question_answer

The height of the chimney when it is found that on walking towards it 50 m in the horizontal line through its base, the angle of elevation of its top changes from\[{{30}^{o}}\]to\[{{60}^{o}}\], is

A) \[25\sqrt{3}m\]               

B) \[25\,\,m\]

C) \[25\sqrt{4}m\]

D)        \[\frac{25}{\sqrt{3}}\,\,m\]

Correct Answer: A

Solution :

Let\[PQ=h\]be the height of chimney. \[A\]and\[B\]are the two points 50 m apart. In\[\Delta \,\,APQ\],                 \[\tan {{30}^{o}}=\frac{h}{AP}\]                 \[AP=h\cot {{30}^{o}}\]                                ... (i) and in\[\Delta \,\,QBP\],                 \[\tan {{60}^{o}}=\frac{h}{BP}\]                 \[BP=h\cot {{60}^{o}}\]                                 ... (ii) \[\because \]     \[AP-BP=50\] \[\therefore \]  \[h(\cot {{30}^{o}}-\cot {{60}^{o}})=50\] \[\Rightarrow \]               \[h=\frac{50}{\left( \sqrt{3}-\frac{1}{\sqrt{3}} \right)}=\frac{50\sqrt{3}}{3-1}\]                 \[=\frac{50\sqrt{3}}{2}=25\sqrt{3}m\]