Two observation points A and B lie on a horizontal plane surrounding a vertical tower. The angles of elevation of the top of the tower from A and B are 30^ and 45^ , respectively. If the line segment AB subtends a right angle at the top of the tower and the distance between A and B is 60 6 meters, then the height of the tower (in meters) is _____.

Step-by-step solution

Let the height of the tower be h , its top be T , and its foot on the horizontal plane be O . In the right-angled triangle TOA , the angle of elevation at A is 30^ . The distance from A to the top of the tower T is the hypotenuse TA . TA = h 30^ = 2h In the right-angled triangle TOB , the angle of elevation at B is 45^ . The distance from B to the top of the tower T is the hypotenuse TB . TB = h 45^ = h 2 We are given that the segment AB subtends a right angle at the top of the tower T , meaning ATB = 90^ . In the