Question

The angle of elevation of the top of a tower is observed to be $60^{\circ }$ ; at a point $40m$ above the first point of observation, the elevation is found to be $45^{\circ }$ . Find the height of the tower is $hm$ then the value of $\left[h\right]$ is where $\left[\cdot \right]$ is the greatest integer function.

Answer 1

Answer: 94

Suppose $AC$ is the tower. The angle of elevation of $A$ at $B$ is $60^{\circ }$ and $BD=40m$ . The angle of elevation at $D$ is $45^{\circ }$ .
$\tan 60^{\circ }=\frac{AC}{BC}=\frac{AE+EC}{BC}$
$\Rightarrow \sqrt{3}=\frac{AE+40}{BC}$
$\Rightarrow AE+40=\sqrt{3}BC...\left(i\right)$
Now, $\tan 45^{\circ }=\frac{AE}{DE}=1$
$\therefore AE=DE=BC...\left(ii\right)$
From Equations $\left(i\right)\&\left(ii\right)$ ,
$AE+40=\sqrt{3}AE$
$\Rightarrow AE=\frac{40}{\sqrt{3}-1}=54.641m$ .
And $AC=\left(40+54.641\right)m$ .
$\therefore$ Height of the tower $=94.641m$ .
And, horizontal distance of the tower
$BC=AE=54.641m$ .