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Let C(x) 5 represent the cost of producing items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling items is P(x) =xp(x) - C(x) (r...

Question

Let C(x) 5 represent the cost of producing items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling items is P(x) =xp(x) - C(x) (revenue minus costs). The average profit per item when items are sold is P(x) /x and the margina profit is dP / dx The marginal profit approximates the profit obtained by selling one more item given that items have already been sold_ Consider the following cost functions and price functions Complete parts (a) through (d) below:c(x) = 0

Let C(x) 5 represent the cost of producing items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling items is P(x) =xp(x) - C(x) (revenue minus costs). The average profit per item when items are sold is P(x) /x and the margina profit is dP / dx The marginal profit approximates the profit obtained by selling one more item given that items have already been sold_ Consider the following cost functions and price functions Complete parts (a) through (d) below: c(x) = 0.02x2 40x + 100 , p(x): 300 , 500 Find the profit function The profit function P(x) = 0.02x2_ 260x - 100 Find the average profit function and margina profit function_ Plx) The average profit function is



Answers

Let $C(x)$ represent the cost of producing $x$ items and $p(x)$ be the sale price per item if $x$ items are sold. The profit $P(x)$ of selling x items is $P(x)=x p(x)-C(x)$ (revenue minus costs). The average profit per item when $x$ items are sold is $P(x) / x$ and the marginal profit is dP/dx. The marginal profit approximates the profit obtained by selling one more item given that $x$ items have already been sold. Consider the following cost functions $C$ and price functions $p$. a. Find the profit function $P$. b. Find the average profit function and marginal profit function. c. Find the average profit and marginal profit if $x=a$ units are sold. d. Interpret the meaning of the values obtained in part $(c)$. $$\begin{aligned} &C(x)=-0.04 x^{2}+100 x+800, p(x)=200-0.1 x,\\ &\bar{a}=1000 \end{aligned}$$

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.

And this problem were given the cost function, the price of X items. And we're told eventually that we want to use 500 items or units. The first part of the problem wants us to find profit. Let's recall the following that if we want to find profit, which I'm gonna signify as a capital p of X, but so that we could distinguish between the capital P in the lower case P I'm gonna use on underlying above and below the p of X to start us out. So capital P of X would be our prophet function. Profit, by definition, is revenue. Mine has cost. Revenue is the amount of money that comes in cost is the amount of money that goes out. So to calculate revenue, we're taking the number of items x times the price of the ex items, Mike. And then the cost, of course, is the sea of X function. So we're taking X times p of X, which in this case is 100 minus our cost function, which, as we input the cost function, we want to remember to insert it inside of parentheses so that we subtract the entire cost function. Simplifying Get 100 eggs that were going to distribute a negative sign. Peru the parentheses to give us positive 0.2 X squared minus if the X and minus 100 combining like terms and reordering the obtained positive 0.2 x squared. Let's rewrite that ence. There we go. So 0.2 X squared 100 x minus 50 X will give us a positive 50 x in a minus 100. So again, I'm using a line about the P the Capital P, to indicate our profit function. In this case again, just a distinguished the capital p from the lower case P for price. So that takes care of the first part of the problem. On the next part of the problem, we need to find the average profit and the marginal profit so again, the average profit and the marginal profit. So I start with the average profit. So by definition, to find the average profit, we're gonna take the profit folk. Sure, again, using the capital P of X been divided by X, which represents the number of items. So in this case, we're gonna take the function we just found in part eight, invited by us or, in other words, 0.2 x squared plus 50 x minus 100 all divided by X. We could simplify this expression term by term by taking each term in the numerator and dividing by X that will yield us the following points Year old two X cross 50 minus 100 Divided by X again, This gives us the average profit for E Z notation. I'm gonna use a AP to represent our average profit. And I'm not using the lines above and below the capital p in this case, since we have the A in front to represent the average profit, next part of this portion of the problem was to find the marginal profit. By definition, the marginal profit is the derivative of the prophet function. We're gonna take our capital p of X and take its derivative, also known as the derivative of P. With respect to X, says our prophet function is a quadratic function. When we take its derivative will obtain a linear function. We're going to use the summit difference rule along with a power rule taking the power of to multiplying it by 0.2 as it's constant, will give us 0.4 rewriting the base and then subtracting one from the expo Knit. What you'll the following 0.4 times ETS, the derivative of 50 x will give us 50. So positive 50 and then lastly the derivative of negative 100 since as a constant, the derivative it zero sorry, marginal profit is given by 0.4 X plus 50. Next, we want to evaluate our average profit function are marginal profit function and the number 500 units. So if we had a production of 500 units so very portion of the problem Yeah, see if I can get the page to move down there ago. So the next portion we need the average profit you've got, you wave it at 500 and again, this is our part. See 1/3 part of the problem and we also need our marginal profit function evaluated at 500. So we're gonna use those individual functions that we found in part B. Let's recall that our average profit profit function waas 0.2 x plus 500 minus 100 over x therefore our average profit evaluated at 500 we're going to substitute 500 for every X in our average profit function. So 0.2 times 500 cost 50 minus 100 divided by 500. Now we're gonna simplify that. I'm going to just pick up a calculator. And since we're talking about money, we want to around this answer to two decimal places. So 20.2 times 500 it's 10. Add 50 to that and then 100 divided by 500 should give us point to Adam is attracting these numbers. We should get $59 80 cents our average profit for our marginal profit at 500 units. Again, we're gonna use the marginal profit function from Part B, evaluating our marginal profit function at 500 begin 0.4 times 500 plus 50 again using a calculator type in 0.4 Times 500 which would give us 20 and then last night, 20 plus 50 IHS $70 and finally willing to explain what these numbers mean. So the $59.80 tells us that on average, if we produce 500 items that each item, on average will cost $59.80. What the marginal marginal profit at 500 means of being $70 needs is that when we produce the 501st item that it will be at a profit of $70. So I think I misspoke. So I'm going to speak that again. So average profit would be the average profit on each item upto 500 items. So again, average profit not costs the average profit of $59.80 per item when we're producing 500 items. And that also means that if were producing 500 items will make a profit of $70 on the 501st item. So again, these air profit amounts and not cost him out. So $70 for an additional idol, So $70 is our profit on the 501st item. Where is $59.80 as our profit per item on average, what we're producing


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