A spherical balloon subtends an angle = ⁻¹ ( 7 9 ) at the eye of an observer. The angle of elevation of the center of the balloon from the eye of the observer is 30^ . If the height of the topmost point of the balloon from the level of the observer's eye is 30 m, then the radius (in m) of the balloon is :

Step-by-step solution

Let the radius of the balloon be r and the distance from the observer's eye to the center of the balloon be d . The balloon subtends an angle at the observer's eye. From the geometry of tangents to a sphere, the right triangle formed by the observer's eye, the center of the balloon, and the point of tangency gives: ( 2 ) = r d We are given = 7 9 . Using the half-angle identity = 1 - 2 ^2 ( 2 ) : 7 9 = 1 - 2 ^2 ( 2 ) 2 ^2 ( 2 ) = 2 9 ( 2 ) = 1 3 Thus, d = r ( /2) = 3r . Let h_c be the height of the center of the bal