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Solve the given problems.To connect the four vertices of a square with the minimum amount of electric wire requires using the wiring pattern shown in Fig. $27.23 .$ Find $\theta$ for the total length of wire $(L=4 x+y)$ to bea minimum.

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 4

Applications

Derivatives

Oregon State University

Harvey Mudd College

University of Nottingham

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in this problem. We have a wire that is going between two poles and is anchored to the ground. Between our pools are two different lengths. 15 and 10 feet. Um, and we want to know an equation for l. Where Ellie goes l one plus l two. So is the total length of that wider. And we want as a function of data they had of being Miss data up here in that quarter. Okay, so to do this, we need to essentially find links of l one and l e felt so I'm gonna start the length of Elba. Um, I am going to use this left hand trained RL, but I can say the toast sign data through my Jason side Hurt my autumn Jason side over my iPod next. So my adjacent side is 15. My hotness is well, there we go. OK, so to stall for a one, I'm gonna multiply my own one on both sides. How one and then I've been divide with co sign off data, and I get l one equals 15 over the coastline data. Okay, so that's Arlington. L one length of L two is a little trickier. Um, so the length of l two will be based on What's the length here is? I'm gonna call this life X. So behind X, I'm going to dio the tangent of data equals X over 15. That's the opposite side over the adjacent side. So 15 times the tensions of data because X I got to multiply both sides by Okay, so now I'm gonna call this section here. Why? Why is going to be 30 minus what are excellence? 15 10 beta. Whatever's leftover off that 30 feet. Okay, so now we know two things we know our, um What are length? Oneness and what? Why is as well as what? Exes. But we don't really need that right now, so we need to do the Pythagorean theorem to find out what? Lt whisks. We have both other sides of the triangle so we can say why Squared Lost 10 square that's on the side equals l two square. Okay, so that's going to be 30 minus 15 10 data squared. Plus 10 squared equals bell two squared. So that is going to be, um, this is gonna be 900 minus nine 100 10 beta minus 2 25 10 squared data plus 100 for that sort. 10 squared equals L two square. So I got this whole section by doing the distributive property, I know I'd would most play my 30 times 30 30 times, 15 10 Takeda 30 textract intended. And then and then negative 15 tam data. Most might by itself. Okay, so now I can say L two is equal to the square root of all this, which is screwed 900 minus 900. And they are minus 25 10 square that a plus 100? No, to close. There's the edge of the screen. So let's get out of that. Um, plus 100. Okay, I can notice that all of the's are divisible by 25 so I'm going to pull out 25 u C l two is equal to scary of 25 times. Um, 40 minus 36. 10 data, um, minus mine squared of theta. So I got this 40 x. I added 90 plus of hundreds of thousands ankles. 100 is 1000. And then I divided about 25. So that's how I got 40. Okay, so here we are with our l two nrl web. So l of data is going to l to without equals 15 co sign data plus the square root when I can factor out this 25 to be just five out from 5 40 for five times the square root of 40 minutes. 36 10 data. Um, Okay. Minus 9 10 square. Perfect. So that is our equation. This is plus, um, last night. There we go. That's what I was missing. Now, why do you never 15 times like the 15. I get positive, make sure you're getting there's a possibility of century. So there we go. Now, we have an equation for l of data. So next question is asking, What's the domain of this? Um Well, I can go through zero I could put zero in, and it goes up until, um, our data hits the other the other corner of the graph. So here. So for example, 15 and 30 feet. If this wire was right at the base of the other one, that is the largest. Our angle, they That can be So where the tangent of data equals 30/15. There is no to that is the largest angle we can be. And that angle is 60 3.4 degrees. You could check that on a tangent graph if we wanted. Okay, so now we want to find El of 45 degrees. Okay, l 45 degrees risk and put it into this equation. So that's gonna be 15 over there, co sign of 45 degrees plus five times the square root of 40 minus 36 tangent of 45 degrees plus nine times the tangent squared of 45 degrees. And I put that all in my calculator, and I got 39 point who, um, 39.2 were and feet there. We got feet. So my question then is what is exe if this is here. So X is going to be this section in our in our triangle. So this is the first triangle. This is this triangle up here. We're looking for X. Well, it's a 45 degree angles 45 45 90 triangle. And therefore it's I saw Silly is. So then acts would be 15 feet. Okay, The next thing we're gonna do is we're gonna graph our function, um, and find the angle that minimizes the amount of wire. We need it. So if I got on the graph, I've already put it in here, and I want to find the angle that minimizes it. So the angle has to be less already. Got Celestine 60 63.4 degrees. So all of this part of the graph that's over 100 degrees the wire be going in the Krays direction only looking within our to me. So does most is really nice. Tell me this is the mid one point. So when x Just our data in this when the data is 50.19 that is our minimal points on this graph. That is where the amount of wire we need is only 39.5 beats. Okay, so the angle winds would need Teoh 50 point active. Okay, so now we want a friend the where that's anchored on the ground. So we're gonna go back over here. A look at our that same triangle. Sofia, on angle of 50 points. 19 degrees. We want to find out how far on the ground is anchored so I can do the chansons. 50.19 degrees Secret X over 15 You multiply both sides by 15. And I got 50 times the tangent of 50 from 19 degrees and we get X. I put that into my calculator. Um, and I get approximately 18 feet from the first base. So that's where we could put our anger. So we have the minimal amount of wire in this will diagram anchor this year, 18 feet away, we have minimized the amount of wire we need.

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