A street light is mounted on a pole. The tip of the shadow of a man who is standing on a street a short distance from the pole h
as an angle of elevation to the top of his head of 30°. A woman standing in the opposite direction of the pole as the man was standing on the same street has a angle of elevation from the tip of her shadow to her head of 41°. If the two people are 40 feet apart, how far is the street light from the head of the woman?
The angle at the lamp between the two shadow tips is ...
180° -30° -41° = 109°
This angle is opposite the 40-ft side of the triangle. The distance (d) in question is opposite the 30° angle, so can be found from the Law of Sines as ...
d/sin(30°) = 40'/sin(109°)
d = (40')·sin(30°)/sin(109°) ≈ 21.152'
The distance from the light to the woman's head is about 21.15 feet.
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We have to assume the heads are at the same height.