Answer:
about 11.5 m
Explanation:
You want the height of a tree when the angles of elevation to its top are 25° and 50° from points 15 m apart.
Tangent
The tangent relation between angles and sides in a right triangle is ...
Tan = Opposite/Adjacent
In the attached diagram, this means ...
tan(25°) = TX/AX
tan(50°) = TX/BX
Solution
The difference between AX and BX is known, so we can rearrange this to ...
AX -BX = 15 = TX/tan(25°) -TX/tan(50°)
15·tan(25°)·tan(50°) = TX(tan(50°) -tan(25°) . . . multiply by tan(25°)tan(50°)
TX = 15·tan(25°)·tan(50°)/(tan(50°)-tan(25°) ≈ 11.5 . . . . meters
The height of the tree is about 11.5 meters.
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Additional comment
The value of the height can be computed by finding each tangent only once if we use ...
TX = 15/(1/tan(25°) -1/tan(50°))
You recognize 1/tan(x) = cot(x) = tan(90°-x), so this is ...
TX = 15/(tan(65°) -tan(40°))