The angle of elevation of the top of a vertical tower from a point on the horizontal ground is 45^ . From this point, an observer walks a distance d meters towards the tower up a slope inclined at an angle of 30^ to the horizontal. From the new position, the angle of elevation of the top of the tower is observed to be 60^ . If the height of the tower is 50( 3 + 1) meters, then the value of d is _____.

Step-by-step solution

Let the height of the tower be h = 50( 3 + 1) m. Let the initial point be A . The horizontal distance of A from the foot of the tower is x = h 45^ = h . The observer walks a distance d along a slope of 30^ . Horizontal distance covered = d 30^ = d 3 2 . Vertical distance covered = d 30^ = d 2 . From the new position, the horizontal distance to the tower is h - d 3 2 and the vertical distance to the top of the tower is h - d 2 . The new angle of elevation is 60^ , so: 60^ = h - d 2 h - d 3 2 3 ( h - d 3 2 ) = h - d