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Certain product, the cemand funclon D(x) 5x and the supply function Sx) - x+14 Ahere the numbe cfuolteof-he DroductKaleindthe equlllbrium nolnt;Graph the supply and...

Question

Certain product, the cemand funclon D(x) 5x and the supply function Sx) - x+14 Ahere the numbe cfuolteof-he DroductKaleindthe equlllbrium nolnt;Graph the supply and demand functions on ame ares tnen shade the regioncorresponding the producer' sucd VSTc Calculateprdcucer 5 qurolue ezolainWnati Means

certain product, the cemand funclon D(x) 5x and the supply function Sx) - x+14 Ahere the numbe cfuolte of-he Droduct Kaleindthe equlllbrium nolnt; Graph the supply and demand functions on ame ares tnen shade the region corresponding the producer' sucd VS Tc Calculate prdcucer 5 qurolue ezolain Wnati Means



Answers

For a particular commodity, the quantity produced and the unit price are given by the coordinates of the point where the supply and demand curves intersect. For each pair of supply and demand curves, determine the point of intersection $(A, B)$ and the consumers' and producers' surplus. (Check your book to see graph) Demand curve: $p=12-(x / 50) ;$ supply curve: $p=(x / 20)+5$.

Right in this question, if we want to find out what the producer surplus of a supply function is given a number of sales that were sold. We are given that the producers surplus that were given that this surplus is given by the following by the following equation. It's given as the integral from zero to some price unit capital X. Of a price minus the producer surplus as a function of X integrated with respect to X. So yeah. Were also given that the were given that the producer surplus function is given by three Plus 0.01 times x squared. And we have sold ted units total. So yeah, yeah. Were also given that the price is given by the producer surplus given by the number of items sold. So by plugging all of this information into the into our formulas, we first have that the price of selling 10 items is given by Ps of 10, which evaluates to which will substitute in later. And we also have our producer surplus function. So let's see what that gives us when we plug in all of this information to our integral. We will get that we have an integral from 0 to 10 of the price function which is PSF 10 minus P. S of X. And we're going to integrate this whole thing with respect to X. This is of course the integral from 0 to 10. Yeah. Of three plus 0.1 times 10 squared minus three plus 0.1 times X squared. And this whole thing is integrated with respect to X. Yeah, Combining like terms we will get that. We have actually only the integral from 0 to 10 of 0.01 times 10 squared -0.01 x squared. Integrated with respect to X. This entire function here. So why don't we see what our new, why don't we see what this evaluates to? This first term here Is simply just one. So in doing the integral, we'll have to integrate we'll have to evaluate x -0.01, divided by three times X cubed From 0 to 10. Yeah. In doing so we will get that. If I put in zero, everything vanishes. So I I just have to put 10 into all of the excess. So in doing so we will get that. This is approximately 20/3 or $6.67 as our producer surplus. Yeah. A graph of the function, a graph of the function of interest is pasted below notice that it is the area that is bounded between these two curves here. I'll just give a rough graph of the sketch. So if this is the number, so if the blue line is the number of items that we sold, the producer surplus is given by the green curve here, and the black area is exactly the producer surplus, and that's how you solve this.

So if we want to find the demand function based on each of these marginal revenue functions, So we might need to write a couple of things down before we start this problem just to kind of guide us and what we're doing. So first, remember that marginal is a very small change of something. So what this is saying is that this marginal function is really just the derivative. You're too of our revenue function. So if we were to integrate our revenue function, we get back. I'm sorry. If we integrate our marginal revenue, we should get the revenue function back. But we don't want to find the actual revenue function. We want to find the demand. So you might recall that revenue is equal to demand times the number of items sold. So our is revenue, these demand. And actually, maybe I'll write this with the actual variables. So our X is equal to D of X, and then this will be times X. So But this tells us, or this implies that our demand function is the revenue function divided by next. And in this case, X is the number of items were selling. So what we first want to do is go ahead and integrate this function. So you go have him scoop this over a little bit. Mm. There's some a little bit. Mm, obviously. Oh, go ahead and put metrical sign here again. And so I want to first integrate this so I can solve for the revenue function. So integrating this with respect to X So doing that gives our revenue function are of X. And now I can go ahead and integrate the right hand side. So remember, if it's not written and you just have a constant, it's implied that you have a constant to the zero power being multiplied. So we'll go ahead and use the uh huh, some and scalar property of integration. And doing that will give so 50 next zero. So just first rewriting the expression next to the two thirds and then giving us a little space so we can write in our integration symbol so integral of DX here an integral DX there. And now both of these are power rules. So this will be 50 x to the zero plus one over the new power which will be one plus some constant of integration. See one that I'll call it and then minus bye X to the two thirds plus one. So two thirds plus one would be five thirds. And then I add another integration. Constant. Call it CT. So let me go ahead and speak to stuff a little bit. Just get myself a little bit more room. Yeah. Now, since we have to integration Constance here, we can go ahead and add them together because having two constants give another constant. So I'll go ahead and call this constant C and AL I can clean up the other part. So the first part will become 50 X and then five, divided by five thirds will be three. It would be three x to the five thirds power plus c, and now this year will be our revenue function. Right. So remember, we're still looking for demand function, So I want to end up dividing this by X eventually. But before we do that, let's solve for what our constant of integration C is so we can find out what our constant of integration is. If we use the second line in this problem where it says call that if no items are sold, the revenue zero. So this is really tired. Asses R of zero is equal to zero. So let's go ahead and plug this and or combined this fact with our revenue equation and doing that will give So zero is equal to all. Go ahead and write, child the expression first and the parentheses for my variable. Yeah, this class c and then I want to plug in zero for each of those. So I end up with zero is equal to so 50 times zero is zero and taking the five thirds power of 00 multiplied by 30 So this is also zero. So we just end up with C is equal to zero. Okay, Now, we can use this fact here to plug into our revenue equation and doing that, and they don't go ahead and actually move all this up here now. So doing that tells me that are of X is equal to 50 x minus three x to the five thirds power. Now to get the demand function, I want to divide this whole thing by X so d o x is equal to our of X over X and then dividing X into 50 x will give me 50 and then dividing X into three x five thirds will give so X to the two thirds power, and this here will be our demand equation.

Okay, so we're given that we have our marginal Regine revenue function, which is our prime of X and rest. If I'd or the man function hard. The man function is his oravax. So we can take the integral A both sides to solve for R of X, which is our man punching. So let's start with that. We'll use our different role just with the fall line into two in a row. Okay, now we see that we have constants. Insider Interval. So let's use our constant multiple rule and Fordham outside. Okay? I'm no using our integral roles. We can solve our internal. This is our power, Olson. It's two to power or to over 3/4 1 which is 5/3 over 5/3, plus a constant kit again stood up to go to 50 x minus three extra are 5/3. Plus, our constant is equal to our demand function or of X. And here we're also noted that if no items are sold in the revenue zero so just means that's are you. Zero is zero. So we can use this to sell for a constant. So that is zero is equal to 50 time throw minus three times zero plus k. So we see that K is equal to zero. So are the man function races K Since that girl is he good to have fallen?

See every digital to see the rhetoric exit because to do point to excess choir plus 10 X plus five divided by XviD eggs You know the vertical s and told I take city close to zero slant from gold is giving us why is it questions you know, point to express train? There's no any white intercept because we cannot put X equals 20 because X is always better than you. That's fine. Or cemented additional points to a skit. The graph then get out. Look, something like this, which has one slanders and jokes. And one particle isn't the minimum value of see every occurs at X equals 25 So number off units for cost to be minimum is it questo fight?


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