Question

From the top of a cliff $200ft$. high, the angles of depression of the top and bottom of a tower are observed to be ${30}^{\circ }$ and ${60}^{\circ }$ respectively. The height of the tower is________

Answer 1

Consider the $△BAP$

$tan{30}^{\circ }=\frac{AP}{AB}=\frac{x}{200}$

$\frac{1}{\sqrt{3}}=\frac{x}{200}$

$x=\frac{200}{\sqrt{3}}$......(1)

Now consider the $△BRQ$

$tan{30}^{\circ }=\frac{RQ}{BR}=\frac{200-h}{x}$

$\frac{1}{\sqrt{3}}=\frac{200-h}{x}$

$\frac{x}{\sqrt{3}}=200-h$

$h=200-\frac{x}{\sqrt{3}}$

substitute the value of $x$ from eq (1), we get

$h=200-\frac{200}{(\sqrt{3}{)}^2}$

$=200-\frac{200}{3}$

$=\frac{600-200}{3}$

$=\frac{400}{3}$
Hence, the height of tower is $h=\frac{400}{3}ft$