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Problem 77 Hard Difficulty

Where should the point $ P $ be chosen on the line segment $ AB $ so as to maximize the angle $ \theta $?

Answer

$$5-2 \sqrt{5} \approx 0.53$$

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BJ

Beatrice J.

December 16, 2020

How can you do this same problem but as to minimize the angle theta?

Video Transcript

in this problem. It is given that. Yeah. Where should the point B be chosen on the line segment A. B. So as to maximize the angle teacher. So we are supposing few values. Right. So here we are taking let us say it is given this value is given. Okay, we are supposing it is not given any problem. But the procedure remains the same For a question three b. Question four and see shape. So this is three. This is total is four And this is seven danger. So now we can say Bp is what let us suppose Bp is X. This is line segment and let us save B A. Mhm It was two for -6. No, this is what? This is part zero. Less than 48 hours to eggs. Less than a request to all. Thanks. Great. Sukhtian alpa done. And uh equals true three by X. Dan vita in question seven x 4 x 4. That's right. Because we are supposing this is as a right and a triangle isn't it? So this is the right angle triangle. So this is perpendicular. This is please because anger is made with the base here and don't confuse because now this is as a perpendicular. The three as a weapon. Different access obviously. So this is what was strangled and second triangles. We have already found notice value of 10 help. And so this will be L. B equals to 10 inverse three by it one. So for vita it will be turning voice seven x 4 -6. Okay, so now we can say that this is this is one line. So alpha beta and double me 1 80°. That is fine. So can we right here the can right phone call a power plus beat up plus tita. It was to fight. Right So now this value of theta becomes We have already found out the value of help and in terms of 10 inverse. So we were right here by -10 in verse. Yeah. Uh huh. Today by L bob three way X. I'm sorry. And this is my race again. Can inverse. Mhm Seven by four minus actually. Yeah. Uh huh. Right so now we will differentiate. So on differentiating com maximum value no D to Dubai. Uh huh. B. X. Should be questioned. So does your video hitter by the X equals group zero. Right. Yes. Yeah I'm sorry I'm sorry I'm really sorry. Uh This should be zero. Right? Because differential for maximum value to differentiate and we we just write it in we were just differentiate and we will find the value that what should be the maximum value for it. So for a maximum value to differentiate. So by the representation of constant five will be zero minus. This will be the oh the differentiation of town university. So this will be one plus one by one plus. Today by X. Sports pleasure. And this will be minus three. Buy Xs five. Right? This is for the first part and for the second part it will be oh one by 1-plus 7 by 4 -6. Full Square. And this will go more to be seven bites four minus X foursquare. So this will be questioned. 20 defensive. So for the maximum value the differentiation should be differentiation of the tablets. Inspector. Actually be it was 20. We are doing so what women get. Mhm. Yeah we will get did he do By the x equals to zero equals you can see. Oh yeah. Oh this will be all right. So this will be zero. Right? So this will finally give us what this will give us today by excess square last night After solving equals two seven by four minus X. Who will square plus for dinner? Yeah. Right so this will be what? Okay. Yeah We are just simply paying. So this will come out to be three and 2 16 -8 eggs plus X. Is square Plus 1 47. It was 27 x square plus 63. This will give us four XS sq Plus 24. X -132. The Industries. Okay This will give us access world plus 6 6 -33. It was 20. Yeah it's just no this will be again exit was too to the point for it. This value will be 3.4 and x equals 2 -9 great. So which one is to be choosing? We will choose A positive value. That is 3.4. So that's going to be so from a we can say 4- excesses 4 -3.400 This will be .5 or nine. Thanks. Yeah. So we can easily say we must choose, okay to the point we must choose going to be the critical, roughly wow from B is 3.4 units, 4 8 units and from it's a Yeah. The question it really yeah .519 units. This is how we met maximize deep. This is how we maximize Yeah, yeah. The value of and you teeth and yet I hope you understand the concept. Thank you for watching.