Question

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

Answer 1

Let $PQ=h$ be the height of tower angles of depression of $Q$ and $P$ are ${30}^o$ and ${60}^o$ as seen from top $B$ of cliff $AB$ of height $200ft$.
Let $AP=x$.
$200-h=x\tan 30=x/\sqrt{3};△BQR$.
$200=x\tan 60=\sqrt{3}x;△BPR$.
Dividing, we get $\frac{200-h}{200}=\frac{1}{3}$
or $3(200-h)=200$
or $4003h$
$\therefore h=\frac{400}{3}ft$.