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Height and Distance
Height and Distance
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Question

Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let π8{\pi \over 8} and θ\theta be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then tan 2 θ\theta is equal to

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Solution

l80=tan⁡θ{l \over {80}} = \tan \theta ..... (i) 2l80=tan⁡π8{{2l} \over {80}} = \tan {\pi \over 8} ..... (ii) From (i) and (ii) 12=tan⁡θtan⁡π8⇒tan⁡2θ=14tan⁡2π8{1 \over 2} = {{\tan \theta } \over {\tan {\pi \over 8}}} \Rightarrow {\tan ^2}\theta = {1 \over 4}{\tan ^2}{\pi \over 8} ⇒tan⁡2θ=2−14(2+1)=3−224 \Rightarrow {\tan ^2}\theta = {{\sqrt 2 - 1} \over {4(\sqrt 2 + 1)}} = {{3 - 2\sqrt 2 } \over 4}

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