Question

Find the angle of elevation made by the top of the tree with the ground.

• A. ${20}^o$
• B. ${25}^o$
• C. ${30}^o$
• D. ${35}^o$

Answer 1

The correct option is C ${30}^o$

Let $AB$ be the tree broken by the wind at the point $C$. Its top $B$ strike the ground at the point $D$ such that $\angle CDA=\theta$ and $CB$ takes the position of $CD$ i.e., $CD=CB=y$ metres. But $AC=x$ metres then the height of the tree$=AB=(x+y)=15m,CD=y=(15-x)$metres
$ADC$ is rt. angles $△$ at $A$
${CD}^2={AC}^2+{AD}^2$
$\Rightarrow$ ${(15-x)}^2=x^2={\left(5\sqrt{3}\right)}^2=x^2+75$ (Pythagoras theorem)
$\Rightarrow$ $225+x^2-30x=x^2+75$
$\Rightarrow$ $30x=150m$
$\Rightarrow$ $x=5m$
$AD=5\sqrt{3}m,AC=x-5m$ and $CD=(15-5)m=10m$
$ADC$ is right angled $△$ at $A$ then,
$\frac{AC}{AD}=\tan \theta$
$\Rightarrow$ $\frac{5}{5\sqrt{3}}=\tan \theta$
$\Rightarrow$ $\tan \theta =\frac{1}{\sqrt{3}}=\tan {30}^o$
$\Rightarrow$ $\tan \theta =\tan {30}^o$
$\Rightarrow$ $\theta ={30}^o$
The angle of elevation made by the top of the tree with ground $=\theta ={30}^o$