Answer:
137.3 m
Step-by-step explanation:
You want the height of a lighthouse when the angles of elevation to the base and top are 62.3° and 65.1°, respectively, from a distance of 550 m.
Tangent
The height of the point being observed can be found from the tangent relation:
Tan = Opposite/Adjacent
where the side Opposite the angle of elevation in the right triangle model is the height of the point being observed, and the Adjacent side is the distance to the base of it (at sea level, in this case).
Height difference
The height of the lighthouse is the difference between the heights being observed:
Opposite = Adjacent · Tan
height to lighthouse top = (550 m) · tan(65.1°)
height to cliff top = (550 m) · tan(62.3°)
The height of the lighthouse above the cliff top is the difference of these heights:
lighthouse height = (550 m) · tan(65.1°) -(550 m) · tan(62.3°)
lighthouse height = (550 m)(tan(65.1°) -tan(62.3°)) ≈ 137.3 m
The height of the lighthouse from the top of the cliff is about 137.3 meters.
