Question

A tree $12m$ high, is broken by the wind in such a way that its top touches the ground and makes an angle ${60}^{\circ }$ with the ground. At what height from the bottom the tree is broken by the wind?

• A. $5.569m$
• B. $1.732m$
• C. $5.916m$
• D. $2.456m$

Answer 1

Answer: (A)

$5.569m$

Height of the tree = $AC+BC=12m$

Let $AC=xm$

In $\Delta ABC$,

$\sin {60}^{\circ }=\frac{x}{12-x}$

$\frac{\sqrt{3}}{2}=\frac{x}{12-x}$

$\sqrt{3}(12-x)=2x$

$12\sqrt{3}-\sqrt{3}x=2x$

$12\sqrt{3}=2x+\sqrt{3}x$

$12\sqrt{3}=x(2+\sqrt{3})$

$x=\frac{12\sqrt{3}}{2+\sqrt{3}}$

$x=\frac{\left(12\sqrt{3}\right)(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}$

$x=\frac{\left(12\sqrt{3}\right)(2-\sqrt{3})}{4-3}$

$x=12\sqrt{3}\left(2\right)-12\times 3$

$x=24\sqrt{3}-36$

$x=24\times 1.73-36$ $(\sqrt{3}=1.73)$

$x=41.52-36$

$\therefore x=5.52m$