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Height and Distance
Height and Distance
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Question

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is 60∘{60^ \circ } and when he retires 4040 meters away from the tree the angle of elevation becomes 30∘{30^ \circ }. The breadth of the river is :

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Solution

From the figure tan⁡60∘=yx\tan {60^ \circ } = {y \over x} ⇒y=3x.......(1) \Rightarrow y = \sqrt {3x} .......\left( 1 \right) tan⁡30∘=yx+40\tan {30^ \circ } = {y \over {x + 40}} ⇒y=x+403........(2) \Rightarrow y = {{x + 40} \over {\sqrt 3 }}........\left( 2 \right) From (1)(1) and (2),(2), 3x=x+403⇒x=20m\sqrt 3 x = {{x + 40} \over {\sqrt 3 }} \Rightarrow x = 20m

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