A spherical balloon of radius 10 m subtends an angle of 60^ at the eye of an observer standing at a height h above the surface of a lake. The angle of elevation of the center of the balloon from the observer's eye is 30^ , and the angle of depression of the reflection of the balloon's center in the lake from the observer's eye is 60^ . The value of h (in m) is :

Step-by-step solution

Let d be the distance from the observer's eye to the center of the balloon. The balloon subtends an angle of 60^ , so the half-angle is 30^ . 30^ = radius d 1 2 = 10 d d = 20 m. Let y be the vertical height of the balloon's center above the observer's eye level, and x be the horizontal distance to the center. Since the angle of elevation of the center is 30^ : y = d 30^ = 20 1 2 = 10 m. x = d 30^ = 20 3 2 = 10 3 m. The observer is at a height h above the lake. Therefore, the height of the balloon's center above the