The angle of elevation of a stationary hot air balloon from a point P on the level ground is 45^ . An observer moves a distance d meters from P towards the balloon along a straight path inclined at 30^ to the horizontal ground. At the new position, the angle of elevation of the balloon is 60^ . If the height of the balloon from the ground is 50( 3 +1) meters, then the value of d is :

Step-by-step solution

Let the initial position of the observer be P(0,0) and the balloon be at a height h . Since the angle of elevation from P is 45^ , the horizontal distance to the balloon is also h . So, the coordinates of the balloon are (h, h) , where h = 50( 3 +1) . The observer moves a distance d along a path inclined at 30^ to the horizontal. The new position Q has coordinates (d 30^ , d 30^ ) = ( d 3 2 , d 2 ) . From Q , the angle of elevation of the balloon is 60^ . The horizontal distance from Q to the balloon is h - d 3 2 .