There is a small island in the middle of a $\displaystyle 100$ m wide river and a tall tree stands on the island. P and Q are points directly opposite to each other on 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 $\displaystyle 30^{\circ}$ and $\displaystyle 45^{\circ}$, find the height of the tree. (Use $\displaystyle \sqrt{3}=1.732$ )

Correct Let $\displaystyle A B=$ height of tree $\displaystyle =h$ \[\angle A P B=30^{\circ}, \angle A Q B=45^{\circ} \]Let $\displaystyle B Q=x$ \[\therefore P B=100-x \]In $\displaystyle \triangle A B Q, \tan 45^{\circ}=\frac{h}{x} \Rightarrow h=x$In $\displaystyle \triangle A B P, \tan 30^{\circ}=\frac{h}{100-x}$ \[\begin{array}{l} \Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{100-x} \Rightarrow 100-x=h \sqrt{3} \\ 100-h=h \sqrt{3} \Rightarrow 100=h(\sqrt{3}+1) \\ h=\frac{100}{\sqrt{3}+1}=\frac{100(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)}=\frac{100(\sqrt{3}-1)}{(\sqrt{3})^{2}-(1)^{2}} \\ h=\frac{50}{\not 2}=50(1 \cdot 732-1)=36 \cdot 6 \mathrm{~m} \end{array} \]