Question

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff. At a point on the plane, 30 metres away from the tower, an observe notices that the angles of elevation of the top and bottom of the flagstaff are 60° and 45° respectively. Find the height of the flagstaff and that of the tower.

Answer 1

Let $OX$ be the horizontal line, $AC$ be the vertical tower and $BC$ be the vertical flagstaff.
We now have:
$OA=30$ m, ∠$BOA={60}^o$ and ∠$COA={45}^o$
Let:
$AC=h$ m and $BC=x$ m



In the right ∆$AOC$, we have:
$\frac{AC}{OA}=\tan {45}^o=1$

⇒ $\frac{h}{30}=1$
⇒ $h=30$ m

Now, in the right ∆$AOB$, we have:
$\frac{AB}{OA}=\tan {60}^o=\sqrt{3}$

⇒ $\frac{h+x}{30}=\sqrt{3}$
On putting $h=30$ in the above equation, we get:
$30+x=30\sqrt{3}$
⇒ $x=30\sqrt{3}-30=21.96$ m

We now have:
Height of the flagstaff = x = 21.96 m
Height of the tower = $h=30$ m