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Suppose the marginal cost is C' (x) = 0.24x2 16.8x + 303 to produce x items. Complete parts a through c(a) Use a graph or a table on a calculator to find the c...

Question

Suppose the marginal cost is C' (x) = 0.24x2 16.8x + 303 to produce x items. Complete parts a through c(a) Use a graph or a table on a calculator to find the coordinates of the vertex of C' (x):The vertex is |(35,9)| (Type an ordered pair:)(b) What do the coordinates represent in terms of production costs?The cost to increase production fromunits is approximatelydollars_(c) Find C'(x) dx and interpret what the quantity means.fc'(x) dxfc'(x) dx is the additional cost if t

Suppose the marginal cost is C' (x) = 0.24x2 16.8x + 303 to produce x items. Complete parts a through c (a) Use a graph or a table on a calculator to find the coordinates of the vertex of C' (x): The vertex is |(35,9)| (Type an ordered pair:) (b) What do the coordinates represent in terms of production costs? The cost to increase production from units is approximately dollars_ (c) Find C'(x) dx and interpret what the quantity means. fc'(x) dx fc'(x) dx is the additional cost if the company increases production from units_



Answers

Marginal cost The cost $C$ (in dollars) of producing $x$ units of a product is given by $C=3.6 \sqrt{x}+500 .$ (a) Find the additional cost when the production increases from 9 to 10 units. (b) Find the marginal cost when $x=9 .$ (c) Compare the results of parts (a) and (b).

We're going to consider the following cost function um which is cmax equals six X. Squared. That's 14 plus 18. And then we also know that this is in thousands of dollars. We want to find the marginal cost That production that will actually cause five. So the marginal cost When we take the derivative so see prime of acts. This is going to be 12 ax plus 14. And then we want to evaluate it at five. So when we evaluate this at five we kept getting 74 And that's gonna be in thousands of $74,000 per unit. Then we want to estimate the cost of raising the production level from 5 to 5.25. So if you look at sea prime of 5.25 Because to 77 Or if we look at sea .5-5 for example, we end up getting this value here. So this is how we can analyze the problem.

So here we're gonna be considering the cost function C f X equals six X squared Plus 14 x plus 18. You want another marginal costs at production level x equals five. So that's going to be c prime of X or exceed Prima five that we're looking for. We know that c prime effects is equal to 12 acts, that's 14. So then when we consider this at x equals five We're gonna have 60 plus 14. So it's going to be 74, Only FC primary five. Then we want to estimate the cost of raising production levels from 5 to 5.25. So it's going to require us to use this right here And then if we take 74 times 0.25 That time I just going to raise the cost for the .25. So that's your final answer.

So we have this cost function C. And we're going to find the marginal cost function M. Which is equal to see prime or the first derivative of our cost function. And so we can say M is equal to the derivative of 205,000 plus 9800 times X. And what we can do first is we can split this derivative up, since if we have the derivative of one term plus another term is equal to the derivative of the first term, plus the derivative of the second term. And now this first term is just a constant. And we know that the derivative of constance is just equal to zero. So we really just need to deal with this derivative of 9800 times X. And what we can do is we can take out the 9800 since it's a constant and multiply it by the derivative of X. And now we just have 9800 times the derivative of X. And we can use the power rule to find the derivative of X here and is equal to one where we have X to the first, so we bring down the one times it by X to the one minus one power, or X to the zero power, which is just one. So are a derivative is equal to 9800. Therefore our marginal cost function is just equal to 9800.

So uh this is a separable differential equation and you can have your D. C. To be equal to you know you just separate the the variables, right? And then you integrate both sides. And then this is gonna be C. Which is a function of X. And this is just gonna be X squared. And this is gonna be 12 X plus some arbitrary constant. Which is like be right now uh suppose get put uh excuse me, zero. And then see to be 1 25 and this is gonna be 1 25 right? You put C two B 1 25 and then your X zero and your X zero. And then you just have your be right here. Right? So B is effectively 1 25 as well. So your calls function here is going to be X squared minus 12 X plus 1 25. Right? So now that is the total cost. If you want to find the average cost. So a C. That is just the total cost over X. By some trying to divide everything here by X. Okay, when I do that, what do I have? I'm just gonna share the denominator onto the enumerators in turn by turn basis. So this is gonna be just X. Because this one is going to cancel one of these. This one is gonna be just 12 and this one is going to be 1 25 over X. Okay. So that is the average cost function. Okay, this is the average cause function. Mhm. Now what is the cost for? Um you know for X equals 50? Right? Do not forget. This is the cause this is the average cost function and this is the original cost function that we have? Uh C. X. Right? Which was X squared minus 12 X plus 1 25. This one the numerator. Right? That is the original cause function that we have. Right. Well, they say the total cost. So what is the total cost of production given? Uh X. Which is a quantity to be, you know, 50. Then whenever I see x, I'm gonna put 50. Mhm. So this is calculator work. I put it in a calculator and you're going to have uh you know, mm hmm. $2025 Right? 2025. So 2000 and $25 went around trying to produce like 50 quantities. I'm having to pay $2025. Okay. Yeah. So how much of this Cause here is fixed, right? This is the total cost, right? The total cost is comprised of the fixed costs and variable cost, fixed cost and variable variable costs. Right? So this is a total cost. It is a combination or it's an addition of face costs and variable cost. So, how much is the face cost? And how much is a variable costs in this total cost? Right here? Well, the fixed cause is this constant right here. If you look at this equation, you can see that this X is a variable. This X is a variable and this is a constant. So this one is fixed. It's not change. But this one change is one of our exchanges, right? When X is 50 is going to change, When X is 100 is gonna change is gonna give you a different value. Right? But this one is always going to be 1 25 no matter what X is gonna take, right? So in this uh total cost equation, this is a variable cost. So out of this total cost variable. I beg your pardon? This is the fixed cost. So out of this total cause the fixed costs is 1 25. All right, So 1 25. So 1 25 of the 2025 year constitutes the fixed cost. So what is going to be the variable cost? Well, that is just going to be the remainder whenever you subtract this from that. Right? So the variable costs, that's A. V. C. Is just 2025 minus of fixed cars, which is 1 25 and that is 1900. Right? So $1900 is the variable cost and the fixed cost is 1 25 and they combined together to give you the total cost right here. Okay? So whenever you're in production and you're producing something right? I suppose you have a machine, right? You have your machine that you use in doing production. Well, the cost of production is always going to be comprised of fix cars and then the variable costs. So if you buy a machine brand new machine, like maybe $10,000 right? One of the machine is being used or not, it's gonna depreciate, right? The value is not going to be the same from the day you you bought it, right? There's something called depreciation that is always going to be applied to the machine. So whether the machine has been is in use or not, it's always gonna depreciate. So you're paying for depreciation, so that is fixed. It is not concerned anybody. What are you using a machine? The machine or not? As long as the machine is with you and you bought it and it's not brand new again, then it's always gonna appreciate. So next time if you're going to sell the machine, it's not gonna be $10,000 anymore. Depreciation is gonna affect it. So you're gonna sell it like we line 25 or something depending on how long the machine has been with you, right? So such a cost is always going to be fixed. It is not matter if you're using it or not. Something that is going to be variable is like a variable costs like electricity. So even the machine uses electricity, right? If you're producing something definitely the machine is going to use electricity. So you're going to pay for the bales, right? But if the machine is not being used, then you pay zero electricity. So you pay electricity bills based on the usage of the machine. Okay? So the machine is in use along the longer you use the machine, the more electricity, but you're gonna you're gonna pay. So that electricity bill is a variable, you know, cost in various with respect to the times or the long the length of time you use the machine. But it appreciation is in variant is is indifferent of whether you use the machine or not. As long as the machine is with you and it's no more brand new, it's always gonna reduce in price. So you're paying for the cost of keeping the machine with you. Uh That is not brand new. Okay, So at the end of the day, an example of a fixed causes depreciation cost, right? It applies to any machine that you have any vehicle, any house, right? It's always applies to it. And then an example of a variable causes like utilities, utility bills, like if the machine uses electricity, if it has been used, if it isn't used, electricity bills are paid or are in current, but if it's not and use then there's language utility bills uh in current, right? So that is the difference. So a fixed cause an example of a fixed cause is depreciation, depreciation costs. And an example of a variable cost is, you know, is, um, utilities, right? Utilities utilities. Another example is salaries, if you have employees, whether you're producing something or not, as long as they're there until the end of the month is supposed to pay them, right? So that causes fixed. So there is numerous examples, but you can just take depreciation and the utilities as examples of fixed and variable costs, respectively.


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