Question

There is a small isle in the middle of a 200m wide stream and a tall palm tree stands on the isle. A and B are points directly opposite to each other on two banks and in line with the palm tree. The angles of elevation to the top of the palm tree from A and B are respectively 30${}^0$ and 45${}^0$.
What is the height of the palm tree?

• A. 93.2 m
• B. 113.2 m
• C. 73.2 m
• D. 53.2 m
• E. None of these

Answer 1

The correct option is C

73.2 m

Let OM be the palm tree of height 'h' metres.

In triangle AOM and BOM, we have -

$tan{30}^0$ = $\frac{OM}{OA}$ and $tan{45}^0$ = $\frac{OM}{OB}$

$\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{OA}$ and 1 = $\frac{h}{OB}$

$\Rightarrow OA+OB=\sqrt{3}h+h$

$\Rightarrow AB=\left(\sqrt{3}+1\right)$h

$\Rightarrow 200=\left(\sqrt{3}+1\right)h$

$\Rightarrow h=\frac{200}{\sqrt{3}+1}m$

$\Rightarrow h=\frac{200\left(\sqrt{3}-1\right)}{2}m$

$\Rightarrow h=100(1.732-1)m=73.2m$

Hence, the height of the palm tree is 73.2 m.