WEBVTT
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in this problem, we want to build a pipeline
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between the factory and refinery Refinery is located across a
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two km river, and it's also two km east
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along the shore of the river from the factory.
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So what we do is we build a pipeline from
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the factory to a point P. On the coast
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and then across the water to directly to the refinery
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. And we want to locate the location P.
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That minimizes the cost of the pipeline. We also
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know that the cost of building along the coast,
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and I'm going to call L1, the pipe that's
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along the coast. The cost for our one is
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400,000 dollars per mm per kilometer. I'm actually gonna
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call Helen Keller dollars per kilometer, and L two
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is$800,000 per km. And so we can make
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an equation for cost. That is C. Is
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equal to 400 times L1 and that's the length of
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the papal on the shore plus 800 don't you?
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And we want to determine the minimum of C.
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So we're actually, we're going to take a derivative
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eventually. But first we're going to try to get
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this all within regarding just a single uh variable.
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So we can take a derivative with respect to that
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variable. And in order to do this, we're
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actually going to use this X, which is the
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amount the distance um eastwards between the doctor and the
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refinery minus the distance of the pipeline, Because both
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L two and L one are quite easily related to
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this distance here, and that is L one here
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, L one plus x Is going to equal two
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km as you can see. And using Pythagoras,
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pythagoras theorem, you know, the L two is
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equal to the square root of X squared plus y
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squared. And because why is the sector is the
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component of lT that goes directly across the river?
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We know that, you know that why is two
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kilometers long? So we can say that L two
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is equal to the square X squared plus two square
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, which is for Now, we're gonna plug in
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these new equations for L one and L two And
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we get C is equal to 400 two minus X
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plus 800. And the square root of X squared
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plus four. And we could not take the derivative
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with respect to X. But we're going to simplify
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attorney at first, I'm going to call say it's
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um 800-400 x. What? 800? This
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is going to be X squared plus four to the
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power of 1/2. Okay, now taking the derivative
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. So, well, that's true. The derivative
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of C with respect to X Going to equal to
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the derivative of 800 with respect to acts which is
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0- the derivative of 400 x. With respect
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to X, which is-400. And then the
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derivative of this thing with respect to X. And
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I'm going to use a changeable for that. So
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first power is derivative with respect to X squared plus
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four of 800. That square plus four. One
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of the two. And the second part is the
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derivative with respect to X. So that's it.
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They live with respect to X of expert plus four
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. And this year is going to be just using
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power rule is 800 times the power which is 1/2
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times expert plus four. And the power you subtract
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one from the power. So this is negative 1/2
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. Multiplied by the different in this respect with respect
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to X. Which is to its power with your
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ex. So this in total is equal to negative
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400 plus 800 times a half times X squared plus
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four to the power of negative 1/2 times two X
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. And I'm gonna simplify this bit this um two
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And this one over to cancel our and I'm gonna
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put this in the bottom seeing as it's a negative
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exponents. So that's equal to negative 400 plus,
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I would say 800 x over square root of X
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squared plus four. And this is how do we
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live three of that was a bit fast. That
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should be all the steps that you eat though.
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So hopefully that is clear. Um Now when we're
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trying to minimize costs, we're going to find the
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critical point which is going to be when the derivative
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is equal to zero. So we're gonna solve for
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zero equal to this negative 400 plus 800 X over
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a square root of X squared plus four. We're
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going to multiply everything by spirit of X squared plus
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four so can be equal to negative 400. The
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spirit of X squared plus four 800 X. And
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I'm going to switch it to the other side.
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I'm also going to divide it by 400. So
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you can say good of X squared plus four equal
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to 800 X over 400. Now I'm going to
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great everything First. I'm actually going to simplify this
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. 800 over 400 is to over one. So
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this is Spirit Ex scrapers boy Is equal to two
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x for everything that's X squared Plus four is equal
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to four X squared. And then I'll move X
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squared to those sides. So it's four is equal
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to four X squared minus X squared we'll be done
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or is equal to three X squared for over three
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equals x squared and you can say X is equal
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to the square root of four or 3. So
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let's just step by step simplifying that and screw it
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. Or four with three is about 1.1 55.
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Nothing got back into the diagram. This length here
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is one point 155. And since we're looking for
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L well you know that X plus L one is
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equal to two kilometers. So I'm going to say
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L one plus 1.155 Deep of the two, And
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we can sell it for L one is equal to
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zero eight for right, So the Point P is
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located 0.845 km east out of the factory. I
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hope this was helpful. Thank you.