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Problem 67 Hard Difficulty

The marginal cost of manufacturing $ x $ yards of a certain fabric is
$$ C'(x) = 3 - 0.01x + 0.000006x^2 $$
(in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to 4000 yards.

Answer

$$\$ 58,000$$

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Video Transcript

alright, we're giving a problem here That describes that describes the marginal cost of a certain fabric in X yards as thief function of CFX first derivative which is equal to three minus 0.1 x plus a 0.0 sure zero six x squared. All right, now when we what we're asked to is find the increasing costs If the production that was raised $2000 to $4000. Basically, we're just trying to determine what is the difference between a cost of 4000 yards minus the cost of 2000. That's the first thing we need to do is take the integral of the sea X derivative first derivative. Because we know if we take the integral of C FX, we can get Sea of X, and then we could just plug in our 6000 find the difference. Okay, so we would take integral of three minus Point is, you were one x loss a 0.0 006 X squared d x, and that will come out to be three x minus a 0.5 I'm sorry I added an extra zero there five x squared and then we have My stylist is acting up. Sorry. She's back now, so we're gonna take the next term. There we have the plus waas 0.0 you know, zero 1234 eating. That's right. That look right goes or she should be to I'm sorry. This will be 00 See, Grow So many zeroes here confusing me. We got 00000 to Okay, that'll be our CFX. Right? So we can go and calculate CIA 4000. Pull out my handy dandy calculator. We're just gonna plug that value in for our ex ISS for the final one Will have won 28 000 and then 4000 square part privacy issue five the negative 80,000 working my way backwards just because I wanted to get rid of these cubes and squares first. So three times. 4000, we know with just $12. All right, so 12,000 minus 80,000. Plus 100 and 28,000 is equal to 60,000. And then we can capture C at 2000 on. See that it will be 6000 here you know, we subtract 2000 squared multiplied that point she was five. Just 20,000. Yeah. Okay. And then we're gonna add a 20,000 huge. What part of a point? Todo It's a 16,000, and that will come out to be 2000. All right, so if we take 60,000, it subtract 2000 just equal to our C at 4000. Attracted by CIA 2000, you get the final value of 58,000 dollar increase. I still see a $58,000 increase in production cost when you raise it from 2000 yards to 4000. All right, well, I hope that clarifies the question. Thank you so much for watching.