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Problem 66 Hard Difficulty

Consider the situation in Exercise 51 if the cost of laying pipe under the river is considerably higher than the cost of laying pipe over land ($ \$ $400,000/km). You may suspect that in some instances, the minimum distance possible under the river should be used, and $ P $ should be located $ 6 km $ from the refinery, directly across from the storage tanks. Show that this is never the case, no matter what the "under river" cost is.

Answer

$6 x+2 \sqrt{b^{2}-a^{2}}$

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Video Transcript

died same with Chiana were asked to consider the situation and exercise 51 if the cost of laying pipe under the river is considerably higher than the cost of laying pipe over land. So recall from exercise 51 Well, we're told that an oil refinery is located on the north bank of a straight river that is two kilometers wide, that a pipeline is to be constructed from the refinery. Yes, storage tanks located on the south bank of the river, six kilometers east of the refinery. And the cost of laying pipe is $400,000 per kilometer overland to a point p on the North Bank, $800,000 per kilometer under the river to the tanks and rest where p should be to minimize the cost of the pipeline. So instead of it being $800,000 per kilometer to lay pipes under the river In this case, that's the seat. That was cool. How the bird was like a play is the burden. Ah, right. We're not actually told exactly what the under river cost is, so we're going to treat it as a parameter settlement. Mhm. Okay, okay, that works so once again to understand this problem, well, we would suspect may be In some instances, the minimum distance possible under the river should be used and that P should then be located six kilometers from the refinery directly across from the storage tanks. However, we should show in this problem this is never the case. No matter what the under river cost is once again to understand, Let's dress a quick diagram. So we have our refinery here on the north bank of the river. Then we have a point p here somewhere still on the north bank of the river. And then we have our tanks here on the south bank of the river. Now the refinery in the tanks are six kilometers apart. There's a commission. The river has a width of two kilometers and then music. And then once again, we'll just call the distance from East Asia. P to the point directly across from the tanks will call this distance X, and therefore this distance from the refinery to P is six minus X Russian and therefore the distance from P directly to the tanks by the Pythagorean theorem is the square root of X squared plus four kilometers. Yeah. Now the cost of laying pipes on land and millions of dollars like some blame this is going to be 0.4 times six minus x. This is in millions of dollars. As for the cost underwater, this we don't actually know. It's going to be some parameter. I'll call it C Times the square root of X squared plus four. There's different ways where C is going to be the cost per kilometer and therefore the total cost, or shall call Big C. Actually, let's choose a different parameter. It's gonna be too hard. She's a parameter a. So they will be the cost per kilometer and so big. See, the total cost is going to be the cost on land 0.4 times six minus X plus the cost underwater eight times the square root of X squared plus four. It's John Slick. Now, once again to find a minimum cost, we want to differentiate our cost function said equal to zero and solve for X. Yeah, it has. Yeah, I'm thinking right, this person. So we have C prime of X. We're going to treat this as a function of X. This is negative 0.4, right? Plus a times X over the square root of X squared plus four. And we want to set this equal to zero. So we have a X is equal 2.4 times the square root of X squared plus four. Mm. To solve this radical equation, we're gonna square both sides. So I get a square two X squared equals 0.16 times x squared plus four. Oh, I want to say and so solving for X, we have a squared minus 0.16 X squared is equal to 0.16 times four, which is 0.64 Yeah, well, hopefully good right here. And so X is equal to because X is the plus or minus square root of 0.64 over a squared minus 0.16 And because we're really the only considering positive X will take X to be the square root. Mhm. Well, this will be 0.8 over the square root of a squared minus 0.16 Yeah, I'm thinking I'm thinking about open doors now. We want to compare well, because this is it's like you don't just Yes, X lies between zero and six. This is the problem of finding absolute extreme of a function on a closed interval so need to compare the cost for X equals 06 and two of the Route three. So the cost of zero if X equals zero Well, this is a 0.4 times six, which is Yeah, 24 plus zero. So just 2.4 and this is in millions of dollars. Sorry. Likewise C of six. Well, this is 0.4 times zero, which is zero plus. Actually, I was wrong about that. This is not 2.4. I was looking at the derivative Yeah, making shoes. So it's 2.4 plus eight times the square to four, which is plus two a now see of six. On the other hand, this is 60.4 times zero, which is zero plus eight times the square root of 36 plus four. This is equal to the square root of 40 which is to root 10. So this is to a routine fancy and finally plugging in a critical point. If C is equal to or X is equal 2.8 over the square root of a squared minus 0.16 Well, this is equal 2.4 times six minus 60.8 over the square root of a squared. Minus 0.16 plus a times. Yeah, well, we're out in Vancouver. Cute. I just followed him on murder box, Square Root of Iowa. June 20. 1st being true. Edmund 0.8 over. Squared of X squared minus 0.16 squared, which is 0.64 over a squared minus 0.16 How we doing on he shows the movie Good. Toronto's moving. Ah, I love this new era of zero, right? Yeah, kind of like me. And stop being this. It feels great. Plus four. I've got back on it. Look, plug it on. This is equal 2.4 times six is 2.4. Okay, minus 0.4 times 0.8 is 0.32 over the squared of a squared minus 16 Yeah, I think that's the most essential. Plus eight times the square root of no I. Now, this is 0.64 plus four times a squared minus four times 40.16 This is minus 64 This is just four over. A squared. Yes, Over a squared minus 16 This is equal to 24 minus 32 over the square root of a squared. Minus 0.16 Plus, then we have to A times a is to a squared over a squared minus 0.16 It feels nausea. Course you have that. Chelsea Clinton. Think quality, actually, Yeah. This guy tried to cancel. Yeah, I saw. Now, it's not entirely clear from the way this is written, but if you use a graphing calculator and compare, you should find that See if six is always greater than like That's so sea of 0.8 over the square root of a squared minus 0.16 So dogs being agents, that's him. The same guy being agents of gentrification. How are using the dogs? Scare black people. So it was a lot more convoluted. This shows that it's never the case, no matter what the under river cost is that people are using. Yeah, like maybe they That's hilarious. He said, Yeah, you're not allowed. Mhm. It is never the case that P should be located. Okay. Oh, that is such mm. Six kilometers from the refinery. And this is what we wanted to show right? This? Yeah,