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Final product: recommendationofthepricesyouthink Writealetterto Malena inwhichyouexplainyour she should use to gain the most profit: Prepare graphs, equations and d...

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Final product: recommendationofthepricesyouthink Writealetterto Malena inwhichyouexplainyour she should use to gain the most profit: Prepare graphs, equations and detailed explanationofthecalculationsyouperformed tofind eachprice Beclearaboethowenu found theprice thatmaximizes profitforeach item; and identifyhow manyofeachitem Malenaneedsto sellinordertoreach herprofitgoalofssooo Your letter must include the following: Descriplionofthetaskinyourownwords revenue,cost,and Iprofit (showyourmathand

Final product: recommendationofthepricesyouthink Writealetterto Malena inwhichyouexplainyour she should use to gain the most profit: Prepare graphs, equations and detailed explanationofthecalculationsyouperformed tofind eachprice Beclearaboethowenu found theprice thatmaximizes profitforeach item; and identifyhow manyofeachitem Malenaneedsto sellinordertoreach herprofitgoalofssooo Your letter must include the following: Descriplionofthetaskinyourownwords revenue,cost,and Iprofit (showyourmathand Howyoufiguredoutthedemand describe your work In words) Your overall recommendation wilh justificalion (show your malh and describe your workinwords) Conclusion that clarifies thal you completed Ihe task to help Malena and your relleclions about what you leared What assumplions can you make? paid Malena' customers You may assume thatall shipping costswll be will be the difference between the total each type of item, Malena's profit For = lesofthatitem)and totalc Icost(amountpaid revenue(amountreceivedfromretall to the wholesale supplier and to eBay) equaltothenumberofthat Youmayalso- assumethatthedemandforanitemis = item sold related tothepriceoftheitem andyoumayassumethis Thedemand foran itemis relationship islinear.



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A company manufactures goods that are sold exclusively by mail order. The director of market research needed to test market a new product. She planned to send brochures to a random sample of households and use the proportion of orders obtained as an estimate of the true proportion, known as the product response rate. The results of the market research were to be utilized as a primary source for advance production planning, so the director wanted the figures she presented to be as accurate as possible. Specifically, she wanted to be $95 \%$ confident that the estimate of the product response rate would be accurate to within $1 \% .$ a. Without making any assumptions, determine the sample size required. b. Historically, product response rates for products sold by this company have ranged from $0.5 \%$ to $4.9 \% .$ If the director had been willing to assume that the sample product response rate for this product would also fall in that range, find the required sample size. c. Compare the results from parts (a) and (b). d. Discuss the possible consequences if the assumption made in part (b) turns out to be incorrect.

According to the ocean, we have given function after physical do three ex miners by the point, the objective function and the work. What is his point? Did he is big one Goldman zero Be one Go Martin See seven. Goma seeks the sound coma nine and the value of the object function off eat were text theories have all be is equal to three into one minus zero is equal to three Have be vehicle tow three into one Minors then is equal to minus seven Have C There is three into seven minus six is equal. Do 15 ab wolf be various three into seven. Finest night music will do Well, we can see that the maximum value off the object function is f. Maximum music will do maximum where you come from three minus seven, 15 12 The maximum value is 15 and similarly, we find out the minimum value comes from these points three minus Alan 15. Well, the minimum value is minus someone. We can say that minimum value minus salan in given point one Co Martin and go Maximum value is 15 comes in the 0.7 coma six

Were asked to answer a question about maximum profit using functions. We're told that the cost per unit in the production of an MP three player is $60 but the manufacturer charges $90 per unit for orders of 100 or less. And to encourage large orders, the manufacturer reduces the charge by 15 cents per MP three player for each unit ordered in excess of 100. For example, there will be a charge of $87 per MP three player Clint Order, size of 120 in part A were given a table that shows the profits p for various numbers of units, orders X, and were asked to use this table to estimating maximum profit table is two rows the first rose, the UNIX X and the second rows of profit. P. You see, the units range from 1 30 of the 170 and the prophet ranges from as low as 3315 to as high as 3375. Therefore, we might guess that the maximum volume would be around 3370. Sorry, maximum profit. I should say not volume. Maximum profit should be around $3375. And this occurs when 150 units have been sold and in part, the whereas to plot the points XP from this table in part a and rest to determine if the relation defined by the old prepares represents P as a function of that is going to be a rough sketch. We have our two axes. A vertical axis is the PXs within positive. Horizontal is the X axis, which will also be positive now the X axis, as I pointed out before ranges from 1 31 70 someone use a little trick. This squiggly line which tells us that we skip all the way from zero up to 130. Now we know that at 1 30 units the profit is 3315. Say this is about here. I'll do the same thing for the y axis squiggly line. That indicates that we jumped from zero up to 3315 dollars. Then at 1 40 units, the profit is 3360 at 1 50 units. The profit is 3375 back here, at 160 units. The profit is 300,000, 360 and at 170 units. The profit is 3315. We get this symmetric looking grab now. If you draw a vertical line and roll it through this graph, you'll find that at no point does the line your sex more than one point of the ground. So by the vertical line test, it follows that profit. P is a function of X, as represented by this relation. Right then in part C were given that he is a function of X, which, given our findings in part B, makes sense and rest to write the function Papi and determine its domain were given the hint that P two profit is equal to R minus. C. There are is the revenue and C is the cost. So let's determine what the revenue is as a function of the number of units. Yeah, so we know the revenue is $90 per unit for orders of 100 or less, so the revenue is $90 per unit. Times X units as long as X is between zero and 100. After all, we can't producing negative number of units. Now if X is greater than 100 we're told the manufacturer reduces the charge by 15 cents per MP three player for each unit ordered in excess of 100. So revenue for these excess is going to be 90 minus 0.15 and X minus 100 19 minus 190.15 times the number of units exceeding 100. Now this is the charge per MP three player going to multiply this by the number of MP three players, which is X, and this is only for X greater than 100. So we have the revenue. Determine the costs. Republic cost per unit is simply $60 so C is 60 times X and the unit therefore, the profit as a function of X. Well, this is a piecewise function. It's equal to 90 x minus 60 x, which is equal to 30 x four x is greater than or equal to zero and less than or equal to 100 and for X greater than 100 equal to 90 minus 900.15 times X minus 100 times X minus The cost. 60 s which this simplifies two and so we have 90 x minus 60 x. I suppose if you wanted to, you could simplify the function or leave it in this form. By the way, we have P as a function of X. Now to determine the domain of this function, let's return to what we were told about this problem. Of course, this only makes sense. If we order a non negative number of units, can't produce a negative number of MP three players so that their name is going to be all real numbers X, such that X is greater than or equal to zero.

So for this problem, they give you a set of data to plot, and I'm gonna not gonna plot that. But we're gonna come up with the model. And so, uh, so we do have profit is a function of X. That is true. And X is the number of units sold, number of units solved. And they tell us that if the number of units sold is less than 100. So if X is less than or equal to 100 we know that it costs them. Uh, costs $60 to produce one of these units, and they sell it for $90. So they're gonna end up. If I look at their profit for this setting, we know that their profit will end up being the 90 times how much they bring in. So they're gonna bring in $90 for each item. However, it cost them 60 times. Uh, X is their cost. So this is how much it will. They'll bring in. So the revenue and then this is minus the cost. And then that's gonna end up being their profit. So we know that that simplifies Donald will simplify that in a minute. But what do we do if if we have an object that is more than 100 and if it costs more than 100 then they say they reduced the price. And so we know that, um, normally the price of an object is $90. That's how much they sell it for. But they're going to reduce that fit by 15 cents for every unit over 100. So we need to find out how money over 100. So if we take how many units and we already know the number is bigger than 100 if we're using this part of the piece wise function. And so let's say they were selling 101 objects, then we would need to subtract away 100 from that. And that would be the number of items that are over 100 and they're going to reduce the price by 15 cents for each one of those items over. So if they had to say this was 110 then 10 times they're going to reduce the price 15 cents. So if we put 110 in here 110 minus 100 would be 10, and we'd multiply it by the 15, and they'd reduce the price by a dollar 50. So this is what the selling prices but we need to sell. Multiply it by X, and that's going to end up giving. This is our revenue well, and I need small room. So let's erase this guy and we'll move him over a little bit because I leave myself enough room. Here we go, so we'll go X is greater than 100. So this is our revenue when you need to attract away the cost, will. The cost is always staying $60 for each item. So now what do we need to do? We need to clean these up, and we've We've satisfied actually what they asked us to do to write a function for this in terms of the number of units sold. But we can clean this up, So let's do that and subtracting these two, we would get 30 acts. That's what the profits going to be as long as you have something that is from zero to 100. Yeah, and then let's clean this up. We'll clean this up just over here, so that I don't have Thio mess with this down here. So we would be distributing this through so we would have 90 minus 0.15 x and then a negative times a negative is positive. And that 0.15 times 100 would move the death, the place I'd become 15 on then that needs to be multiplied by X and then we need to attract away 60 X. All right, so we can combine these two together and those two ed upto 105 and then we're going to distribute the X. We have 105 Oops, 105 x and then x times. This gives me minus 0.15 x squared and then we're subtracting away 60 x. So now we can combine these two together and then I'll write this final answer down here. So we have negative 20.15 x negative 0.15 x squared. And then again combining these two X values together, we're gonna get plus 45 X and that's going to be when X is bigger than 100. And so really, we have whole numbers for our domain. Yeah, because we can't have negative numbers and you can't sell portions of units. So really, our whole numbers would end up being our domain. And this is a function. And you could even check this function by going in and plugging in 1, 10, 1, 21 30 as the table shows to verify that this will will actually work. Now there may be a cap on how many, because realistically, they probably just can't make you know billions and billions of these things. So they're going to be restricted on how maney they can actually manufacture. So there's probably a cap on how largest would be. But you definitely do wanna have whole numbers, something where you're not in allowing decimals to come in. It's not just really numbers, but you will need a piece wise function.

So to start this problem, we're told to find the prophet function. Peak was fairly simple. We're told that p of X, um when referring to the prophet is equal to, uh, x times the price function minus the cost function. So all we have to do is combine these, uh, combined all this information that we know combined these functions that we know and we will find that p of X is equal to a negative 0.6 x squared plus 100 x. And again, this is all coming from the idea that we're gonna multiply 100. We're gonna most play X times the price function, which we already have. And then we're going to subtract, uh, once we multiply that by the price function, we're going to subtract the cost function. So that's where the minus 800 is coming from. And once you combine all your life terms and everything this but you're going to get so we have our price function now for Part B, though we're being asked to find the average profit function and the marginal profit function. So the average profit function is the same thing as the prophet function divided by acts, the number of items. So our average profit sumption is going to be a negative zero, actually 0.6 x and that's because we divide the first term by acts. Then we divide the second term by acts which just leaves us with 100 plus 100 then we're going to be subtracting 800 over X. So this is our average, uh, profit function. And then we're also asked to give the marginal profit function so that is going to be and the vax And for this, the marginal profit function involves, um, the derivatives. So the marginal proper function is the derivative of the prophet function. So we see if we take the derivative of our original profit function, we're going to get negative. 0.12 plus 100 soured me, the derivative of the proper function, which is also the marginal profit function. Then for part C, what we're asked to do is to find the average profit and the marginal profit if X is equal to a and we're told that X A is equal to 1000. So this is a simple as placing in our input values. So we have a of 1000 and a 1000. When we evaluate it, using our average profit function, we're going to get a negative 0.6 times 1000 which would be negative 60 plus 100 minus 0.8. Because what happened when you divide 800 by 1000 All right, And then we just have to add those up in the truck and we get 39.2 and then for the marginal profit function evaluated at 1000 units or 1000 items is also fairly simple. Very, very somewhere. I get what we did before. We're just going to input 1000 as our X and that will give us. This is supposed to be 0.12 X. So here, what we're gonna end up getting is a negative 120 plus 100. So this is gonna equal negative 20 and report be. We have these numbers, which is great, but we have to know what they understand in order for the mass actually means something. So, for part D, exactly, it's asking us is to interpret these values in part C. So the average the A of 1000 tells us that for each of the 1000 I didn't sold average of 39.2 with profit is that means that we will sell our 1000 items. There's a profit of 39 point to her item, and that's for a of 1000 then for of 1000 slightly different. And we know that the 1001st item sold, but actually have a loss of 20 or a profit of negative. So using them words we have here and our profit function that we start off with, we can find values that are actually able to be interpreted.


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