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Q)A person standing on the bank of river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60° when he mobes 40 m away from the bank . he finds the angle of elevation to be 30° . find the height of the tree and the width of the river .
Answers
Solution:
Let the height of tree (RS) be 'h' metres.
The man standing on the bank of river be Q.
After moving from the bank, the new point be P. Ultimately, the PQ = 40m
Angles of elevation of the top of tree from points P and Q are 30° and 60° respectively.
Let width of river (QR) be 'x' metres.
In a right angled triangle QRS,
Tan 60° = RS/QR
=> √3 = h/x
=> x = h/√3 ---(eq - 1)
Next, In right traingle PRS,
Tan 30° = RS/PR
=> 1/√3 = h/(x + 40)
After cross multiplication,
=> √3h = x + 40
=> x = √3h - 40 ---(eq - 2)
From both equations,
=> h/√3 =√3h - 40
=> h = (√3h - 40)√3
=> h = 3h - 40√3
=> h - 3h = -40√3
=> -2h = -40√3
=> h = 40√3 / 2
=> h = 20√3
Height of tree is 20√3 metres.
Substitute eq - (1) or (2)
=> x = h/√3
=> x = 20√3 / √3
=> x = 20m
Therefore, the width of river is 20m.
