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If a cylindrical tank holds 100,000 gallons of wa…

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Problem 51 Hard Difficulty

The cost (in dollars) of producing $ x $ units of a certain commodity is $ C(x) = 5000 + 10x + 0.05x^2 $.

(a) Find the average rate of change of $ C $ with respect to $ x $ when the production level is changed
(i) from $ x = 100 $ to $ x = 105 $
(ii) from $ x = 100 $ to $ x = 101 $

(b) Find the instantaneous rate of change of $ C $ with respect to $ x $ when $ x = 100 $. ( This is called the \textit{marginal cost}. Its significance will be explained in Section 3.7.)

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Video Transcript

Alright, so here we have the cost of producing a commodity in dollars based on the number of units. We have a nice equation for it. We've got a couple of things to do. First of all, we're going to find the average rate of change. So the average rate of change Um from x equals 100 units. two x equals 105 units. So So basically the average rated changes just rise over run. So it's going to be our cost at 105 units minus our costs at 100 units Over the run. Which is just the difference between 105 and 100 units. So we plug into our calculator. So this is where you're going to really want to have your calculator ready. First thing we're gonna do is plug in 105 for each of these excesses. When we do that in our calculator, we get 6601.25 for plugging in 100 5. If we then plug in 100 for each of the excess and plug that in our calculator we get 6500. Um and then that's over five. So that reduces down to be um $20.25 per unit. So the average rated changes $20.25 per unit. So that was the answer to our first part. So that was excellent. Actually the first part has two different times. Let's do our second time. our second time goes from 100 to actually just 100. What so much closer gap here and let's clean this up a little bit. Okay so we're gonna do average rate of change again. So rise over run this time. 101 minus C. Of 100 Over 11 -100. Okay so let's do this down here. This will equal when we plug in 101 65 2005 -160 500 again over one. So this cleans up to B-20 .05 per unit. Okay. So we can see that when we're close to 100. If our gap is really small we're getting closer to A pure $20 per unit. Whereas when we made her get bigger it's a little bit bigger. $20.25 per unit. So finally we're going to actually do it by derivative. We're gonna find it by we're going to find the actual marginal cost, marginal cost and that is actually the derivative. So we're going to take the derivative by power role, derivative of 5000 is zero. Then we'll get derivative 10 x. Is 10. And then let's see .05 times Well, I'll just show you the work .05 drift of X squared is two X. So we end up with 10 plus 0.1 x. If we then find the marginal cost at a 100 then we'll get 10 plus 100.1 times 100 Which is actually is equal actually equal to 20. So we can kind of see we are getting close to 20. But by doing the actual derivative, we realized that at 100 the marginal cost is truly 20. Whoa. Get my pen to work. It's not wanting to circle. Okay, Well, for some reason it's not wanting to circle. But anyway, there it is. Our answer is $20 per unit. Um and yeah, it's not circling so we'll just end it there. Okay? It's acting funny. Oh, there we go. Finally got it to go. Okay. Um Alright. Have a kind of go anyway. Have a wonderful day. See you next time.

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