question_answer

                        There is a small island in the middle of a 50 m wide river and a tall tree stands on the island. P and Q are points directly opposite to each other on the two banks, and in line with the tree. If the angles of elevation of the top of the tree from P and Q are respectively \[{{60}^{o}}\] and \[{{30}^{o}},\] find the height of the tree.                      

A) \[23.56\text{ }m\]                   

B)  \[21.65\text{ }m\]      

C)         \[22.69\text{ }m\]        

D)         \[22.65\text{ }m~\]   

Correct Answer: B

Solution :

Let the height of the tree be h i.e., In \[\Delta \,PAT,\]                                     \[\tan {{60}^{o}}=\frac{h}{x}\Rightarrow \sqrt{3}=\frac{h}{x}\Rightarrow h=\sqrt{3}x\] In \[\Delta QAT,\] \[\tan {{30}^{o}}=\frac{h}{50-x}\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{50-x}\]                         \[\left[ \because \,\,\,x=\frac{h}{\sqrt{3}} \right]\] \[\Rightarrow \] \[\sqrt{3}h=50-\frac{h}{\sqrt{3}}\Rightarrow h=\frac{50\sqrt{3}}{4}=21.65m\] \[\Rightarrow \]The height of the tree is \[21.65m\]