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)

Answer 1

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.