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Numerade Educator



Problem 68 Hard Difficulty

The frame for the kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be?




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Video Transcript

for the kind to be made From six pieces of wood. The four exterior pieces have been cut with the lens indicated in the figure. Mhm. Right. So these are the four exterior pieces we are talking about. That is A A. And B. B. Right? So it is us that these are cut in with the lens indicated in the figure to maximize the area of the kite. How long should the diagonal pieces speak? Right. So we need to find the length of diagnosed A. C. We have supposed let us say this is A. B. C. D. Right? For a guide. Now we have supposed X. X. Here this is X. Let us say this is why. And this is it. Right. So this becomes very simple now, so we need to find the length of time, signal A. C. And needed to maximize the area. So let us see how do we do So? So Here the length of the site is given that it's 80 A. D. A B. That is A D. A B are equal given that is A and B. C. Do you see is given to us that is B. Small B. And we have supposed Bte equals two. B. D. Is if it is accident, it will be also if that becomes towards right. So now A C. Is equal to Q. Z equals two. Let's just say it is, can we request to buy places like. Alright, so now this is for sure that a small A. That this side A is less than I remember this. So this is very important. Alright. So now on applying pythagoras students to do the right angle triangle a soda. Can we do So yes. So this is let us say this is O. A. O. D. Is the right angle triangle. So in this we will apply the path of a restaurant. We will get 80s where that is our high port news. Right? So it is squared equals to a oh a square plus or D squared. That that implies a square. That this small A square as it was two Y squared plus X squared. So I will be in terms of a what will it will be, it will be one. So we will take all these values in towns off a right we will try to. So unlike uh in terms of A. And next year right A B. And X. Here see Okay so it will find further value of said then it will be also in terms of A B C A B X only. Right? So that it becomes very simple. Now we will apply this category stare um to the right angle triangle again, O C. D. Again. So O. C. D jane. So here it becomes serious square. That is high partners. A city here. And this is a right angled triangle already. So this becomes city squared equals two Oc square. This or the square. So this will give us a small B squared equals C squared the sexes square. So in terms of being next we will get that. It was too be square root over B squared minus Access world. So these are the equation one. And we have supposed great. So, because why we are doing so well into because why place that we need to find here. That is why place it is nothing. But that is easy. And we need to find a mandatory. So that's why we are doing so. Right? So now on the length of the diagonal, A C. Is equal to Q. That is my place there. So this will give us from equation to one and two. We will put the value for buying there in terms of A X. N. B. Right? So this will become Q equals to that is length of the diagonal eight. It was too wrote over a square minus X squared less route over B squared minus X squared. So the diagonal BD is equals true. B Is the question to works. Yes, because BD BD is here explicit. That makes two weeks. Right? So this will be representatives be here. Right? So now we will just write the area of the kite of the diagnosed baby. Right? Except okay, So this will be this will be nothing. But the area will be uh area will be lent half into base in too high. Great. So happened to base into hiatus. Happened to be D into B. D. Into a C. Right? So that is Bds represented by P. And S is represented by cuba. Remember this. So we will just put all the values we have calculated above. We can just write the value of P Q. Right? So when we will solve for a that is the function of x ray a square and be A and B are constant here. Great. So we are calculating here. So area becomes X into route over a squared minus X squared plus wrote over this where minors access. Right? So we know this is very important formula. Next step is to maximize the area of kind. So we know that to maximize anything. We just differentiate that value and put that equals to zero. So we will get the maximum value. So this is very important and very common thing. We do indeed mathematics. Right? So here it well differentiated and differentiate shared the area because we want to maximize the area. So we will just get the value of this. You can say this value of X. So that the area will be maximum. So that is what we need to find it. Yes, so a dash X is equal to zero. Now we will differentiate the value. So after differentiation we will finally get the equation route over a square minus x squared plus route over b squared minus x squared that will be into one minus access choir by route over a squared minus x squared room moment be square minus x squared That will give us. So These are the two or two factors you can say yes we have factories did. So this value should be close to zero as well as this value should be then we'll see which one should be correct. one of this will be around and another one week. Right so let us right for the first part that is route over a square minus X squared plus route over b squared minus x squared because 20 when we'll put in this way so the a squared plus a squared becomes sequence two piece which is not correct. Thanks because we have a zoom a is less than the right now let's see for the second part that is one minus x squared by a route over a square minus X squared into route over these were minus access. So then well then we will do equals to zero. We will get this value right we will solve for the situation. After solving the situation. After going through steps we'll get something which is equal to route over you can say a square this is a square I'm sorry of route over a square minus x squared into route over b squared minus X squared equals two here, excess spread. So we will just square this both sides we will get this extra power for here. In the right hand side After solving with all that steps this extra power. Extra power gets canceled. Extra powerful and extremely powerful from the left hand side to write and say it gets cancelled date. So no, what else could be done now here the equation finally becomes is a squared b squared minus a squared X squared minus b squared x squared equals to zero. Right? So when we will solve for it We will get x equals two route over a square. Be square upon a square. Let's be spread. Okay, so this is it person. Right. So finally we will get the value of X. That is X. Is equal to a B divided by ruled over a squared plus B squared. So this is one. Right? So for two weeks but he is nothing but length of dying diagnoses. Remember this length of the diagnosis VD that is peace. It was two weeks that is to into two MB. You can see the wavy by a square. Let's be spread it really Right. So we have got this finally and therefore bidding. Right. So this is our first answer. No, we will calculate for diagonal. Yes. Right. So now for diagonal A C. We will write Q equals to wipe this said and we know by the value of finds it. Now we will put now we know we also know the value of X. Right? We have calculated above. So we will put that value yes. From the equation you can see film equation four equation Yes four. Right. So now yeah what will be the further steps we will solve for this equation. We will be getting here a to the power phone. Let us say to the power for divided by route over. Not not recover. It will be square. So square square will be the square root and a square A square root square off the square root will be Same. Only Baba is 1 1. Right? So here is squared plus b squared. This is rude. Over 88 to the power four divided by a squared plus B squared plus wrote over. All right over B to the powerful upon a square Plys who spread. Okay, okay. Is it the same? Yes. So now it becomes a Squire divided by suit over is square plan B square close B squared divided by route over is purpose. This Right. So this finally gives us the bad news that is going to work a square piece. So this is nothing. But this is a time length of diagonal. Easy so length of guidance. Easy. Right. So this is also a final answer. We have got the value 40BT as well as a C to maximize the area. I hope your industrial the concept. Thank you for watching. Mm hmm.