 Hello and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, a 1.2 meter tall girl spots a balloon moving with a vent in a horizontal line at a height of 88.2 meters from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60 degrees. After some time the angle of elevation reduces to 30 degrees. Find the distance travelled by the balloon during the interval. This is the figure 9.13. Let us now start the solution. We know balloon is moving with a vent in a horizontal line at a height of 88.2 meters from the ground. Let us name the two positions of the balloons as D and A. Let us name these points as, now we know AB is equal to CD is equal to 88.2 meters. We know distance of the balloon from the horizontal line that is ground is same, so we can write in the given figure AB is equal to CD is equal to 88.2 meters. Now we also know that angle of elevation of point A from point A is equal to 60 degrees. So we can say angle D is equal to 60 degrees, angle of elevation of point A from point A is equal to 30 degrees. So we can write angle AB is equal to 30 degrees. Now we have to find out distance travelled by the balloon during the interval that is we have to find out BC. Now let us draw a simple diagram to represent this figure. This is a simple diagram AB is equal to CD is equal to 88.2 meters, angle DEC is equal to 60 degrees and angle AEB is equal to 30 degrees. Now we have to find out BC. First of all let us consider triangle AEB. In right triangle AEB is equal to AB upon BE. We know tan theta is equal to perpendicular upon base. Now in this triangle theta is 30 degrees, perpendicular is AB and BE is base. So we can write in right triangle AEB tan 30 degrees is equal to AB upon BE. Now you know tan 30 degrees is equal to 1 upon root 3. We also know that AB is equal to 88.2 meters. So substituting 88.2 for AB we get 1 upon root 3 is equal to 88.2 upon BE. Now multiplying both the sides by root 3 we get 1 is equal to 88.2 upon BE multiplied by root 3 multiplying both the sides by BE we get is equal to 88.23 meters. Now let us consider triangle DEC. In right triangle DEC we know tan 60 degrees is equal to DEC. We know tan theta is equal to perpendicular upon base. Here value of theta is equal to 60 degrees, perpendicular is DC and DC is base. So we can write tan 60 degrees is equal to now we know tan 60 degrees is equal to root 3 and DC is equal to 88.2 meters. So substituting corresponding values of tan 60 degrees and DC in this expression we get root 3 is equal to 88.2 upon EC. Now multiplying both the sides by EC we get root 3 EC is equal to 88.2 dividing both the sides by EC we get 88.2 is equal to EC. You can see BE is equal to EC plus BC. So we can write BE is equal to BC. Now we know BE is equal to 88.2 root 3 meters and EC is equal to 88.2 upon root 3 meters. Now substituting corresponding values of BE and EC in this expression we get 88.2 root 3 is equal to 88.2 upon root 3 plus 88.2 upon root 3 from both the sides of this expression we get 88.2 root 3 minus 88.2 upon root 3 is equal to EC. Now subtracting these two terms by taking their else here we get 88.2 multiplied by 3 minus 88.2 upon root 3 is equal to BC simplifying further we get 264.6 minus 88.2 upon root 3 is equal to BC. Now subtracting these two terms we get 0.4 upon root 3 is equal to BC. Now rationalizing that denominator of this term we get 6.4 root 3 upon 3 is equal to BC. Now simplifying we get 58.8 root 3 is equal to EC. Now we can simply write it as BC is equal to 58.8 root 3 this can be further written as 58.8 upon 10 multiplied by root 3 we will cancel common factor 2 from numerator and denominator both and we get 294.3 upon 5 is equal to BC. So we get BC is equal to 294.3 upon 5 meters. So the distance traveled by the value during the interval is equal to 294 upon 5 multiplied by root 3 meters. So we can write distance traveled by the value during interval is equal to 294 upon 5 multiplied by root 3 meters. So this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.