 Hello friends and welcome to this problem solving session and we are taking up today the question of the day and the question was this, the diagram below shows a fenced triangular enclosure in the middle of a large grassy field, the points A and C. So you can see there is a point A and there is a point C here are three meter apart. So this distance is 3 meter and a cow G here is the cow is tied by a 5 meter long rope at point A on the outside edge of the enclosure. So this enclosure is, you know, so there is a, there must be a boundary something like that and there is another boundary here. So this is a triangular enclosure, enclosure means something, some space which is closed or bounded by some boundaries, now the area of field that can be reached by the cow is 44 meter square, so that is given. You have to find out the value of this angle theta, how much is this angle theta, that is this cone, this angle which is getting formed here, so you have to find that angle out now. Let us try and solve this very easy problem, it is a simple area related problem. So the concept would be how much area would be covered by the cow in the sense cow is tied with the rope AG which has a fixed length, so obviously it cannot breach the circular periphery, is it? So if you can see I have drawn a simple diagram, now A is the center and AG will be 5 meter clearly by because the rope length is 5, so it is tied with the 5 meter rope. So obviously if you see this side is the enclosure, this is extending this side also, this is extending this side also like that. So the area which I am going to shade now is what the cow will be able to reach, is it? Just tell me, this is the same semicircle first of all, so this entire semicircle the cow will be able to cover because it will not be able to breach the radial limit that is 5 meter, so till 5 meter it will be able to cover everything, but it will not be able to come here because there is a boundary, this is the boundary, so it will not come here. And then after it hits the point C that is this point here, then what will happen? There will be a new center now, so the center shifts to this Y because now if the cow wants to move in this part of the side of the field, then from what will happen is this will be the radius 2 meter, 5 minus 3 correct, so you can see this is 3 meter and this total was 5, so this will be 2 meter. Now if the cow wants to move then it will be sweeping area with radius 2 only, I hope that is clear because that's how it will it will be sweeping this area now with 2 meter, it will not be able to go beyond that because the enclosure AC, this AC will obstruct its movement, so I hope this is clear. So, now what I will do is I will declutter the entire diagram so that we are able to wait a minute, so I will just remove these lines, so some bit of it and this also, this I do not require all of this, so the diagram is self-explanatory now, so now let us find out the area, so area is anyways given, what is that? 44, right, so what is 44, let me also clear this mess here, okay, yes and let me do again, so this is 3, no problem, now let us try to find out the area, so do you not think that area, total area, and which area I mean, I mean area swept by the cow will be area of area of semicircle and semicircle with radius r, so I am writing like that with 5, right, so this is semicircle 1, so I am calling this 1 as well, 1 plus area of sector, area of sector with sector, so this sector friends, isn't it, let me call it 2, 2, so area of sector 2, okay, so let us find out each one of them one by one and then try to find out the area or the angle value, okay, now area of semicircle is very easy, so area of semicircle, so 1 I will write is equal to pi into radius square that is pi square, right, divide by 2, why divide by 2 because area full circle is pi r square, so pi r square by 2 will be the semicircle, correct, and this is in meter square, so let this calculation be like that only, right, so you can also say 25 upon 2 pi, fair enough meter square, now let us try to find out 2, so let me redraw the sector once again, so if you see this is somewhat resembling our sector there and this if this is theta, this one is theta, then clearly this one is pi minus theta, pi is 180 degrees, so pi minus theta, okay, so pi is 180 degrees, so pi minus theta, fair enough, now what is the area of the sector, so area of sector is if you recall the formula, it is nothing but theta in radians or other in this case I will write angle because theta is something else here, so angle of sector, angle of sector divided by 2 pi, now angle of sector has to be in radian guys if you are taking 2 pi, otherwise you have to take in degrees both numerator and denominator, so I am taking in radians, angle of sector divided by 2 pi into d pi into r square, full circle area, now circle of and in this circle what is the radius, this is the small r, okay, so hence now let us calculate, so pi minus theta upon 2 pi into pi r square, so pi into 4, 2 square is 4, so 4 meter square, so clearly you can see pi and pi will disappear, this 2 and this will go correct, so now I hope you are clear that this is 2, 5 minus 3 that is 2, so fair enough, so what is the area final, so final area is pi minus theta into 2, so 2 pi minus theta meter square, okay, now total area is given, total area is equal to 1 plus 2 is equal to 44 meter square, correct, so now let us deploy the values and find the answer, so let us try to calculate now, so this implies 25 upon pi 2, sorry 25 upon 2 pi plus 2 into pi minus theta is giving you 44, right, multiply both sides by 2, so you will get 25 pi, 25 pi plus 4 pi minus theta is equal to 88, I multiplied this entire equation by 2, right, so that I get rid of this 2, now if you simplify this is nothing but 29 pi, right, 29 pi minus 4 theta is equal to 88, okay, so what is theta, theta will be equal to 29 pi minus 88 upon 4, right, let us try to calculate this, this is 29 into 22 upon 7 minus 88 upon 4, so somewhat, so I am taking approximately 22 by 7, though you could have taken 3.14 as well, so no problem, so now, so 29, you can take 22 common, if you see this is 29, 22 common and you will get 29 by, 29 by 28, 7 into 4 minus 4 by 4, so I hope you are getting the calculation, so there was this 4 also, so I have, so there was this 4 also, so I have included that within the, right, so this is 22 into 29 by 28 will be 1.0, correct, 0.03, so 0.03 and then 5, yeah, that is it, n minus 1, so I will take only 3 digits, so this is 22 into 0.035, that means this will be, how much, so 35, 35 into 22, correct, that will be around 0.77 radians, okay, so this should be the answer, 0.77 radians, so this is how you have to solve this, you can convert it into degrees also, but that is not required here, so angle is 0.77 radians, right, so I hope you understood this solution, so let me zoom out and show you in one frame, this is what is the solution.