Question

The angles of elevation of the top from two points on either side of a tree are ${30}^{\circ }$ and ${60}^{\circ }$. Find the height of the tree if the two points are $52m$ apart.

• A. $39m$
• B. $22.5m$
• C. $65m$
• D. $225m$

Answer 1

The correct option is B $22.5m$

Let $B$ and $C$ are two points, such that distance between them is $52m$ and ${}^{\prime }{h}^{\prime }$ be the height of tree.

Now, in $△ABD,$

$\Rightarrow \tan 30°=\frac{AD}{BD}=\frac{h}{x}$

$\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{x}$

$\Rightarrow x=\sqrt{3}h⟶\left(1\right)$

In $△ACD,$

$\Rightarrow \tan 60°=\frac{AD}{DC}=\frac{h}{52-x}$

$\Rightarrow \sqrt{3}=\frac{h}{52-x}$

$\Rightarrow 52-x=\frac{h}{\sqrt{3}}$

Put $x=\sqrt{3}h$ from equation $\left(1\right),$ we get

$\Rightarrow 52-\sqrt{3}h=\frac{h}{\sqrt{3}}$

$\Rightarrow 52\sqrt{3}-3h=h$

$\Rightarrow 4h=52\sqrt{3}$

$\Rightarrow 4h=13\sqrt{3}$

$\Rightarrow h=22.5$

$\therefore$ the height of the tree $22.5m.$

Hence, the answer is $22.5m.$