Consider the following cost functions.
c. Interpret the ...

Question

Consider the following cost functions.

c. Interpret the values obtained in part (b).

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 to consider the cost function for manufacturing a product given by D of Y is equal to 800 plus 0.03 Y, where 0 is less than or equal to Y, which is less than or equal to 3000. Calculate the average cost and the marginal cost when Y is equal to 1500. We're given four possible choices as our answers. Your choice A, the average cost is $$0.53 and the marginal cost is 0.05. $$Your choice B, the average cost is $$0.53 and the marginal cost is 0.03. $$For choice C, the average cost is $$0.56 and the marginal cost is 0.03. $$And for choice D, the average cost is $$0.56 and the marginal cost is 0.01. $$Now we're asked to calculate both the average cost and the marginal cost. So we'll start off with the average cost, and so recall that your average cost is our cost function, which is CFY, divided by the number of units, which is Y. So we want to find the average cost when Y is equal to 1500. This means I'm gonna be looking for C of 1500. Divided by 1500. So this is going to be equal to using our function for C of Y to calculate C of 1500, we will have 800. Plus 0.03, multiplied by 1500. And that whole quantity is divided by 1500. So using our calculator to evaluate this expression, this is going to be equal to 0.56, and here we've just rounded to two decimal places. So this here is our average cost. Now, the other quantity that we need to calculate is our marginal cost. And so, recall that your marginal cost is calculated by taking a derivative of our cost function. So here we're going to be looking for the prime of Y. And this is going to be equal to the derivative with respect to Y, of the quantity of 800. Plus 0.03, multiplied by Y in quantity. And this is going to be equal to. When I take the derivative of our first term, I'm gonna be taking a derivative of a constant since 800 is a constant. And so anytime I take it, take a derivative of a constant, I get 0. When I take the derivative with respect Y of the second term, which is 0.03 Y, I get back 0.03. So here are derivative, the of Y is a constant, so that means when Y is equal to 1500, I get back the value of 0.03. So our marginal cost in this case is going to be 0.03. So now I have both quantities calculated, I can compare those to our answer choices, and we see that the correct answer choice corresponds to answer choice C. So I hope that this explanation has been useful, and I'll see you in the next video.

Consider the following cost functions.
c. Interpret the values obtained in part (b).
C(x) = 500+0.02x, 0≤x≤2000, a=1000

Key Concepts

Cost Function

Interpretation of Values

Domain of the Function

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