WEBVTT
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Okay, so the problem gives us a marginal cost
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function here. I've called the marginal cost function See
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, Prime of X because for all the economics formulas
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, you can really think of the word marginal as
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saying derivative. So in this case, it's the
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derivative of the cost function that I'm calling C of
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X and then an initial cost. So it cost
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for a zero units of$100,000. Here were first
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asked to find the cost of extracting the 1st 50
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tons of ore from the mine. And then we're
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asked for the cost for the next 50 tons.
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So in both the cases were going to use the
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net change the room, which I've indicated on the
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right side, kind of up in the Corp.
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So for the 1st 50 tons, it's pretty straightforward
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. We just need the value sea of 50 which
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will be the cost of extracting 50 tons. And
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so going by what the formula says, we have
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that as CEO zero plus the integral from zero 2
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50 And now this marginal cost function was given in
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thousands of dollars per tonne. So what I'm going
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to do is multiply the whole thing by 1000 to
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make it come out as dollars so it matches would
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see of zero is so for that we will get
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600 plus eight x dx. So this is going
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to be 100,000 plus 600 X evaluated from zero and
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50 plus eight over to X squared. Evaluated that
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zero on 50 and these two zeroes cancel because when
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you plug in zero for the X in either of
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those terms, it's just going to equal zero,
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so we don't have to evaluate those. So now
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here we get that as 100,000 plus 600 times 50
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which is 30,000 plus eight over two, is four
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times 50 squared, which is 10,000 which gives us
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the final result. That sea of 50 is 100
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and$40,000. So that's the cost for the first
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50 comes. So now going here, we now
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have the cost for the next 50 tons. Now
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for what? Then? That changed. The room
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up here says we can modify it a little bit
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because we want the value c 100 but we want
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the cost of the next 50 tons, so want
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to subtract out? See a 50 going back to
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the change The room. That's equivalent what we have
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right now as subtracting FAA from both sides. So
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what we're left with over here is just going to
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be the integral from 50 100 again are see prime
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of X function, but multiplied to get it into
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dollars. So 600 plus eight x dx So evaluating
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this integral well yet 600 x 50 to 100 plus
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eight over to X squared 50 The 100 which here
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gives us 600 then 100 minus 50 plus four times
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Ah, 100 squared minus 50 squared is going to
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be 7000 500 and then that's going to be 50
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. So altogether this is going to be 30,000 plus
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30,000 which is equal to$60,000. And that right
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there is our answer for the next 50 tongues