Question

A man 2 m tall, standing on top of a 25 m building observe the angle of depression of the top and bottom of the next building to be ${30}^{\circ }$ and ${60}^{\circ }$ respectively. The height of the next building is ___ m.

Answer 1

The diagram can be visualised as shown in the figure.

Let the height of next building be h.

In ΔADE, tan${30}^{\circ }$ = $\frac{(27-h)}{DE)}$

$\Rightarrow \frac{1}{\sqrt{(}3)}$ = $\frac{(27-h)}{DE}$

$\Rightarrow DE=\sqrt{3}(27-h)$ ----------------(i)

In ΔABC, tan${60}^{\circ }$= $\frac{27}{BC}$

$\Rightarrow \sqrt{(}3)$ = $\frac{27}{BC)}$ -

$\Rightarrow BC=\frac{27}{\sqrt{3}}$------------------(ii)

DE = BC, Hence equating (i) and (ii) , we have

$\sqrt{3}(27-h)$ = $\frac{27}{\sqrt{3}}$

$\Rightarrow 3(27-h)=27$

$\Rightarrow 81-3h=27$

$\Rightarrow 3h=54$

$\Rightarrow$ h = 18 m