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MTH170-01 Business Mathematics (Winter 2021) Homework: Assignment #6 (Chapter 11) Score_ 0l 2 pls 11.3.7 auina Aarned n bonus a hr wok Rich pluced t1o natwurtha Mon...

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MTH170-01 Business Mathematics (Winter 2021) Homework: Assignment #6 (Chapter 11) Score_ 0l 2 pls 11.3.7 auina Aarned n bonus a hr wok Rich pluced t1o natwurtha Monu Invetlnunt aluount o( Ino tonts Jinin ? Hine nljor Mhdmv nlnned much iretest wa: Poi? Themnloinnt Eia banu: Is sl [Rourid Ire; fnal 4nzwer Ie / 02/030 cent niudc o nqun d ueatme Ju*0aetni0l2hcont € Chdiou $250Mo tid 0FmonmDeenn

MTH170-01 Business Mathematics (Winter 2021) Homework: Assignment #6 (Chapter 11) Score_ 0l 2 pls 11.3.7 auina Aarned n bonus a hr wok Rich pluced t1o natwurtha Monu Invetlnunt aluount o( Ino tonts Jinin ? Hine nljor Mhdmv nlnned much iretest wa: Poi? Themnloinnt Eia banu: Is sl [Rourid Ire; fnal 4nzwer Ie / 02/030 cent niudc o nqun d ueatme Ju*0 aetni 0l2h cont € C hdiou $250 Mo tid 0F monm Deenn



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Use the following table to work Problems 20 and 21 . ProPainters hires students at $\$ 250$ a weck to paint houses. It leases equipment at $\$ 500$ a week. The table sets out its total product schedule. If ProPainters paints 12 houses a weck, calculate its total cost, average total cost, and marginal cost. At what output is average total cost a minimum?

We were asked to use linear programming to solve this problem. We're told that some institution is making menus with foods A and B and that food A has one unit of fat crowns, two units of carbs per ounce and one unit of protein per ounce and food be has one unit of fat grounds, one unit of carbs per ounce, one unit of protein. We're told the lunch must have at least six units of fat per serving, no more than seven units of protein and at least 10 units of carbs when were given that the price of food A is 12 cents per ounce and from B is eight cents per ounce were asked to find how many ounces of each food should a serving contain so that dietary requirements are still met at the least cost. So what we're trying to do is minimize cost in our constraints. Are these dietary requirements so but x equal the amounts of food a announces. So this is the number of ounces of a met. Why be the number of ounces of food be then we're told, hence the total cost C is the total cost of the meal this is going to be equal to. We were given that it's 12 cents per ounce for a so 0.12 times X plus and then eight cents per ounce times why and we see that this is a linear equation and this is going to be the objective function. We're given the constraints that's the lunch must provide at least six units of fat. So we have that number of units of fat. She's going to be one times X plus one times. Why to be greater than or equal to six. No more than seven years of protein in the number of units of protein is going to be X plus one times why it's less than or equal to 10. Sorry, seven, and we have to have at least 10 units of carbs. The number of units of carbs is two times X plus one times. Why has to be greater than or equal to 10. These are constraints, and we see that we have a system of linear inequalities and so we can use linear programming to find the minimum of Z. So what's draw? A graph now implicit in all of this is Theoden Schimmel constraints that X must be greater than or equal to zero. And why must be greater than quarter zero? Because we can't serve a negative amount of food. So we know that the solution to this linear inequality will be in the first quadrant and al graft the first inequality in red. To graph this autograph X plus y equals six. So we see that we have a X intercept and 60 in the Y intercept at 06 and the line satisfies the equality. So a draw solid line through these points you know you zero the origin is test point have that zero is neither greater than or equal six. So it follows that everything to the right of the red line is a solution. Graph your quality to in green. To do this autograph, X plus y equals seven. This has an X intercept at 70 and the Y intercept at 07 Since the solution, I'll dry solid line through these points, and because plug in the origin I get zero is less than equal to seven. Everything to the left of the screen line is a solution. Al graph the third inequality in blue. To do this cell graft to X plus y equals 10. If I take why to be zero, I get that X is five. So there's an X intercept at 50 and if I take extra zero, I get that y equals tens is a y intercept at 0 10 she's all the way up here a little bit hard to see. And because this line sat spicy inequality, I'll draw a solid blue line between these points. Plug in the origin I have that zero is neither greater than or equal to 10. And so everything to the rights of the Blue Line is a solution. And so I will draw the solution to the system. In black. We obtained this black quadrangle with Vergis is at 60 70 and to find the other two Vergis ease when you defined the points of Intersection. So first, let's find the point intersection between the Green Line and the Blue Line. This is the solution of this system. X plus y equals seven and two x plus y equals 10. Solve this system are simply subtract of second equation from the first, so they get negative. X equals seven minutes. 10 is negative. Three with X equals three and back. Substitution gives us that Y is equal to seven minus three, which is four. And so we have that. This vortex is three. For which, from our picture, here's to make sense. Find the last Vertex. This is the intersection of the blue and red lines, so it's saw different system X plus y equals six and two x plus y equals 10. Solve this system will subtract the second equation From the first I get negative. X is equal to negative four or that X is equal to four. It's my back substitution have that wise? He was six minus four, which is to and so this last protects is point for to which also seems to match the picture pretty well. Now let's find value of Z. It all of these various ease. So we have at 60 Z is going to be equal to 0.12 times 6 12 times six is 72. This is 0.72 and at 70 Z is equal to seven times 70.12 It's going T e Ford of 100 or 0.84 at 34 z is equal to this is 0.12 times three, which is 0.36 less point. 08 times four. She is 0.32 So this is equal 2.68 And finally, at 42 z is equal 2.12 times four, which is 0.48 plus point. 08 times two, which is 20.16 which is equal to 0.6 four. And we know by a the're, um Yet if the objective function has a minimum over this region and black, the minimum will occur it and at least one of murder sees. And so, by our calculations, we have that that minimum is going to be 0.64 or 64 cents at the point for two. And so we have it to minimise cost institution Sure has four ounces of food a in the lunch and two ounces of food be

I'm going to if more deal cost function in the Rev infraction and then the break even, um, in a secret in the way you're investing in a new play in the costs in groups and overhead off 30,000 at last, production cost off to bring five day performance. So if I'd fix the cost, so it means that if we say it's present it a member off solved about before Mills is off the number performances that are sold out. They were given that the coast off. No, I'm over it cost up the development. So cost generally is if I'm channel is opposed to all my hands less Ah, variable cost the unit right? So this is it wants to overrides on 50 thousands. So it's £80 isn't glass cost per unit is 2.5 people for mills good and our performances. It's then the water thing is all much. We realize paper for mills, so for four minutes we cannot realize 3125 So it's 3125 depending on the number of special events and chief ex. So break even point is a quiet way, Thank you sequence to costs which means that our live in you 31256 and our fixed costs will total cost on TNT 1000 last true planned five ah x and they look group like James Ah, 3.3125 My last 2.5. It's ah 6 to 5. It's is the most 10 2000 divided by a 6 to 5 to make x the subject of the formula. So X equals two, uh, divided by three and boats off the 2000. Oh, so it's $30 it's divided by 6 to 5 peoples to put 8000. No, this is the most support. Eight. Yeah, So you have to run for 2 80 events in order the, uh projects to break given and said, like your enemy is the number off units, but that required for the company to make no profit or a profit of self. And the breakeven revenue with that took not point is we know already is equals. 23125 x So 3125 times for the age. Uh 3125 times. What? It is under 50,000. So if you want to put it in an order. Tea. Pair it well before the age representing our X in 150,000 in presenting our wife on you.

Alright. In this problem, we have a business owner who is looking to maximize her profit and we know some information about the prices. Sheikhoun sell her product her sweatshirts for in the quantities at which she can sell. And we know some information about her revenues and costs. So how we're going to approach this is go through and for each person quantity, we're gonna find the total revenue which will be priced times quantity, to see how much money she's bringing in from sales. And then we're going to find the total costs. And it's important to note that we have two different types of costs. We have a fixed cost of $1000 per week for her Web service. Then we have a marginal cost of $20 per sweatshirt being sold. So we have to make sure we're multiplying that marginal cost by the quantity and we're always adding on that fixed cost. So here's our equation for costs here, fixed cost plus marginal cost, times quantity and then once we have those, we confined our profit simply by subtracting all of our costs from all our revenue, and that will be our economic weekly profit. So I've got no started here. This first one is a little self explanatory. You're not going to make much of a profit because you're selling at a price of zero. Um, but to go through the process, we have that price. Times quantity for revenue will be zero, and we're still incurring costs because you have that fixed costs of $1000 a week and you still have to pay $20 for each sweatshirt. And our quality of sweatshirts is 100. So, um, 100 plus another 1000 plus 2000 or costs are 3000 zero minus 3000 are profit is negative 3000. You probably wouldn't want to operate in this situation. We'll keep going to see if that increases at a price of $20 you can sell 80 sweatshirts revenue price times quantity, 1600 in our costs, we still have a $1000 fixed cost. This time we're adding on our marginal cost of 20 times. 80 is our quantity. So 1000 plus 20 times 80 is 2600 as our costs, um so our revenue 1600 minus our costs 2600 give us a profit of negative 1000. So we're still operating at a loss here. So we'll move on to the next one at a price of 40 weaken cell 60 sweatshirts. So our revenue here will be priced times quantity 40 times 60. And that is a revenue of 2400, which is looking good Better than before andare costs. We still have our fixed costs of 1000. We still are marginal cost of 20. But we're gonna multiply that 20 now by 60 and we're going to get a total cost of 2200. We at that all together. And so we have revenue of 2400 cost of 2200. Our profit is now 200. So it's good we're in. The positives were making a profit, but we'll keep going to see if we can do better at a price of 60 Weaken sell 40. Our revenue is 60 times 40. Um, and so we're gonna get the same revenue is before we multiply those. It's gonna be 2400 for a revenue. But our costs will change because their quantity is changing. So we still that $1000 fixed cost, and we have that $20 marginal cost. But we're multiplying that by just 40 this time, Um, s O that means our cost will be 1000 plus 20 times 40 which gives us 1600 and so are our profit will be 2400, minus 1600 which gives us $600 of profit. That is looking good, but we'll keep going, just in case we can do even better. Um, at a price of 80 quantity is 20. So 80 times 20 gives us a revenue of 1600 and our costs will be 1000 for fixed costs plus marginal cost 20 times the quantity of 20. Um so that gives us 1400. And so our profit is again going down to 200 and it's always good to check. Do this last one. But I'm not gonna bother. It might not be totally worth your time because the quantity is zero. So it's gonna be kind of similar to that for a situation where you're not bringing in any revenue, but you still have cost, so you know you're gonna have a negative profit here. So now that we have all of our profits. We can look at where that is highest, and we can see that it is here when we're making an economic profit of $600 per week. So that is our total weekly economic profit. And that is that this profit maximizing point. So we can now define, um, this point where the price of each sweatshirt it's is $60 and the quantity sold is 40. These are our profit maximizing price and quantity, and again, economic profit is $600.

All right. So we are going to, ah, create Thibodeau. Oh, it cost the model and 11 model for this situation. Where combined Manufacture, smoke and nose. It is a fixed cost off agent 1000. It cost 20. Don't ask approaches each. Kano, this enterprise 8000 become Oh, insourcing this exercise except present the number off cardinals produced and sold. So you already on occasion, some guide lights let x equal to cardinals song, uh, and just punches then. So the old days so cost. As we all know, the total costs is certainly a function or fixed cost. Ah, cost the units multiplied by number off Carlos. Or just so so this is equal to 50 cause it's giving is 18,000 cost. The unit is to Andy. They only cardinals told fortunes destroying the X so until don't cost is more. The word is that Is there anything you is simply Ah, the sending Pryce, which is a 2000 multiplied by the number of cardinals produced in sort, which is X then, um, the break even point, uh, Rick, he invented point is equal to, uh they pointed way. Thank you. Mayan US coast is equal to sell. Oh, our eagle. So this means that venue is the war to cost. So what is it that Grady Eiji. It's what is caused. 18,000 glass. Karen The X Grouping leg James Student. It's Comes to the other side into gears, lawsuits the X 18,000 and defined by since d So are its equals. Two so young. So that's the number off you needs that supposed to be purchased in solved in love for the company break area and one doesn't break it and point 1,000,000. You mean a position away? We are neither making a profit only loss where pro feuds is. He watches they'll where cost easy hostility.


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