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Question
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 30º and 45º, find the height of the tree. (Use `sqrt(3)` = 1.732)
Solution
Let OA be the tree of height h m.
In ΔPOA, ∠O = 90°
tan 30° = `("OA")/("OP")`
⇒ `1/sqrt(3) = "h"/("OP")`
⇒ OP = `sqrt(3) "h"` ...(i)
In ΔQOA, ∠O = 90°
tan 45° = `("OA")/("OQ")`
⇒ `1 = "h"/("OQ")`
⇒ OQ = h ...(ii)
Adding equations (i) and (ii), we get
OP + OQ = `sqrt(3) "h" + "h"`
⇒ PQ = `"h"(sqrt(3) + 1)`
⇒ 100 = `"h"(sqrt(3) + 1)`
⇒ h = `100/(sqrt(3) + 1)`
⇒ h = `(100(sqrt(3) - 1))/((sqrt(3) + 1)(sqrt(3) - 1))`
⇒ h = `(100(sqrt(3) - 1))/2`
⇒ h = 50 (1.732 – 1)
⇒ h = 50 × 0.732
⇒ h = 36.6m
Thus, the height of the tree is 36.6 m.
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