question_answer

A straight tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of \[{{30}^{o}}\] with the ground. The distance from the foot of the tree to the point, where the top touches the ground is 10 m. The height of the tree is _____.

A)  \[10\sqrt{3}\,m\]       

B)  \[\frac{10\sqrt{3}\,}{3}m\]   

C)         \[10(\sqrt{3}+1)m\]    

D)         \[10(\sqrt{3}-1)m\]

Correct Answer: A

Solution :

Let AB be the tree broken at point C, Let \[AC=x\text{ }m\]and \[CD=CB=y\text{ }m.\] Then, height of the tree \[=(x+y)m.\]                         Now, in right  \[\Delta ADC,\] \[\frac{AC}{AD}=\tan {{30}^{o}}\Rightarrow \frac{x}{10}=\frac{1}{\sqrt{3}}\] \[\Rightarrow \]  \[x=\frac{10}{\sqrt{3}}m\] And,  \[\frac{AD}{CD}=\cos {{30}^{o}}\] \[\Rightarrow \] \[\frac{10}{y}=\frac{\sqrt{3}}{2}\Rightarrow y=\frac{20}{\sqrt{3}}m.\] \[\therefore \] Height of the tree \[=(x+y)\] \[=\left( \frac{10}{\sqrt{3}}+\frac{20}{\sqrt{3}} \right)=\left( \frac{30}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}} \right)=10\sqrt{3}m\]