A mining company estimates that the marginal cost of extracting $ x $ tons of copper ore from a mine is $ 0.6 + 0.008x $, measured in thousands of dollars per ton. Start-up costs are $100,000. What is the cost of extracting the first 50 tons of copper? What about the next 50 tons?
$60,$ or $\$ 60,000$
Applications of Integration
Okay, so the problem gives us a marginal cost function here. I've called the marginal cost function See, Prime of X because for all the economics formulas, you can really think of the word marginal as saying derivative. So in this case, it's the derivative of the cost function that I'm calling C of X and then an initial cost. So it cost for a zero units of $100,000. Here were first asked to find the cost of extracting the 1st 50 tons of ore from the mine. And then we're asked for the cost for the next 50 tons. So in both the cases were going to use the net change the room, which I've indicated on the right side, kind of up in the Corp. So for the 1st 50 tons, it's pretty straightforward. We just need the value sea of 50 which will be the cost of extracting 50 tons. And so going by what the formula says, we have that as CEO zero plus the integral from zero 2 50 And now this marginal cost function was given in thousands of dollars per tonne. So what I'm going to do is multiply the whole thing by 1000 to make it come out as dollars so it matches would see of zero is so for that we will get 600 plus eight x dx. So this is going to be 100,000 plus 600 X evaluated from zero and 50 plus eight over to X squared. Evaluated that zero on 50 and these two zeroes cancel because when you plug in zero for the X in either of those terms, it's just going to equal zero, so we don't have to evaluate those. So now here we get that as 100,000 plus 600 times 50 which is 30,000 plus eight over two, is four times 50 squared, which is 10,000 which gives us the final result. That sea of 50 is 100 and $40,000. So that's the cost for the first 50 comes. So now going here, we now have the cost for the next 50 tons. Now for what? Then? That changed. The room up here says we can modify it a little bit because we want the value c 100 but we want the cost of the next 50 tons, so want to subtract out? See a 50 going back to the change The room. That's equivalent what we have right now as subtracting FAA from both sides. So what we're left with over here is just going to be the integral from 50 100 again are see prime of X function, but multiplied to get it into dollars. So 600 plus eight x dx So evaluating this integral well yet 600 x 50 to 100 plus eight over to X squared 50 The 100 which here gives us 600 then 100 minus 50 plus four times Ah, 100 squared minus 50 squared is going to be 7000 500 and then that's going to be 50. So altogether this is going to be 30,000 plus 30,000 which is equal to $60,000. And that right there is our answer for the next 50 tongues