To start - what do you understand by the term profitability as related to revenue and cost?

 

[mvickyru]

Re: profitability

Profitability I see it as how much or how productive a product or services was to a company inn relation to how much revenue it gave and how much cost to produce or present the product or service.

 

[o_O]

Re: profitability

So, in terms of R(x) and C(x), profitability = ___ ?

 

[mvickyru]

Re: profitability

could it be the derivative to the profitability function?

 

[o_O]

Re: profitability

What exactly could be the derivative of the profitability functiion?

 

[mvickyru]

Re: profitability

Revenue - Cost is usually considered P (profit), so I assume in this situation it would be P= R - C
Am I correct?

 

[o_O]

Re: profitability

So, the derivative of the profitability function, call it P'(x), is equal to R'(x) - C'(x). So, how can you use P'(x) to tell you when P(x) is beginning to decline?

 

[mvickyru]

Re: profitability

do you look for the antiderivative?

 

[o_O]

Re: profitability

Well, remember what the derivative of a function tells you. If a function is increasing, then its derivative is greater than zero. If a function is decreasing, then its derivative is less than zero. So, how would that apply to this case? You can tell what happens to P(x) just by looking at P'(x).

 

[mvickyru]

Re: profitability

ok so I would look at P'(x) when it is decreasing?

 

[o_O]

Re: profitability

Not quite. You are looking for when profitability, i.e. P(x), begins to decline. What would you look for in P'(x) to find when this happens?

(If it helps, think about f(x) = -x[sup:1shen2pd]2[/sup:1shen2pd]. It is an increasing function until a certain point and then starts to decrease. How would you use f'(x) to find out when this occurs? This is essentially what you're looking for for this particular problem in terms of P'(x).

 

[mvickyru]

Re: profitability

Would I look for the limit? . This is the part I don't quite understand.

 

[Deleted member 4993]

Re: profitability

No - say it was increasing before some

x = x_1

(P'(x) is positive) - then after x = x_1 it starts to decrease (P'(x) is negative) - then P'(x) must be what value at x = x_1?

 

[o_O]

Re: profitability

Hmm. Guess you're not quite getting it.

What I was getting at was that profitability starts to decline when \(\displaystyle P'(t) < 0\).

Picture that example I gave you f(x) = -x[sup:1fr4w1kp]2[/sup:1fr4w1kp]. It starts increasing until x = 0 and starts decreasing afterwards. Even if you didn't know what f(x) looked like, you could've found out WHEN it starts to decrease just by looking at when f'(x) < 0. Same principle applies to your problem.

 

[stapel]

Are you familiar at all with how the first derivative of a function relates to the slope of the graph of the original function...?

Eliz.

 

[mvickyru]

No I am not familiar. Basically we are still in logarithms and the instructor wants us to do this problem. She says that because we already took calculus we should know everything and just apply it to business. (this problem is from a business calculus course)
That is why I don't know how to do it.

 

[stapel]

Are you familiar at all with how the first derivative of a function relates to the slope of the graph of the original function...?

No I am not familiar.

Oh.... :shock:

I'm afraid you're being asked to work with concepts that you haven't apparently yet studied. (Obviously you haven't taken the expected year or so of prerequisite calculus that the instructor has assumed you all have, or you'd be very familiar with first-semester differential calculus. But you've apparently never studied this at all.) The reason you're not understanding the hints, set-ups, and explanations you're being given here is that you don't have the month or two of background material. No wonder you're confused! :?

If, as you say, the instructor is assigning homework over material which has not yet been covered and with which she refuses to help, you might want to consider having a serious talk with your academic advisor. Something is very wrong with the picture you present, and I'm afraid there is nothing we can do to put things aright. Sorry!

You may also want to consider hiring a local qualified tutor, and setting aside a few hours a week (daily would be best) to dedicate to concerted re-teaching. With diligent effort and face-to-face help, you may be able to get caught up inside just a few weeks. :idea:

Good luck!

Eliz.