Q31 of 46 Page 1

From the top of a hill, the angles of depression of two consecutive kilometer stones due east are found to be 30 � and 45 �. Find the height of the hill.

We have


Let height of the hill be h km, which is asked in the question.


Since, the stones are consecutive so CD = 1 km.


In ∆ABD,


[∵, ]


⇒ [∵, tan 30° = 1/√3]


⇒ x + 1 = h√3 …(i)


In ∆ABC,


[∵, ]


⇒ 1 = h/x [∵, tan 45° = 1]


⇒ h = x …(ii)


Substituting equation (ii) in equation (i), we get


x + 1 = x√3


⇒ x√3 – x = 1


⇒ (√3 – 1)x = 1


⇒ .


Rationalizing it,


⇒ [∵, (a + b)(a – b) = a2 – b2 and so, (√3 + 1)(√3 – 1) = 3 – 1]


⇒ [∵, √3 = 1.73]


⇒ x = 1.365


Thus, height of the hill is 1.365 km.


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