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A manufacturer can produce digital recorders at the cost of $50 apiece. It is estimated that if the recorders are sold for p dollars apiece, consumers will buyq = 1...

Question

A manufacturer can produce digital recorders at the cost of $50 apiece. It is estimated that if the recorders are sold for p dollars apiece, consumers will buyq = 120 - precorders each month1, Express the manufacturer's profit P as a function q.2. What is the average rate of change in profit obtained as the level of production increases from q 0 to q = 2023. At what rate is profit changing when q = 20 recorders are produced? Is the profit increasing or decreasing at this level of production

A manufacturer can produce digital recorders at the cost of $50 apiece. It is estimated that if the recorders are sold for p dollars apiece, consumers will buy q = 120 - p recorders each month 1, Express the manufacturer's profit P as a function q. 2. What is the average rate of change in profit obtained as the level of production increases from q 0 to q = 202 3. At what rate is profit changing when q = 20 recorders are produced? Is the profit increasing or decreasing at this level of production? Make sure to include all of the details in your response and upload it at the following link



Answers

A company produces and sells shirts. The fixed costs are 7000 dollars and the variable costs are 5 dollars
per shirt. (a) Shirts are sold for 12 dollars each. Find cost and revenue as functions of the quantity of shirts, $q$ (b) The company is considering changing the selling price of the shirts. Demand is $q=2000-40 p$ where $p$ is price in dollars and $q$ is the number of shirts. What quantity is sold at the current price of $$ 12 ?$ What profit is realized at this price?
(c) Use the demand equation to write cost and revenue as functions of the price, $p .$ Then write profit as a function of price. (d) Graph profit against price. Find the price that maximizes profits. What is this profit?

Okay, So part a wants us to find a total manufacturing costs. I'm gonna call that t. Um, so t of lower case team eyes gonna equal r m I just cost of materials plus the cost of benefits, because the total cost is just gonna be to t plus one plus 0.0 point one T square plus two, which is equal to in standard form 0.1 t squared plus two t was three. Okay, part being by the function that represents how much more the operating costs our than the cost of materials. Um Okay, so, uh, let's call it difference. So the difference in the cost? Um, it wants to know how much more the operating costs are. So it's gonna be see if team to cost of benefits minus the cost of the materials. Okay, so that's gonna be 0.1 t squared. Plus two minus. Ah to t plus one. So sweet plus two U minus one, which is gonna be a close one. Okay. Apart. See, Says what was the cost of operations in the 10th month? Ah, after operations began. So that's gonna be the total cost because the parts He just wants the cost of operations in the 10th month. So it's just gonna be see of 10 because he was in months. So it's just gonna be 0.1 times 10 squared, plus two. So the total cost it's in, uh, thousands of dollars. So 100 times 1000.1 is 10 says coming $12,000. Okay, Part D, uh, one ones. How much less where the operating costs and the cost of materials in the 10th month. Okay, so that's just gonna be our difference. And so are different. Should give us a negative number because it says that materials is greater. And so we're just gonna flip that negative. So negative difference in the 10th month is together Tiempo to 10 is gonna be equal. Teoh, uh, 0.1 times 10 squared minus two times 10 bus one. And really, we have to take the negative of whatever we get there. Let's give it a 10 minus 20 which is negative. 10 plus one is negative nine. And so because they have to flip it, the total difference is gonna be $9000. Okay, Part E from the fraction represents the profit earned by the company. So profit it's gonna equal the revenue minus the total cost of manufacturing. Because the revenue waas ah, 10 rooty. I'm gonna subtract our total cost, which was 0.1 t squared. What's to t plus three? Okay, to simplify that, we're gonna have negative 0.1 T square minus two t minus three plus 10 rooty and then, uh, so find the profit in the fifth month and 10th months. Okay, so p of five, we're gonna just plug in five. So we have negative 0.1 times five squared minus two times five minus three plus 10 reply. Okay, so let's go plug that into a calculator. This is a little profit after five months is gonna be 6.8 6000 Okay? And the profit After 10 months, we're just gonna plug in 10 instead of five. So let's go back to your calculator. Change all those tens that you just put in or five. You just put into tens. That's the profit After 10 months is gonna be negative. Um, 1.3 8000. And so it says to comment on it. Um, so it looks like we're going down. So as the company goes into, um, further down the line, it looks like it starts losing more and more money after five months is doing pretty well. But, uh, after 10 months, we're starting to lose money. Okay, Thank you very much.

Alright for this problem, we are given weekly price demand and cost equations for a company that makes cameras, we want to first maximize the weekly revenue, then maximize the weekly profit. So to calculate the maximum weekly revenue, we need to set up the revenue function. So that's going to be the number of units sold times the price per unit. So that will be 400 x minus 0.4 X squared here. Which then means that we want to take the derivative with respect to X. So we will look that should have been 400 x first of all. Alright, taking the derivative with respect to X, then we should get 400 -0.8 x. We want to sell from in this equals zero. So that means that we'll have 0.8 x equals 400. Or That's going to be the same thing as x equals 50. So all we need to do now to figure out what the maximum revenue is is plug that back in. I'm going to pause and type that in the calculator here. All right. So that is going to be 19,000 now for part B about the prophet. Well, the profit is going to be the revenue minus the cost. So we'll have 400 x minus 0.4 X squared -160. So we'll need to do 400 -160. Which should give us I believe it's going to be 340 um Than -2000. Yeah, one second here. Oops. That should be to 40 knots them. Not 3 40. All right. Now we take the derivative of that with respect to X. So that will give us 240 minus 0.8 X. Solving that equals zero. So X is going to be 240 over 0.8. So that's going to be 300 one second here. I realized I missed a zero up here. In part A Okay. So the that also means I need to fix up what the answer is for the revenue, I believe. Yeah. It's not going to be as simple as just adding a 01/2. Alright. That revenue there should be 100,000 and then for the profit. All right, the maximum profit is going to be 34,000.

And this problem were given a a profit question. Accompanies increasing the production Of a product at the rate of 25 units per week. The demand that caused functions for the product are given by the function piece equal to 50 -0.01 x. And C. Is equal to 4000 plus 40 X minus 0.2 X squared. We're told to find the rate of change of the Prophet will respect the time When weekly sales or X is equal to 800 units. And the first problem we know that the first step to this problem, we're giving that P mm is equal to 50 minus zero point 01 X. Right? And we could find a total revenue are as next times peak. Given that access to this uh weekly up quickly amount of units sold. So our becomes X. Times P. That's the total revenue. So we have X times 50 minus zero point 01 X. That gives us the distributive property. 50 X. Uh huh minus zero points 01 X. Squared. Okay. Yeah. Okay that's a total revenue. Now we can find the prophet S. P. Being equal to the total revenue minus the cost C. So 50 X minus zero points Sarah one X squared minus C. And were given that C. Is equal to That function. 4000. Mhm Plus 40 X. Okay minus zero points 02 x squared. Now we could basically add all like terms. So we get the 50 plus 40 becomes 90 hacks. Mm We're left with P is equal to negative 0.1 X squared minus 0.2 Like square. It gives us 0.01. Next Square zero point 01 Yeah X squared. Okay. 15 -40 x. gives us a 10 x. Mhm minus 4000. Now we can take that the french are of dp D. T respect respect D. P. D. T. So we take a different show we have yeah French Shelf 0.01 I spread you know the two multiplies by that? We have zero points 02 X. So next is 10 plus 10 minus zero. This becomes 0.2 oh X plus 10 dx DT. So okay so D P becomes 0.02 Experts, 10 d. x. d. t. In this problem we also know that the rate of the rate of its 25 units per week, The rate of production is 25 units per week. So we know D. X. D. T. It's 25. Mhm. So now we can calculate the rate of change of the prophet with respect to time When X becomes 800 units. So we could calculate dp D. T one Axis 800, implement that into I'll function or the french show p we have mhm Zero points zero wine times 800 I'm sorry, into this function in two D P. So we know DP is zero points 02 times 800 plus 10 times dx DT Which is 25. So doing that. Yeah, We get 6:50. Uh huh. And if if you were to graph this, Yeah. Yeah, if you were to graphic just a quick picture of what the graph fel look like, right mm Your graph will look like something like this. This was done using more from alpha. So you could actually input this into your calculator and there you have it.

Hello. Problem is about the new musical bark that has ticket prices a German by the situation here. Uh, que is the number of attendees people that born thinking. So the number of people that buy ticket price is determined by the We could say production level here. Okay, let's go through the questions. Question A party has several questions. What is the price? Ah, when there are 3000 people attending, Okay, food. We just the substitute queue with 3005 p times three thousands calculations we give. So the price is $10 per person, $10. Um, but the total revenue those will ruin you. It's also a question under the part of a, uh, revenue function. Are you being here? Is Pete I'm secure, which is 10 times 3000 Took revenue is the key $1000 on the third question is, if the press falls $20 per person, um, for that same cue off three thousands good total revenue. We'll be, uh, 20 times 30,006 followed by four zeros, 16 of those. But of course it can be because when peace 20 then Q is not 3000 so we find If P equals $20 what is the attempts? That is the first thing they have to. So you find the to find the cure. We sold the equations for Cubans is 2020 bulls 70 miles north, 700.2 You we transfer this through the other side, we add no point, no give and we footprint you ever by subtracting it. So we have Europe under three, kid 20. Uh, this is 50. We divide with 012 50 by zero going. Which gives us the attendance off 2500 people when the price is when the prices 20 goals doesn't driven you with the revenue for transit. Ozone. Uh, since 2500 Pete, I'm scared. 25 00 Richards to top spin club is 50 followed by three zeros on There we go questioning me. Done So we have several answers here. We have be answer or what was pee pee waas than those here for a dense off. So we have questioning covered. Let's go to be They wanted to write the revenue function as a function off. Cute. Now, revenue equals press for unit times production level or number of a tendency. But then but then these So p we extract p room. The given equation for being since the TV right, 70 minus 70 miles. Loved ones there to you times cute. We want to run the quadratic first. Perhaps this will be minus about 0.2 squared bluffs. 70 on. We have been covered. The revenue dysfunction off. Cute risk. Is that b you see as words, Cube. Use the revenue. Excellent is the question c the Maximus affection we find where the derivative off that action we are about a year. Where does that any clearer? So Ah, the French thing. This expression here the This will be two times going on for Q to the power to us one Scared m 70. This is there, but cute. And we want that to be zero to find the cute transfer. This to be a flood certainly equals. We're here for you. Divided by German of four four is the cure. Use your calculators to find the 1750 is like you. Well, this is a critical point. We can apply the seconds through the test. Fine work. The second roots of Q is, uh right. We're you're too the friendship this because this is the first, the rooms are back. You. So this term will be the river of 70 0 This constant and the derivative off this one is going to be for negative. Now. Uh, since this is a ghost infection, the seconds the riveting are as 1007. 50 is also going to be no point for, which is listening zero Therefore, we have a maximum pally. So there we have the answer to part. See years the maximum that tens a part of me. The attendance of maximizes driven you. He's going to be 1750 attendees. The Mmm. What proof? What prince should be charged when the attendance is 1750? Um, the average is the size of it here. But what it says P is the terminal by 70 minus. You're going to ki you. So we substitute cute. What about 750? 71 snow suit tops 1750 Go to a calculated rage. Gives us bird be five $35 per person. Right? People's credit for Leslie Byrne. E asks what is Yeah, maximum revenue. As we said in but see revenues? Rex Allies, Lanky Rico's 50. We also have the revenue function as a function of Q. So our 1715 I mean, this isn't easy. Away our equals picked on secure. We found that people found a cure. We might as well use that. So it's 35 times, which gives us, huh? 2 15 61,250 goals. There we have it way have a the C d. Bingo. Hope this helps, but


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