 Hi and welcome to the session. Let us discuss the foreign question. Question says, two poles of equal heights are standing opposite each other on either side of the road, which is 18 meter wide. From the point between them on the road, the angles of elevation of the top of the poles are 60 degrees and 30 degrees respectively. Find the height of the poles and the distances of the point from the poles. Let us now start with the solution. First of all, let us draw a simple diagram to represent the problem. Now, here two poles are AB and CD. We know two poles are of equal height, so AB is equal to CD. Now, AB and CD are standing opposite to each other on either side of a wide road, which is 80 meters wide. So, BC is equal to 80 meters. So, we can write, let AB and CD be the two poles of equal heights, that is AB is equal to CD. Now, AB and CD are standing opposite to each other on either side of a wide road. So, road is 80 meters wide implies BC is equal to 80 meters. Now, E is the point between AB and CD, such that angle of elevation of A from E is equal to 30 degrees. Really, we can see this is the line of sight and this is horizontal line. So, angle AEB is equal to 30 degrees and we are also given that angle of elevation of D from point E is equal to 60 degrees. Here also, DE is the line of sight and EC is the horizontal line. So, angle DEC is equal to 60 degrees. So, we can write, angle AEB is equal to 30 degrees and angle DEC is equal to 60 degrees. Now, let us assume that AB is equal to CD is equal to H meters. Now, first of all, let us consider triangle ABE. In triangle ABE, we know tan 30 degrees is equal to AB upon BE. So, we can write, in triangle ABE tan 30 degrees is equal to AB upon BE. We know tan theta is equal to perpendicular upon BE. In triangle ABE, AB is the perpendicular and BE is the base. Now, substituting corresponding values of AB and tan 30 degrees, in this expression we get 1 upon root 3 is equal to H upon BE. We know value of tan 30 degrees is equal to 1 upon root 3 and here we have assumed that height of AB and CD is equal to H meters. Now, multiplying both the sides of this expression by BE, we get BE upon root 3 is equal to H. Multiplying both the sides of this expression by root 3 now, we get BE is equal to H root 3 meters. Now, let us consider triangle DEC. In triangle DEC, tan 60 degrees is equal to DC upon EC. We know tan theta is equal to perpendicular upon base. In this triangle, DC is perpendicular and EC is the base. Now, substituting corresponding values of tan 60 degrees and DC, in this expression we get root 3 is equal to H upon EC. We know tan 60 degrees is equal to root 3 and DC is equal to H meters. Now, this implies, multiplying both the sides of this expression by EC, we get root 3 EC is equal to H. Now, dividing both the sides of this expression by root 3, we get EC is equal to H upon root 3 meters. Now, we know BC is equal to 18 meters and BC is equal to BE plus EC. So, we can write BE plus EC is equal to BC. Now, BC is equal to 80 meters and BE is equal to root 3H and EC is equal to H upon root 3. So, substituting corresponding values of BE, EC and BC, in this expression we get H root 3 plus H upon root 3 is equal to 80. We have already shown above that BE is equal to H root 3 meters and EC is equal to H upon root 3 meters. Now, adding these two terms by taking their LCM, we get 3H plus H upon root 3 is equal to 80. Now, multiplying both the sides by root 3, we get 4H is equal to 80 root 3. Now, dividing both the sides of this expression by 4, we get H is equal to 20 root 3 meters. So, we get height of both the poles is equal to 20 root 3 meters. Now, we have to find distance of this point from pole AB and pole CD that is we have to find BE and CE. Now, we know BE is equal to H root 3 meters. Now, substituting value of H here, we get 20 root 3 multiplied by root 3 is equal to BE. Now, simplifying we get BE is equal to 60 meters. Let us now find out EC, EC is equal to H upon root 3. We have already shown it above. Now, we know H is equal to 20 root 3. So, we get EC is equal to 20 root 3 upon root 3. Now, root 3 and root 3 will cancel each other and we get EC is equal to 20 meters. Now, our required answer is height of the poles is equal to 20 root 3 meters. Distance of the point from the poles are 60 meter and 20 meter respectively. Clearly, we can see distance of point E from pole AB is 60 meters and distance of point E from pole CD is equal to 20 meters. And sum of these two distances is equal to 80 meters. We know 60 plus 20 is equal to 80 meters. So, this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.