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) = 500+0.02x, 0≤x≤2000, a=1000

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says consider the cost function CFX is 0 to 300 plus 0.03X, or 0 is less than or equal to X, which is less than or equal to 1500. Determine the average cost and the marginal cost when X is equal to 800. We have four possible choices as our answers. For choice A, we have the average cost as being 0.41, and the marginal cost being 0.03. For choice B, we have the average cost being 0.35 and the marginal cost being 0.03. For choice C, we have the average cost being 0.41 and the marginal cost being 0.02. And for choice D we have the average cost being 0.35 and the marginal cost being 0.02. Now we need to determine both the average cost and the marginal cost when X is equal to 800. So we're going to begin by looking at the average cost, and to recall that your average cost is going to be our cost function, CFX. Divided by the number of units, which is just going to be X. So we'll substitute in 800 for X. This is going to be equal to D of 800. Divided by 800. Which is going to be equal to, and here we'll use our function CFX to determine CF 800. So we'll have 300. Plus 0.03, multiplied by 800. And that whole quantity is divided by 800. So, using our calculators to um calculate this quantity, this is going to be equal to 0.41. And here we just rounded to two decimal places. So that is going to be our average cost. Now, the other quantity that we need to calculate is our marginal cost. And for this, we need to take a derivative with respect to X of our cost function. So we'll, you will be calculating the of X. And this is going to be equal to the derivative with respect to X of the quantity of 300. Plus 0.03 X. So when we take a derivative of the first term there, 300, we're taking a derivative of a constant, so that's going to be zero. Then we take the derivative of the second term, we'll be taking the derivative of 0.03 X with respect to X, and that will give us 0.03. And so here we have the prime of X being a constant. So when X is equal to 800, we're still going to get 0.03 for that result. So our marginal cost here in this case is going to be 0.03. So now that I have both of our quantities calculated, we compare them to our answer choices, and we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

Consider the following cost functions.
b. Determine the average cost and the marginal cost when x=a.
C(x) = 500+0.02x, 0≤x≤2000, a=1000

Key Concepts

Cost Functions

Average Cost

Marginal Cost

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