Question

There is a small island in the middle of a 100 m wide river and a tall tree stands on the island . P and Q are points directly opposite to each other an two banks and in the line with the tree . If the angle of elevation of the top of the tree from P and Q are respectively ${30}^{\circ }$ and ${45}^{\circ }$ find the height of the tree

Answer 1

Let OA be the tree of height h metre.
In triangle POA and QOA, we have
tan ${30}^{\circ }=\frac{OA}{OP}$ and tan ${45}^{\circ }=\frac{OA}{OQ}$
$\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{OP}and1=\frac{h}{OQ}$
$\Rightarrow OP=\sqrt{3}handOQ=h$
$\Rightarrow OP+OQ=\sqrt{3}h+h$
$\Rightarrow PQ=(\sqrt{3}+1)h$
$\Rightarrow 100=(\sqrt{3}+1)h[\because PQ=100m]$
$\Rightarrow h=\frac{100}{\sqrt{3}+1}m$
$\Rightarrow h=\frac{100(\sqrt{3}+1)}{2}m$
$\Rightarrow h=50(1.732-1)m=36.6m$
Hence, the height of the tree is 36.6 m.