Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+...

Question

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000.

b. Interpret the meaning of your results in part (a).

Accepted Answer

Welcome back, everyone. In this problem, a company has a cost function for producing Z smartphones given by C of Z equals -0.01 Z2 + 500 Z + 8000. Determine the average cost and marginal cost of producing 5000 smartphones and explain their significance. Now let's start by trying to figure out the average cost of producing 5000 smartphones. Now, when we talk about average, recall that the average would be equal to the total divided by the amount. So in this case, our average cost function, let's say C of Z average, is gonna be equal to our cost function C of Z divided by the number of phones that are produced. In this case Z. Now, when Z is equal to 5000, then that means our average cost C of 5000 average, is going to be equal to C of Z negative 0.01 multiplied by 5000 squared. Plus 500 multiplied by 5000. Plus 8000. Divided by 5000. No, when we rate the sort, then we get our average cost to be $451.60. So thus $451.60 is the cost per smartphone when 5000 smartphones are produced. So you know what, let me actually write that out here. Cost per smartphone. When 5000 phones are produced. So we figured out the first part of our question. We, we have the average cost. For a smartphone. Now what is the marginal cost? Let's move on to the second part of our problem. Well, the marginal cost is the change in total cost that arises when the quantity produced Z is incremented by 1 unit. In other words, it represents the cost of producing one additional unit of a good and technically what we mean here. Is that the marginal cost, if we think about math, the marginal cost is the derivative of the cost function with respect to Z, the cost to add just one more unit. So if we determine the derivative of C of Z, then we can substitute 5000, the number of smartphones that are being produced to find the cost it takes to produce one additional smartphone, OK? No. Differentiating. Differentiating C of Z. To find the marginal cost. OK. Then, remember C of Z is -0.01 Z2 + 500 Z + 8000. So its derivative is going to be equal to derivative with respect to Z of that function. OK. Which is going to be equal to -0.02 Z plus 500, OK? We differentiate our first term using the poor rule, well, the 1st and 2nd term, and then the derivative of a constant is zero, OK? So no. At Z equals 5000. Remember those were the amount of smartphones that were already produced. Then the marginal cost. It's gonna be equal to -0.02 multiplied by 5000. Plus 500. And that would be equal to $400. So what does this mean? Basically, it means that $400 is the cost. Of producing one additional smartphone. One additional smart phone. OK. When Buy those on smartphones. Are already produced. Are already produced. Thus, this is going to be our marginal cost 400 to to summarize. And $45451.60 is the average cost. Thanks a lot for watching everyone. I hope this video helped.

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000. b. Interpret the meaning of your results in part (a).

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Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000.
b. Interpret the meaning of your results in part (a).