Question

From the top of a hill, the angles of depression of two consecutive kilometre stones due east are found to be 30° and 45°. The height of the hill is ______.

Options

• `1/2 (sqrt(3) - 1)` km

• `1/2 (sqrt(3) + 1)` km

• `(sqrt(3) - 1)` km

• `(sqrt(3) + 1)` km

Answer 1

From the top of a hill, the angles of depression of two consecutive kilometre stones due east are found to be 30° and 45°. The height of the hill is `underlinebb(1/2 (sqrt(3) + 1) km)`.

Explanation:

Let AB be the hill.

From right ΔBAD, we have

`(AD)/(AB) = cot 45^circ`

⇒ `x/h = 1`

⇒ x = h   ...(i)

From right ΔBAC, we have

`(AC)/(AB) = cot 30^circ`

⇒ `(x + 1)/h = sqrt(3)`

⇒ `x = (hsqrt(3) - 1)`   ...(ii)

From (i) and (ii), we have

`h = (hsqrt(3) - 1)`

⇒ `h(sqrt(3) - 1) = 1`

⇒ `h = 1/((sqrt(3) - 1))`

⇒ `h = {1/((sqrt(3) - 1)) xx ((sqrt(3) + 1))/((sqrt(3) + 1))}`

⇒ `h = 1/2 (sqrt(3) + 1)`