Answer:
113.75 m
Step-by-step explanation:
You want the distance between two observation points if the point S directly south sees the top of a 50 m mast at an elevation of 38°, while the point W directly west sees it at an elevation of 28°.
Tangent
The tangent relation can be used to find the distance from each observation point to the base of the mast.
Tan = Opposite/Adjacent
Here, the sides of the right triangle modeling the elevation are ...
Opposite = mast height = 50 m
Adjacent = distance to mast
tan(elevation angle) = (50 m)/(distance to mast)
South
The distance from the south observation point to the mast is about ...
tan(38°) = (50 m)/s
s = (50 m)/tan(38°) ≈ 63.997 m
West
Similarly, the distance from the west observation point to the mast is about ...
w = (50 m)/tan(28°) ≈ 94.036 m
Between
The distance between these observation points is the hypotenuse of a right triangle on the ground with its right angle at the mast. It can be found using the Pythagorean theorem:
s² + w² = d²
d = √(63.997² +94.036²) ≈ 113.75
The distance between points S and W is about 113.75 meters.
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Additional comment
In the attachment, we have flattened the geometry so we could draw it on an x-y plane. The triangles SOMs and WOMw actually lie in vertical planes with OM being a vertical segment in the +z direction. Even though we have rotated these triangles to the x-y plane, the distances OS and OW remain the same, as does the distance SW.
The calculator display in the second attachment shows the distance calculated as the length of a vector with orthogonal components 50/tan(38°) and 50/tan(28°). This gets the result with about the least number of keystrokes.
