Question

The angles of elevation and depression of the top and the bottom of a tower from the top Of a building, 60 m high, are ${30}^{\circ }$ and
${60}^{\circ }$ respectively. Find the difference between the heights of the building and the tower and the distance between them.

Answer 1

Let AB be the building and CD be the tower.

In right $\Delta$ABD.
$\frac{AB}{BD}=tan{60}^{\circ }$
$\Rightarrow \frac{60}{BD}=\sqrt{3}$
$\Rightarrow BD=\frac{60}{\sqrt{3}}$
$\Rightarrow BD=20\sqrt{3}$

In right ∆ACE:

$\frac{CE}{AE}=tan30°$

$\Rightarrow \frac{CE}{BD}=\frac{1}{\sqrt{3}}$ (∴ AE = BD)

$\Rightarrow CE=\frac{20\sqrt{3}}{\sqrt{3}}=20$

Height of the tower = CE + ED = CE + AB = 20 m + 60 m = 80 m

Difference between the heights of the tower and the building = 80 m – 60m = 20 m

Distance between the tower and the building = BD = 20√3 m.