Find the height of the tower (XY) given the following:
To find the height of a tower(XY), measurements were taken from a baseline AB. It was found that AB = 50m. angle XAY = 42.6 degrees, angle XAB = 60 degrees, and angle ABX = 81.65 degrees.
The problem given is one that involves measurements made in all the three dimensions and this should be kept in mind while solving the problem.
The tower is a vertical line with the base being X and Y being the topmost point. The line AB is a horizontal line. It is given that the angle of elevation of the line from the point A to the point Y is 42.6 degrees. The height of the tower can be determined if the distance of the point A from the point X is known. This is determined using the length of the line AB and the measure of angle XAB = 60 degrees and ABX = 81.65 degrees. The measure of angle AXB = 180 - 60 - 81.65 = 38.35. As AB = 50 m, use the law of sines to determine the length AX.
`(AX)/(sin 81.65) = (50)/(sin 38.35)`
=> `AX ~~ 79.73`m
The height of the tower is equal to h where `h/79.73 = tan 42.6`
=> `h ~~ 73.31` m
The height of the tower is 73.31 m
XY is the height of the tower (use X as the base)
AB is a 50 m baseline, some distance away from the tower.
Angle XAY = 42.6 (which is the vertical angle from ground level to the top of the tower)
Angle XAB = 60 (and is the horizontal angle between the line segment AX and AB
Angle ABX = 81.65 (the horizontal angle between the line segment AB and BX.
Find Angle AXB = 180 degrees - 60 degrees - 81.65 degrees = 38.35 degrees
Use law of sines to find distance AX:
sin 38.35/50 = sin 81.65 / AX and AX = 79.7 m
Use AX and the vertical angle to find the height of the tower:
42.6 degrees is the vertical angle between line AX and XY
Tan 42.6 = XY/79.7
XY, the height of the tower is 73.3 m Answer