question_answer

A tree of 10 m is broken by a storm in such a way that its top touches the ground at a distance of 6 m from its root. Find the height at which the tree is broken.

A)  \[\frac{17}{5}\,m.\]                  

B)  \[\frac{8}{5}\,m.\]

C)  \[\frac{17}{10}\,m.\]                            

D)  \[\frac{16}{5}\,m.\]

Correct Answer: D

Solution :

Height of tree \[AB=10\text{ }m.\] So, \[AD=CD=(10-h)\,m.\] In \[\Delta \,DBC\], \[\sin \theta =\frac{h}{10-h}\]                             .?..(i) \[\cos \theta =\frac{6}{10-h}\]                             ??(ii) From equation (i) and (ii)- \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta ={{\left( \frac{h}{10-h} \right)}^{2}}+{{\left( \frac{6}{10-h} \right)}^{2}}\] \[{{(10-h)}^{2}}={{h}^{2}}+36\] \[{{h}^{2}}+100-20h={{h}^{2}}+36\,\,\,\Rightarrow \,\,20h=64\] \[\therefore \,\,\,h=\frac{16}{5}\,m.\] \[\therefore \]   Height at which the tree is broken \[=\,\,\,\,\,\underline{\frac{\mathbf{16}}{\mathbf{5}}\,\mathbf{m}}\]