Consider the following cost functions.
b. Determine the ...

Question

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = − 0.01x²+40x+100, 0≤x≤1500, a=1000

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Consider the cost function. CF X is. Equal to 0.02 X to the power of 2 plus 50 X plus 2004 0 is less than or equal to X and X is less than or equal to 2000. Determine the average cost and the marginal cost when X is equal to 1500. Awesome, so it appears for this particular problem we're asked to solve for two separate answers. Our first answer, we're trying to figure out the average cost, and our second answer, we're trying to figure out the marginal cost when both the average cost and the marginal costs are at a value of when X is equal to 1500. So now that we know what we're ultimately trying to solve for, let us read off our multiple choice answers to see what our final answer pair might be, noting that we'll read our average cost first and then our marginal cost value second. So A is 20.13 and 10. B is 20.13 and 10, C is 26.13 and 30, and D is 27.1 and 40. Awesome. So first off, let us start to solve for the average cost, which I'll denote as AC, and this is when, once again, X is equal to 1500. So, with that said, that's our first part that we're trying to solve for. So let's take a moment here to write down all of our known variables, all of our known information. Once again, we're told that X is equal to 1500. We're also told the cost function, which is denoted as C of X, so C of X is equal to 0.02 x 2 plus 50 X plus 200. Fantastic. And let us also recall and note that the average cost function or the average cost equation. Which average cost, and it's more specifically it's AC of X, so this function for the average cost is going to be equal to C X divided by X. Fantastic. So once again, the average cost function is equal to C of X divided by X. So now what we need to do is we need to substitute in our value of X into our cost function, and then we need to substitute, once again, our second step will be to substitute X into our average cost function. So, Let's do that, shall we? So it's solved for our average cost, so the average cost, so AC of X, where we're plugging in a value of 1500 for X will be equal to our cost function. So it's gonna be -0.02. And then we plug in 1500 in for X, and don't forget that this will be squared because it's negative 0.02 multiplied by X2. So when you take 1500 square plus 50 multiplied by 1500 plus 200, all divided by our value of X, which once again is 1500. Fantastic. So, when we plug that into our calculator, we will find that our average cost is going to be approximately equal to 20.13 when we round to two decimal places. Hooray, we did it. We solve for our first answer. And once again, all we did is we took our our, basically we're taking the average cost function we're evaluating X as 1500 when we plug that value in, and when we multiply everything out and we evaluate the function like essentially when we plug this into our calculator and we round to two decimal places, we should find that the average cost is approximately equal to 20.13. So now we can start to solve for the marginal cost. So now we're trying to find the marginal cost, which we'll call MC, and this is when once again also X is equal to 1500. Awesome. So with that said, We are using all the same known information that we have, but we also at this point, we need to recall and note that the marginal cost equation, or rather the marginal cost function, so MC of X is going to be equal to DC by DX. So essentially we're going to be differentiating the cost function in order to help us find the marginal cost. So we need to take DC by DX, which is equal to D by DX where D by DX is just a fancy way of saying we're differentiating with respect to X. Of -0.02 multiplied by X to the power 2 plus 50 X plus 200, and when we perform the differentiation, we will find that it's equal to 0.04. X plus 50, fantastic. So now what we need to do is we need to substitute in the value of X equals 1500 into our marginal cost function. So we need to take MC of 1500. is equal to DC by DX evaluated at X is equal to 1500, which is equal to 0.04 multiplied by 1,500s we're plugging in 1500 in for X plus 50. Which is gonna be equal to -10 when we plug in that expression into our calculator, and that is our final answer for the second part for our marginal cost. Hooray, we did it. We solved for both answers that we needed to solve for. So without a doubt, our final answer has to be multiple choice answer, letter B, which once again is average cost is 20.13, and the marginal cost is -10. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Consider the following cost functions.
b. Determine the average cost and the marginal cost when x=a.
C(x) = − 0.01x²+40x+100, 0≤x≤1500, a=1000

Key Concepts

Cost Functions

Average Cost

Marginal Cost

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