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Consider the revenue functionR(p) = p(12 - p - p3)(a)Compute andl interpret R (b)Compute R (p) Show alworking out on paper; take picture and attach Attach File Bro...

Question

Consider the revenue functionR(p) = p(12 - p - p3)(a)Compute andl interpret R (b)Compute R (p) Show alworking out on paper; take picture and attach Attach File Browse Local Files Browse Content CollectionqueSTION 2o Tne derivative Gf flx) . g(x) is equal to f'(x) 96x) + I(x).9 '(x) Then KMTcuel IFalse

Consider the revenue function R(p) = p(12 - p - p3) (a)Compute andl interpret R (b)Compute R (p) Show alworking out on paper; take picture and attach Attach File Browse Local Files Browse Content Collection queSTION 2o Tne derivative Gf flx) . g(x) is equal to f'(x) 96x) + I(x).9 '(x) Then KMTcuel IFalse



Answers

Refer to Exercise 56 a. Find an expression for the revenue function $R$ for the Sicard sports watch. Hint: $R(x)=x d(x)$ b. Find $R^{\prime}(x)$. c. Find $R^{\prime}(8), R^{\prime}(10),$ and $R^{\prime}(12),$ and interpret your results.

Let us have this problem right here, it's a differential equation. Okay, And let us do to 25 minus three X. Okay, So this is a separable differential equation. I can separate it once again, like we've been doing throughout some 7 10 tutorials before, so this is gonna be integral. And then this is gonna be our right. And this is gonna be to 25 X. And this is gonna be 3/2 X squared plus some arbitrary cost and see. Right? So this is a revenue. The revenue function. Okay, So this is an arbitrary constant and I want to take it away by finding a particular X. Right? So suppose I put X equals zero and I equated to zero. So this is gonna be zero equals 2 to 5 X zero X zero plus C. Right? So this is zero, this is zero and this is zero. So it means that C. Is equal to zero. So over here the sea is going to be zero. So then the revenue function finally is going to be 2 to 5 X minus three. Halfs X squared

So we have those E Yeah, 310 you know, minus four X. Suppose we have this. So this is gonna be D. R A calls 310 D X. Okay. Okay. And we're just gonna integrate both sides and you know that this one is going to give you are because you're integrating one with respect to uh this are right here and you're integrating this with respect to X right here. So this is gonna be 310 minus X squared? Right? Two X squared actually. And then plus an arbitrary cost. And see, okay, now that has put extra quarter zero. Put X equals zero. Put are equal to zero, What do you have? So X equals zero, R equals zero. And then you can see that C is also going to be zero because this is zero, this is zero, this is zero. So C is going to be zero. So finally our is going to be 310 X mhm minus two X squared. So this is your function that satisfies this differential equation and satisfies the initial conditions, right?

Uh, revenue function echo 2 500 times the lock off X plus one. And then we want to find a match in our revenue. It's echo Judah, different oceans. So it recorded. Our experts aren't. Monies are ex dividing by lunch. And here were given us equal to one. Now, if I was, you get equal Jew. Ah, off express one. When it's our ex dividing my watch you could you one here And then we should get equal to express one gonna spy And I wish you get equal. What? You have experts one. So we get in the 500 lock off the express Jew and no minus 500. Knock on the express one. Now we have this in common 500 him to confront their final inside outside. And we have a lock on the express to minus that lock off. Actually, this one And here we can combine a jeweler into one single one. No, I'm the x plus Judy even ever Express one and no AA. We want you know what happens if the ex getting larger and larger if expert to infinity and then we see that this one hand would tend to the lock of one like a one nickle Jew zero. So we're suing Ghenda 500 times, Drew and therefore the matching or ever knew a good Jew is zero.

Okay, This question gives us our of tea, which is the cumulative revenue earned by a certain store t weeks after release. So it wants us to find our of five, are prime of five and the relative rate at five and interpret all of these. So are five is just 3 50 Ellen of five because we just substitute five in for tea. And if we do that out in our calculator here we see the 3 50 Eleanor five is approximately equal to 5 63.3 So how can we interpret this? Well, this is saying that five weeks after the release, the store's revenue waas and then are five, which was 5 63.30 Because again, this function just tells us how much revenue the stores making up to that point. So five weeks after the release, the store's revenue was $563.30 total. Think we should add that key word, and they're so they know we're talking about cumulative. And then the next question asks us about our prime. So let's take a derivative. So the 3 50 stays in front, and then the derivative of a natural log is just one over tea. So our prime of tea is just 3 50 divided by t. So this means that are prime of five is equal to 3 50 divided by five, and that's equal to 70. And the units of this are $70 per week. So this is saying that five weeks in the total cumulative revenue is going up at a rate of approximately $70 per week. So the revenue is increasing at that point because we have a positive sign on our tribute of there. It's increasing at a rate of $70 per week. So again, this follows our standard interpretation for derivatives here because we start out at five weeks with this much revenue, and the derivative tells us how that revenues changing. So in this case, our revenue is going up $70 per week at that point. So the last question wants us to consider the relative rate, which is our prime of five divided by our five, and this tells us the relative rate of change. So how much the changes compared to how much is already there. So in our case, this is 70 divided by are five, which he said was 3 50 Ellen of Science. And this is equal to one over five Ellen of five. And if we convert this to a decimal, we get a rate of approximately 0.1 to 4. So what does this number mean? So this means that the rate of revenue gathering is approximately 12.4% of the current revenue, so at this rate, they're going to be gaining an additional 10% of their revenue.


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