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1. In the manufacture of a product, fixed costs per week are Tk.4000. If the marginal cost function isMC=0.000001(0.002q2-25q)+0.2;where c is the total cost of prod...

Question

1. In the manufacture of a product, fixed costs per week are Tk.4000. If the marginal cost function isMC=0.000001(0.002q2-25q)+0.2;where c is the total cost of producing q pounds of product perweek, find the cost of producing 10,000 lb in 1 week. 2. Find the area between the curvey1=x2-4x+8and y2=2x from 0≤x≤3

1. In the manufacture of a product, fixed costs per week are Tk. 4000. If the marginal cost function is MC=0.000001(0.002q2-25q)+0.2; where c is the total cost of producing q pounds of product per week, find the cost of producing 10,000 lb in 1 week. 2. Find the area between the curve y1=x2-4x+8 and y2=2x from 0≤x≤3



Answers

Cost The weekly cost $C$ of producing $x$ units in a manufacturing process is given by
$$C(x)=60 x+750$$.
The number of units $x$ produced in $t$ hours is $x(t)=50 t$
(a) Find and interpret $C(x(t))$
(b) Find the number of units produced in 4 hours.
(c) Use a graphing utility to graph the cost as a function of time. Use the trace feature to estimate (to two-decimal- place accuracy) the time that must elapse until the cost increases to $\$ 15,000$ .

In this question were given the accompany profit is in p is given by this formula here where X. Is the units of items that they sell per week? Now P. N. X. R. Representing in thousands. Yeah of in public. Now our job in a is to find the marginal profit dP over the X. Mhm key. So using Kocian rule we square the denominator. We repeat the denominator at the top in we Different shape than you married to 200 x. When a different should get 200 put a minus sign. Now freeze the numerator which is 200 x. And we differentiate the denominator 100 plus X square. When we differentiate we will get to X. Now when you tidy up you will get 200 Dumps 100 -X. sq over 100 plus X square square. Okay, in part b mhm. We want to find the time point. The time rate of change of profit dP DT. Mhm. Now we know that the P. D. T. Can be written as DP and we can write the T over here. Now the variable that is linking them will be X. So be the Xia. And the extra yeah so we want dX DT we have X. Is equal to 4-plus 2 T. When differentiate X. With respect to T. I get to. So the pdt will be D. P. D. X. And so I will put in 200 100 -X. sq over 100 plus X. Square square. Okay thumbs the X. DT here will be too Okay so we want to put in the X. We want to put in 4-plus 2 T. So that's what we will do. Bring the two in front. We'll get 400 100 minus my ex is four plus two T. Mhm Square over 100 plus four Plus 2 T. Square. And the whole thing square. Alright In Patsy, how fast are the profits changing when key goes to it? So basically we want to find DP over DT when tea was to eat. So D. P. Over D. T. When he was to eat. We just saw it into all the teacher. So you find it when you start it into all the T. Here you'll get zero Remember Teas in $1,000 per week. So this means the profits uh falling because he's a minus sign X. 400 $80 per week. And we are done, this is

So given our demand function, we know that the revenue function um will be found Based on a question 13, we know we want to find revenue function that's going to be a negative zero 00 six X squared Plus 1 80 x. And then we already know that our cost function Is going to be equal to 0.12345 two X. Cubed -0.02 x squared plus 1 20 X plus 60,000. This ends up being our cost and revenue function. Then we end up getting this is cubed right here. So then our profit function and V. P. Of X equals and uh revenue function minus the cost function. So we end up getting this graph right here. Then we want to compute, we want to find the marginal cost. So we have see prime of X. And then we have marginal revenue to our prime of X. We get right there, seed crime of 2000. We can have we can also our prime of 2000 and then the profit. We can compute all these. Um and that gives us our different grass and the different behaviors that we end up seeing as a result.

Hello. This poem is five parts to it, a B city. What we are given is that the company has a cost functional. C off Q equals 4000 plus two Q dollars on dhe. Its revenue function is are cute. Use 10 Q dollars. Q. Is the quantity produced for costing? Forever New is the quantity. So, uh, okay, wish you a the question of fixed cox fixed costs, fixed costs. We look at the way we look at the and of course function. Uh, has two turns. The 1st 3rd does not have any Q involved here and the second term has queuing. Also, this term will give us the variable cost times the number off units produced. So we put if cube equals zero. That means we haven't produced any. Uh, I think is you. The cost of producing zero items will be two times 00 plus 4000 4000. So the answer to question a fixed costs are $1000. Okay. Questioning marginal coast marginal costs. Murder because him c equals marginal cost equals the derivative off the cost function by the variable. Here, uh, it's years are unit off. See, which is doors. Two units. Oh, Q On the meaning of the marginal cost is how much both cost increase when we increase the the unit production by one, which is in fact here. This is only near a function this is going to be soon We consider is going to be too, because this is a slope off the graph of doing your function. But let us do the let us do the differentiation first. Emcees sorry. Him sees a function of here. So it is the derivative seen by the words you, which is the derivative of the first times zero. Because this costume ah, the riveted off two times skewed is too. To what? It's $2 additional growth. The, you know called, huh? Conditional. Michael produced. Okay. Uh, question C, He's extract from the revenue functional on. Did ask what is the price, huh? Okay, Now we in general, not in general. But the revenue function is defined as the price, per I think times the number about him. So So what do we see here? We see that this can only be big. Friend said price is be easy. Thing on The answer is 10 Gore's her Okay. What should the we need to Graff? Okay, I need some preparation. All think Paul's okay? Let's, uh, drew the horizontal axis on which we will place quantity on. We're going to be graphing. We're going to grab, see if Q on DDE C, Q on and the are here. The same access. So we get the door amount here, and this will be a quantity. Okay, Breath arc, you will be easier. Are you? Let's find some for zero quantity. The art is going to be zero. So this will be on the graph of our on dhe with C that we take one of these to be 102 34 Let's say this is 100 500 678 9000 units. And so, um, 500 units, 10 times 100 with lee, five thousands. So let's place this to be 1000 to 34 Let's put this to be 5000 which makes this one thousands. And the points of the graph of the revenue is 500,000 and revenue over is linear and fastest through these two points. So this would be revenue. Ah, the costs ocean. The cost function is a linear function. And the sorry when cute zero we're cost is for thousands. So we have seen tricep, uh, vertical lists of all 4000 witches here on Let's take 500 here as well. When the quantity is 500 then we have two times 500,000 plus 4000 is 5000 which comes in handy because we are we then no. But a certain breaking point is going to be here. So this is the cost off. They want to give the point. Here off intersection is five hundred's by thousands. And this will be our zero, where the break even point is the point where the cost of Q equals the revenue off you. Let's see, what is the cost of producing Crime 100 units? It is what doesn't and what is the revenue Looking at the black breath revenue again? This point he's 5000 equals revenue 5000. Okay, now, if we have a look at what happens with profit, the prophet function is the function, the difference between the revenue and the cost. It's going to be positive when revenue is greater than the cost. When the graph off the revenue is above. Let's hear the two breaths are, uh, the same points so we could say our is equal to you here. And this is the point where the rocket is made. He is Pines equals zero again in this part of the graph, when cues greater than 500 in this part of the grand, the revenue function is above the cost function. So this is where the profits will be possible. Hear from 0 to 500 quantity, we have the cost being above the cost being above the revenue. So we are subtracting greater number from a smaller number, which results in the profit being negative. And that's one of the questions. How do we know that when the once he produced is greater than the breaking point that the company will make a profit? The reason is because in the areas for, uh, Hugh greater than here zero, which is we can't be certain, but we can't, uh, until we actually calculated. But for the time being, we're taking the Q zero. That is the breaking point is 500. Uh, the revenue is greater than the cost, so profit is positive. What does it mean basil? The company is making money before the five country profits Negative. It means that the company's making off. Finally, Part B is to calculate when he's the revenue function equal the cost function. Uh, we heat the week. The breaking point here By chance, we found 500 to yield arrive in your 5000 and also 500 to yield 5000. But should we have chosen that, say, where a quantity of 1000? We would have gotten two different points and it wouldn't have been so obvious Where this breaking point? What's So we do this outbreak Lee we saw what is Q zero p zero will be the solution to the equation where revenue equals cost. Okay, so what is the function of revenue? The revenue is 10 Q. And that should equal thing here should equal 4000 plus 4000 plus to you. Subtract took you from either side. We have eight year that equals 4000 and divided by eight. Hugh equals 500 which confirms our graphical biographical solution, and this souls befo the whole train. Okay. Ah, wanted the people's 500 is just one more thing What is the revenue cost? So are 500 equals. We calculate that here equals $5000. Hm. Hey, we have So be so. See the Andy. We have the whole problem, so I hope it helps.

So in this problem, we're house too. Um, the first were given the a company's Merchant of Revenue. Um, that is given by that's given by this, uh, creation and our it's measured in thousand dollars. Xs measured in thousands of units and were asked, how much money should the company expect from the from a production run out next week or two. Three thousand experience. So, to find out, we want to integrate this marginal revenue and evaluated from X equal to zero two executed three. So here isthe what? Don't look, thanks. So in. Since we're integrating this, this would be our people to the integral from Syria to three two, minus two over Express, one script. So the first stuff that I'm going to do here is since I have you two here and two here, I think I can make this simpler if I factory it out. So I'm going to read it to out here. It's a constant outside the interval, this does not change. And now the interview function here. So what they did would be one minus just one over X press print squid. That's one Claire date. Okay, so, um, another thing that I would like to do to set this up to make it better or easier to integrate is this. I am going to write this, uh, binomial as the power. So the seventeen so instead of raiding, is dead. So I have here, I have here one over this. So I'm going to write this with the negative exponents on top. So it's gonna look like this. X press one squid. Great. So now I can play on minus square. Very good. Okay, so now I can apply. So for this first term, I am going to use the Powerball so I can think of this us Long Times X to the syrup or X to the syrup. And I'm going to use the parable. So that is bit of integrating X rays to some power, and integral will be. I'm going to add one to the exponents, and then I'm going to divide by this new slips by this new experiment. So in this case, back So I have two times thiss not change. Since I'm already integrating, I no longer need this. This one becomes X rays to the serial classman. That is one which I don't need to write. Explain it one and same with this. I don't need to write over one. So it's just the x No, for this, this is a power. But we have something else in here, so we need something a little bit different. So I'm going to use the inverse change room that happens when I have in the inter grand function I have if function G instead of functions have multiplied by the fight derivative of thie inside function. So if I have this set up, I can go ahead and into great this one. This outside function trading G s its own, like a variable. So in this case, I have this to break it down. Since I, uh X plus one minus two, the outside from the outside function would be x two minus two. On the inside function would be express what, no derivative of thiss iss again. Who are you? Soon Pole one plus one is the derivative of X derivative of one zero. So it's just so wakened since it's just one. Technically, Lincoln do this and it's true, although we don't need to, but, um so weak. So from what we have here. We already have the set up exactly like this. Which means we can integrate the outside function cleaning inside function aspect. So remember, intent function is X plus one and it is the power. The outside is the power function. So we do this rates up and said function X plus one and now minus two. We add one to it. That's minus one. And we defined by this new explore in minus one break. So that isthe song. What? The function integrates too. I can see this one here and then since it's a definite and tomorrow this will be this will be evaluated from X equal to zero two three. So we stirred. No. Before we start, um, I would like to this Plus this minus one. In this minus, they can throw out to be positives. And then instead of having minus one, I'm going to rewrite this again, Um, with this in the body. So we have two times good status not going anywhere. X plus. And now we have one over X plus one race to the one which is just this. So now we have Oh, and we still have to keep going So we have X equal to Cyril until three. No, Teo, to evaluate this expression and peace. And the central, we have a mme. First, we evaluate this on the top when Mexico's two three so that it's two times I agree. One over great press corn and we subtract the same expression evaluated at X equals to Syria. So we have cereal plus one over zero plus one Get. Okay, this is zero. This's equal to one. This is still tree. And this isthe who won over four, So Oh, and and missing a break it Okay, um, so we keep this, too, when we have great plus one for it. That should be third team thirteen over four minus this one. And that should be two times. Mm. Nine over four. So we have a two on top in a forum bottom. So we divide by two to get one, and we divide by two to get to. So we have nine over too. Courtney, that's great. That nine over two, which is equal to four point, but often so no. Um, since we are talking about since our it's measured in thousands of dollars then and on that horse and then this is four point five, uh thousand dollars or rewrite that in another cut. Really, pops are would be dollar sign four point by thousand is four thousand five hundred said this would be the value that we are looking for.


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