Question

From the top of a cliff $200$ metres high, the angles of depression of the top and bottom of a tower are observed to be ${30}^o$ and ${60}^o$. Find the height of the tower and calculate the distance between them.

• A. Height:$=156$; Distance$=119.7m$
• B. Height$=133\frac{1}{3}$; Distance$=115.46m$
• C. Height$=220$; Distance$=112.76m$
• D. None of these

Answer 1

The correct option is B Height$=133\frac{1}{3}$; Distance$=115.46m$

Height of cliff = 200 m
Height of tower = $h$
Let the distance between the tower and cliff = $w$
Angle of elevation of top of cliff from top of tower = ${30}^{\circ }$
Angle of elevation of top of cliff from bottom of tower = ${60}^{\circ }$
$\tan \angle$ of elevation = $\frac{height}{distanceonground}$
$\tan 60=\frac{200}{w}$
$w=\frac{200}{\sqrt{3}}$
$w=115.47$ m
$\tan 30=\frac{200-h}{w}$
$w=200\sqrt{3}-h\sqrt{3}$
$\frac{200}{\sqrt{3}}=200\sqrt{3}-h\sqrt{3}$
$200=600-3h$
$h=\frac{400}{3}$ m