{Use of Tech} A different interpretation of marginal cost Supp...

Question

{Use of Tech} A different interpretation of marginal cost Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, planners have determined that the total cost C(x) of producing 25,000 gadgets in batches of x items at a time is given by C(x) = 1,250,000+125,000,000 / x + 1.5x.
c. The meaning of average cost and marginal cost here is different from earlier examples and exercises. Interpret the meaning of your answer in part (b).

c. The meaning of average cost and marginal cost here is different from earlier examples and exercises. Interpret the meaning of your answer in part (b).

Accepted Answer

Welcome back, everyone. In this problem, an electric company produces 20,000 tablets per year in batches of x items at a time. The total cost in dollars of the production of 20,000 tablets in batches of X items at a time is given by the function c of X equals 2.5000000 plus 250,000,000 divided by X plus 1.3 X. And we want to determine and interpret the meaning of the average and marginal cost when x equals 10,000. Now, we're trying to figure out the average and marginal cost and what they mean for our production function. We already know that the total cost function is given, and we also know that it produces 20,000 tablets per year. 20,000 tablets per year in batches of x items at a time. So let's try to figure out the average and marginal cost so that we'll be able to try and make sense of what they really mean. Now the average cost the average cost let me put that in red. K. The average cost of 1 tablet would be equal to the total cost of production divided by the total number of tablets. Okay? And that makes sense because if we divide the total cost by the total number of tablets, it gives you the cost for 1. So let's see if we can figure that out. We already know that the total cost is given by the function c of x and the total number of tablets is 20,000. So, therefore, that means the average cost, let's call that the average of c of x equals our total cost, c of x, divided by 20,000. So our function, it that means we're gonna divide everything in our function by 20,000. So So that's 2,500,000 divided by 20,000. 250,000,000 divided by x divided by 20,000. So So that would be 20,000 x. And 1.3x also divided by 20,000. And when you divide through by 20,000, you find that the average cost function is now 125 +12500 divided by x plus 0.00065x. Now to find the average cost, we know that x, which represents the number of batches at a time, is 10,000. So we can substitute this value of x into our average cost function. Because now that means at let me make a note at the side. At x equals 10,000, our average cost would now be equal to 125 plus 12,500 divided by 10,000 plus 0.00065 multiplied by 10,000. And when we put these values into our calculator, we'll get the average value for a batch of 10 thousand to be equal to $126.90. So that's our average value. Next, let's try to solve for the marginal cost for our production. Now the marginal cost the marginal cost of 1 tablet Marginal cost of 1 tablet is equal to let me make a note here. It equals the first derivative of the function. So it equals the first derivative of c of x. So if we differentiate c of x and then calculate that value when x equals 10,000, we'll be able to find the marginal cost. Now since c of x equals 2,500,000 plus 200 let me let me write it out. C of x equals 2.5000000 plus 250,000,000 divided by x plus 1.3x, then we can rewrite 250,000,000 divided by x as 250,000,000 multiplied by x to the negative one, the power of negative one. Now we can go ahead and differentiate. So differentiating our function, when we differentiate a constant, it equals 0 differentiating 250,000,000 multiplied by x to the power of negative one. We can use the poor rule, which tells us that we subtract 1 from the power and multiply by the previous power. So that would be 250,000 multiplied by negative one. So that's negative 250,000,000, not 1,000. Sorry. Negative 250,000,000 multiplied by x to the power of negative 2. And when we differentiate 1.3 x, it leaves us with 1.3. So that's the function for our marginal cost. So that means at x being equal to 10,000, then our marginal cost would now be equal to negative 250,000,000 multiplied by 10,000 to the power of negative 2, that is divided by 10,000 to the power of 2, plus 1.3. Now, when we put this value into our calculator, eventually, we'll find that that marginal cost equals negative $1.20. Therefore, our marginal cost of 1 tablet is negative $1.20. But now how do we make sense of these negative values? Because we can't we we we we can't have a negative payout as it were. Well, when the batch size is 10,000, that means the average cost of producing 20,000 tablets is $126.90 per tablet. And that makes sense because we're talking about a cost. Likewise, when the batch size is increased by 1, the total cost of producing 20,000 tablets will decrease by $1.20. Thanks for watching everyone. I hope this video helped.

Key Concepts

Total Cost Function

Average Cost

Marginal Cost

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