Standing on one side of a 10 meter wide straight road, a man finds that the angle of elevation of a statue located on the same side of the road is X. After crossing the road by the shortest possible distance, the angle reduces to Y. Find the height of statue.
How do we solve this question applying trigonometry?
Thanks in advance.
Best Answer
Letting the height of the tower be $h$, and the initial distance to the tower be $D$ we arrive initially at the equation:
$tanX = \frac{h}{D}$
After crossing the road, the equation becomes,
$tanY = \frac{h}{\sqrt{D^2 + 10^2}}$
Equating the two yields,
$DtanX = \sqrt{D^2 + 10^2}tanY$
Thus,
$D^2tan^2X = D^2tan^2Y + 100tan^2Y$
Thus,
$D = \sqrt{\frac{100tan^2Y}{tan^2X - tan^2Y}}$
Now we have $D$ we can substitute back into the initial equation to find $h$ as,
$h = tanX\sqrt{\frac{100tan^2Y}{tan^2X - tan^2Y}}$