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Heights and Distances
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Heights and Distances- Example
Example: There is a small island in the middle of a 100 m wide river and a tall tree stands on the island. P and Q are points directly opposite to each other on two banks, and in line with the tree. If the angles of elevation of the top of the tree from P and Q are respectively 30 and 45, find the height of the tree.
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Solution: Let OA be the tree of height h metre.
In triangles POA and QOA, you have
tan 30 = OA/OP and tan 45 = OA/OQ
1/3 = h / OP and 1 = h / OQ
= 3 h and OQ = h
= 3 h + h
= (3 + 1)h
100 = (3 + 1)h [PQ = 100 m]
h = 100/(3 + 1)
h = 100(3 - 1)m / 2
h = 50 (1.732 - 1)m = 36.6 m
Hence, the height of the tree is 36.6 m
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